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Consistent Weak Forms for Meshfree Methods: Full Realization of h-refinement, p-refinement, and a-refinement in Strong-type Essential Boundary Condition Enforcement

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Highlights • Two weak forms are introduced that are consistent with meshfree approximations • Higher order optimal h-refinement previously unavailable • p-refinement previously unavailable • New ability to increase accuracy called a-refinement Abstract Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial due to the fact that field variables are not guaranteed to coincide with their coefficients at nodal locations. A common approach to overcome this issue is to strongly enforce the boundary conditions at these points by employing a technique to modify the approximation such that this is possible. However, with these methods, test and trial functions do not strictly satisfy the requirements of the conventional weak formulation of the problem, as the desired imposed values can actually deviate between nodes on the boundary. In this work, it is first shown that this inconsistency results in the loss of Galerkin orthogonality and best approximation property, and correspondingly, failure to pass the patch test. It is also shown that this induces an O(h) error in the energy norm in the solution of second-order boundary value problems that is independent of the order of completeness in the approximation. As a result, this places a barrier on the global order of accuracy of Galerkin meshfree solutions to that of linear consistency. That is, with these methods, it is not possible to attain the higher order accuracy offered by meshfree approximations in the solution of boundary-value problems. To remedy this deficiency, two new weak forms are introduced that relax the requirements on the test and trial functions in the traditional weak formulation. These are employed in conjunction with strong enforcement of essential boundary conditions at nodes, and several benchmark problems are solved to demonstrate that optimal accuracy and convergence rates associated with the order of approximation can be restored using the proposed method. In other words, this approach allows p-refinement, and h-refinement with p th order rates with strong enforcement of boundary conditions beyond linear (p > 1) for the first time. In addition, a new concept termed a-refinement is introduced, where improved accuracy is obtained by increasing the kernel measure in meshfree approximations, previously unavailable.
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Consistent Weak Forms for Meshfree Methods: Full Realization of
h-refinement, p-refinement, and a-refinement in Strong-type Essential
Boundary Condition Enforcement.
Michael Hillman∗† Kuan-Chung Lin
Highlights
Two weak forms are introduced that are consistent with meshfree approximations
Higher order optimal h-refinement previously unavailable
p-refinement previously unavailable
New ability to increase accuracy called a-refinement
Abstract
Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial due to the fact
that field variables are not guaranteed to coincide with their coefficients at nodal locations. A common approach
to overcome this issue is to strongly enforce the boundary conditions at these points by employing a technique
to modify the approximation such that this is possible. However, with these methods, test and trial functions do
not strictly satisfy the requirements of the conventional weak formulation of the problem, as the desired imposed
values can actually deviate between nodes on the boundary. In this work, it is first shown that this inconsistency
results in the loss of Galerkin orthogonality and best approximation property, and correspondingly, failure to pass
the patch test. It is also shown that this induces an O(h) error in the energy norm in the solution of second-order
boundary value problems that is independent of the order of completeness in the approximation. As a result, this
places a barrier on the global order of accuracy of Galerkin meshfree solutions to that of linear consistency. That
is, with these methods, it is not possible to attain the higher order accuracy offered by meshfree approximations
in the solution of boundary-value problems. To remedy this deficiency, two new weak forms are introduced that
relax the requirements on the test and trial functions in the traditional weak formulation. These are employed in
conjunction with strong enforcement of essential boundary conditions at nodes, and several benchmark problems
are solved to demonstrate that optimal accuracy and convergence rates associated with the order of approximation
can be restored using the proposed method. In other words, this approach allows p-refinement, and h-refinement
with pth order rates with strong enforcement of boundary conditions beyond linear (p > 1) for the first time. In
addition, a new concept termed a-refinement is introduced, where improved accuracy is obtained by increasing
the kernel measure in meshfree approximations, previously unavailable.
keywords Meshfree methods, essential boundary conditions, refinement, transformation method
1 Introduction
Galerkin meshfree methods [9] are a unique class of numerical methods based on a purely point-based discretization.
They offer advantages in classes of problems where mesh-based finite elements encounter difficulty, such as those
involving extreme-deformation, multi-body evolving contact, fragmentation, among others; they also offer other
attractive features like arbitrary smoothness or roughness uncoupled with the order of accuracy, ease of discretization,
ease of adaptivity, and intrinsic enrichment [3,7, 9, 24]. However, their implementation is less trivial than the finite
Kimball Assistant Professor, Department of Civil and Environmental Engineering, The Pennsylvania State University, University
Park, PA 16802, USA.
Corresponding author; email: mhillman@psu.edu; postal address: 224A Sackett Building, The Pennsylvania State University, Uni-
versity Park, PA 16802, USA.
Graduate Student Researcher, Department of Civil and Environmental Engineering, The Pennsylvania State University, University
Park, PA 16802, USA.
1
element method. For instance, careful attention needs to be paid to numerical quadrature, and enforcement of
essential boundary conditions (cf. [9]). The focus of this work is the latter issue.
Enforcement of essential (or Dirichlet) boundary conditions is non-trivial in Galerkin meshfree methods since the
nodal coefficients of shape functions do not coincide with their field variables at nodal locations in the general case.
Colloquially, this is described as lacking the Kronecker delta property, or weak-Kronecker delta property (although
an even weaker condition is sufficient to impose values at nodes on the boundary as will be discussed). Therefore,
unlike the finite element method, essential boundary conditions cannot be directly enforced on the shape functions’
coefficients. Several techniques have been proposed to overcome this difficulty.
In general, these methods can be classified into two categories: (1) strong enforcement of essential boundary
conditions at nodal locations [1, 11,27,31,34], and (2) weak enforcement of boundary conditions, such as the Lagrange
multiplier method [4], the penalty method [34] and Nitsche’s method [15, 29]. In the first category, the idea is to
modify the approximations such that nodal coefficients correspond to field variables on the essential boundary. For
the second, these methods allow test and trial functions which do not need to satisfy any particular requirement
related to the essential boundary, and instead impose boundary conditions weakly, i.e., in the sense of a distribution.
The first method proposed for enforcing essential boundary conditions in meshfree methods was the Lagrange
multiplier approach used in the element free Galerkin (EFG) method [4]. While this circumvents the aforementioned
difficulties in a relatively straight-forward manner, additional degrees of freedom are introduced, and the stiffness
matrix is also positive semi-definite. The choice of the approximation for these multipliers is also subject to the
Ladyzenskaja-Babuˇska-Bezzi (LBB) stability condition, which is an inf-sup condition necessary for well-posedness of
the discrete problem [2, 6]; an approximation to the multiplier that is not ”well-balanced” with the discretization of
the primary variable will not yield a stable solution. Shortly after, a modified variational principle [26] was proposed
to overcome these shortcomings. In this method, the idea is to substitute the physical meaning of the Lagrange
multiplier (the constraint ”forces”) in terms of the primary variable back into the weak form; thus, the problem does
not involve any additional degrees of freedom, and is not subject to the LBB condition. However, this method does
not guarantee stability either as it is equivalent to using a penalty value of zero in Nitsche’s method, while a minimum
penalty value is necessary for stability [17].
The penalty method is also a straight-forward way to enforce essential boundary conditions, which augments
the potential with a weak penalty on the constraint. However, the solution is strongly dependent on the value of
the penalty parameter: lower values lead to large errors on the essential boundary, while large values lead to an
ill-conditioned system matrix [15]. Nitsche’s method can be viewed, in some sense, as a combination of the modified
variational principle and the penalty method. The solution error is much less sensitive to the value of the penalty
parameter than the penalty method, as the penalty parameter plays an alternate role of ensuring solution stability
rather than enforcing boundary conditions. Nevertheless, an extremely large or small parameter also leads to the same
issues as the penalty method [15]. A reliable way to select the parameter is based on an eigenvalue problem related
to the discretization [17]. However, an important corollary is that the parameter depends on the discretization,
and for meshfree methods that have a variety of free parameters, this entails the distribution of points, order of
approximation, kernel measure, kernel function, etc. In the authors’ experience, it is difficult to choose a suitable
penalty parameter (to maintain desired convergence rates) a priori for accuracy higher than linear. More details on
the effect and choice of the penalty value for these methods can be found in [15, 17].
So far, the methods discussed are all in the class of weak enforcement of essential boundary conditions. Strong
methods have been developed as well, which modify the approximation such that their enforcement is similar to the
finite element method. The transformation method also known as the collocation method was first introduced in [11].
This method constructs the relationship between nodal coefficients and their field values in order to achieve the
Kronecker delta property in the approximation. This however requires the inverse of a somewhat dense system-size
matrix to solve the problem at hand. This technique was independently derived and discussed by several researchers
later [1,27, 31, 34]. To avoid inverting a dense system-size matrix, techniques have been introduced to greatly reduce
the density of the final system matrix after transformation procedures [12, 34], which has been termed the mixed
transformation method. It is worth mentioning the work in [12] offers convenient and simple implementations of these
transformation methods with row-swap operations on the system matrix. Using these techniques is equivalent to
employing Lagrange multipliers to enforce the essential boundary constraint point-wise at nodal locations [12].
Alternatively, approximations can also be constructed so that direct imposition of essential boundary conditions
can be performed without inverting any matrices. These techniques are most convenient for explicit dynamic cal-
culations for obvious reasons. Approaches include coupling of meshfree shape functions with finite elements near
the essential boundary [5, 21, 33], employing singular kernel functions for nodes on the essential boundary of the
domain [12], and constructing moving-least squares approximations with the interpolation property via primitive
functions [8]. Forcing the correction function to be zero on the essential boundary has also been introduced [16],
which yields the interpolation property (for a discussion on this aspect of meshfree approximations see [28]), but
this technique is difficult to use in high dimensions and complex geometry. More recently a conforming kernel ap-
2
proximation has been introduced which possesses the weak Kronecker delta property, and can thus strictly satisfy
the requirements on the test function (and for simple boundary conditions, the trial function) in the weak formu-
lation [20]. Finally, outside of these two classes of methods, a novel way to impose boundary conditions using
D’Alembert’s principle was introduced in [18].
The most common method employed in the literature appears to be strong enforcement at nodal points. So far,
to the best of the authors’ knowledge, there has been no published work examining the accuracy of higher-order
meshfree approximations used with these strong-form type methods, except one paper [8]. There it was reported
that while using quadratic basis to approximate a function can yield expected convergence rates, employing it in the
Galerkin equation results in only first-order accuracy, a discrepancy which was attributed to a lack of verifying the
desired conditions for test and trial functions in between the nodes.
In this paper, this assertion, and the effect of this discrepancy in the strong-type approach is closely examined,
where it is shown that the requirements on test and trial functions in the weak form are indeed not verified between
nodal locations. And, in fact, the difference between the desired values is of order hon the boundary (his the
nodal spacing), independent of the approximation order p. It is further shown that this discrepancy results in failure
to pass the patch test, and loss of Galerkin orthogonality. Patch tests performed demonstrate that the L2norm
of the error in the domain is restricted to order O(h2) due to these inconsistencies, and order O(h) in the energy
norm, regardless of the order of approximation employed. Correspondingly, much lower rates of convergence are
obtained than expected for meshfree basis functions of order higher than linear (p > 1), and the rate of convergence
is limited to that of employing approximations of linear consistency. To remedy these deficiencies, two weak forms
are introduced that allow for larger spaces of test and trial functions. When employed with the strong-type methods,
optimal convergence rates (for sufficiently regular solutions) are obtained. This technique thus allows, for the first
time using strong methods, p-refinement, and h-refinement with pth order optimal rates beyond linear. Further, it
is shown that the proposed method provides improved accuracy by increasing the kernel measure ain the meshfree
approximation, previously unavailable, which is termed a-refinement.
The remainder of this paper is organized as follows. The reproducing kernel approximation is first introduced
in Section 2 as a basis for examination of a typical meshfree method, and issues with strong essential boundary
condition enforcement are discussed. In Section 3, two weak forms are introduced which allow the enlargement of the
approximations space to include meshfree approximations constructed under the strong-type enforcement techniques.
Numerical procedures are described in Section 4, and numerical results are then given in Section 5 to demonstrate
the effectiveness of the proposed methods. Section 6 provides concluding remarks.
2 Background
2.1 Reproducing kernel approximation
In this work, the reproducing kernel is chosen as a model approximation that does not strictly meet the requirements
of the commonly used weak statement of a problem that includes Dirichlet boundary conditions.
Let a domain ¯
= Ω be discretized by a set of Np nodes S={x1,· · · ,xNP|xI¯
}with corresponding
node numbers η={I|xIS}. The pth order discrete reproducing kernel (RK) approximation uh(x) of a function
u(x) is defined as [11, 25]:
uh(x) = X
Iη
Ψ[p]
I(x)uI(1)
where {Ψ[p]
I(x)}Iηis the set of RK shape functions, and {uI}Iηare the associated coefficients.
The shape functions (1) are constructed by the product of a kernel function Φa(xxI) and a correction function
C[p](x;xxI):
Ψ[p]
I(x)=Φa(xxI)C[p](x;xxI).(2)
The correction function is composed of a linear combination of monomials up to order p, which allows the exact
reproduction of these monomials and pth order accuracy in the approximation (1). In matrix form this function can
be expressed as:
C[p](x;xxI) = H[p](xxI)Tb[p](x) (3)
where H[p](x) is a column vector of complete pth order monomials and b[p](x) is a column vector of coefficients. The
coefficients are obtained by enforcing the following reproducing conditions:
X
Iη
Ψ[p]
I(x)H[p](xI) = H[p](x),(4)
3
or equivalently,
X
Iη
Ψ[p]
I(x)H[p](xxI) = H[p](0).(5)
Employing (2)-(5), the RK shape functions in (1) are constructed as:
Ψ[p]
I(x) = H[p](0)T{M[p](x)}1H[p](xxIa(xxI) (6)
where
M[p](x) = X
Iη
H[p](xxI)H[p](xxI)TΦa(xxI) (7)
and is called the moment matrix. Without modification, the approximation is in general non-interpolatory, that is,
uh(xI)6=uI. A simple demostration of this property is given in Figure 1.
0 2 4 6 8 10
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
-6
-4
-2
0
2
4
6
8
data
RK approximation
Figure 1: Example of a meshfree approximation of data uI=xIsin(xI).
2.2 Strong enforcement of essential boundary conditions at nodal locations
2.2.1 Model problem: Poisson’s equation
Without loss of generality, in this work we consider the strong form (S) of Poisson’s equation as a model boundary
value problem, which asks: given s: Ω R,h:hR, and g:gR, find u:¯
Rsuch that the following
conditions hold:
2u+s= 0 in Ω (8a)
u·n=hon h(8b)
u=gon g(8c)
where 2 ∇ · ∇, and hand gdenote the natural boundary and essential boundary, respectively, with
gh=,Ω = gh, and ¯
Ω=ΩΩ.
2.2.2 Conventional Galerkin approximation
A weak form (W) of the of Poisson’s equation (8) can be constructed that seeks uH1
g,H1
g={u|uH1(Ω), u =
gon g}such that for all vH1
0,H1
0={v|vH1(Ω), v = 0 on g}the following equation holds:
a(v, u)= (v, s)+ (v , h)h(9)
where
a(v, u)=Z
v· ∇udΩ,(10a)
(v, s)=Z
vs dΩ,(10b)
(v, h)h=Zh
vh .(10c)
4
With approximations vhof test functions vand uhof trial functions u, with vh= 0 on gand uh=gon g,
a proper Galerkin approximation to (9) can be constructed which employs finite-dimensional subsets SgH1
gand
S0H1
0, and seeks uh∈ Sgsuch that for all vh∈ S0the following equation holds:
a(vh, uh)= (vh, s)+ (vh, h)h.(11)
In approximations which possess the Kronecker delta property, and in particular the weak Kronecker delta prop-
erty, a subset of H1
0is usually easily constructed. For instance, in linear finite elements, the boundary of the
computational domain is defined by element edges where nodal values are linearly interpolated, so enforcement of
a value of zero at nodes on the boundary ensures vh= 0 on g. For any method with the weak Kronecker delta
property and the partition of unity, the same argument follows. For construction of a subset of H1
g, a common choice
is to let the approximation interpolate values of gon the essential boundary, and Sgis also subset of H1
g, or closely
resembles a subset of H1
g.
For meshfree methods which generally do not posses these properties, it is apparent from these discussions that
the construction of subsets of H1
0and H1
gis non-trivial.
2.3 Strong nodal imposition in meshfree methods
Strong imposition of essential boundary conditions at nodal locations is a popular choice in meshfree methods to
(approximately, as will be shown) construct admissible test and trial functions for the conventional weak formulation
(9). Essentially, these entail a modification to meshfree shape functions such that nodal degrees of freedom on
the essential boundary coincide with their field variables. For this to be the case, the Kronecker delta property is
not actually necessary [8, 12], and instead the set of modified shape functions {ˆ
Ψ[p]
I(x)}Iηonly need to verify the
requirements:
ˆ
Ψ[p]
J(xI)=0 Iηg, J η\ηg(12)
and ˆ
Ψ[p]
I(xJ) = δIJ Iηg, J ηg(13)
where δIJ is the Kronecker delta function, and η\ηgis the complement of the set of node numbers ηg={I|xISg}
for nodes Sg={xI|xIg}located on the essential boundary. The above means that all ”inside nodes” should not
contribute to the approximation at ”boundary nodes”, while all ”boundary nodes” need to verify the delta property
at nodal locations on the boundary.
It is important to note that (12) and (13) only verify the prescribed conditions at nodal locations, but not in
between nodes. Therefore one may enforce boundary conditions on nodal coefficients, as is done in the literature,
but cannot ensure proper approximation spaces are constructed.
In contrast, the above properties are distinct from the weak Kronecker delta property, where only boundary shape
functions contribute to the approximation on the entire essential boundary:
ˆ
Ψ[p]
J(x)=0 xg, J η\ηg.(14)
From the above, it is apparent that approximations with (14) will have little issue with constructing proper subsets
(or very close approximations) necessary for the weak formulation (9). Meanwhile for meshfree approximations with
only (12) and (13), and not (14), as is most common, constructing proper subsets is not possible.
2.3.1 Test function construction
Using these modified shape functions, in an attempt to construct a test space satisfying S0H1
0, the following
approximation is typically employed:
vh(x) = X
Iη\ηg
ˆ
Ψ[p]
I(x)vI(15)
where {ˆ
Ψ[p]
I(x)}Iηis the set of modified shape functions with properties (12) and (13), and {vI}Iη\ηgare coefficients
of the test function.
Due to (12) and (13), the test functions verify vh(xI)=0Iηg. However, for these meshfree approximations,
the value of vh(x) is in the general case, non-zero between nodes on the essential boundary and therefore violates the
construction S0H1
0.
To illustrate this, consider a domain ¯
Ω=[1,1] ×[1,1] discretized uniformly in each direction by 9 nodes with
9×9 = 81 nodes total. A linear RK approximation (p= 1 in (15)) is employed using a cubic B-spline kernel function
with a normalized support of 3. A test function with the arbitrary coefficients set to unity is constructed using
5
the transformation method, with Ω = g. As seen in Figure 2, the test functions are in fact non-zero between
nodes along gwith the employment of (15). According to the norms computed in Table 1, the “error” (defined
as non-zero values on the essential boundary) does converge at about a rate of one (O(h)) in the L2(g) norm, yet
the magnitude of the error (in L(g)) stays about the same regardless of the discretization. According to [30],
the L2(g) error should be O(h3/2), however it seems to be O(h) when observed numerically, at least for meshfree
approximations.
Figure 2: Example of a test function in meshfree methods using the transformation method.
Table 1: Norms of error for boundary conditions imposed by test and trial functions, p= 1, varying h.
L2(g)L(g)
htest rate trial rate test trial
0.5000 0.01821 - 0.03615 - 0.03516 0.08443
0.2500 0.01014 0.84513 0.02137 0.75856 0.03645 0.10125
0.1250 0.00523 0.95507 0.01119 0.93372 0.03630 0.10495
0.0625 0.00262 0.99535 0.00573 0.96561 0.03628 0.10688
Next, the same setup is tested with p= 2 and p= 3, since a ”linear” error occurs for the previous test, and linear
basis was employed. The same norms are computed, shown in Tables 2 and 3, respectively for the two cases. Again
an O(h) error is observed, and it is seen that this error is apparently independent of the order of approximation.
Later, it will be shown that this error can be directly related to the error in the energy norm of the problem—which
will limit the rate of convergence for higher order (p > 1) approximations. This will then be confirmed numerically.
Table 2: Norms of error for boundary conditions imposed by test and trial functions, p= 2, varying h.
L2(g)L(g)
htest rate trial rate test trial
0.5000 0.01105 - 0.01935 - 0.02249 0.05458
0.2500 0.00352 1.65251 0.00640 1.59508 0.01476 0.03715
0.1250 0.00172 1.03368 0.00330 0.95787 0.01441 0.03723
0.0625 0.00086 1.00177 0.00176 0.90788 0.01440 0.03990
6
Table 3: Norms of error for boundary conditions imposed by test and trial functions, p= 3, varying h.
L2(g)L(g)
htest rate trial rate test trial
0.5000 0.00666 - 0.00975 - 0.01241 0.02231
0.2500 0.01016 -0.60978 0.01614 -0.72816 0.04419 0.09271
0.1250 0.00317 1.68073 0.00702 1.20050 0.02588 0.07773
0.0625 0.00163 0.95634 0.00353 0.99222 0.02653 0.07701
Finally, as a test, the kernel measure ais varied, with p= 1 and h= 1/4 fixed; the results are shown in Table 4.
One can first observe that if a1 then the error (not shown to full significant digits) is machine precision; in this
case the RK approximation closely resembles a bilinear finite element discretization. Then, as the kernel measure
increases, the error on the boundary increases as well. It is generally expected that in the solution of PDEs, that
increasing the measure of an approximation will increase the accuracy of the solution; however this is not observed
in practice, and an ”optimal” value is observed in meshfree methods [25]. The increasing error on the boundary can
explain that there exists two competing mechanisms: increasing error with increasing adue to failure to satisfy the
requirements of test functions, and increasing the accuracy of the approximation with increasing a.
Table 4: Norms of error for boundary conditions imposed by test and trial functions, h= 1/4, p= 1, varying a.
L2(g)L(g)
atest trial test trial
1.01 0.00000 0.00000 0.00000 0.00000
1.50 0.00118 0.00207 0.00592 0.01363
2.00 0.00483 0.00873 0.01821 0.04438
2.50 0.00863 0.01693 0.03007 0.08018
3.00 0.01014 0.02137 0.03645 0.10125
3.50 0.01085 0.02303 0.04106 0.11543
4.00 0.01207 0.02563 0.04533 0.13379
2.3.2 Trial function construction
Strong enforcement at boundary nodes uh(xI) = g(xI) is also typically introduced, and in an attempt to construct
SgH1
g, the following approximation is employed:
uh(x) = X
Iη\ηg
ˆ
Ψ[p]
I(x)uI+gh(x),(16)
where
gh(x) = X
Iηg
ˆ
Ψ[p]
I(x)gI,(17)
the values {uI}η\ηgare the trial functions coefficients, and gIg(xI) is the prescribed value of g(x) at an essential
boundary node xI∈ Sg. Because of the properties (12) and (13), the trial functions verify uh(xI) = g(xI)Iηg.
While essential boundary conditions for trial functions are verified at nodal locations, the condition uh=gis again
not enforced between the nodes. Figure 3 depicts a linear function prescribed as g(x) = x+ 2yand approximated
by (17) using the same discretization that was employed for the test function. Again it can be seen that along the
boundary, the solution is collocated only at nodal points. As shown in Table 1, the L2(g) norm of the difference
between gand ghalso converges at a rate of approximately one (O(h)) just as the test function, while the magnitude
of error (in L(g)) also stays roughly the same, despite refinement. It should be noted that even though linear
bases are employed, the function is not exactly represented due to the influence of the interior nodes on the value of
the meshfree approximation on the essential boundary between nodes. That is, it should be clear from Figure 3 that
the RK approximation under the transformation framework does not possess the weak Kronecker delta property.
7
Figure 3: Approximation gh(x) in meshfree methods using the transformation method.
Next, p= 2, and p= 3 are tested, with the same norms computed and shown in Tables 2 and 3, respectively.
Again an O(h) error is observed, and it is seen that this error in representing the essential boundary conditions is
also apparently independent of the order of approximation. The kernel measure ais again varied, with p= 1 and
h= 1/4 fixed, and the results are shown in Table 4. Again for a1 the boundary conditions are represented quite
well, as the RK approximation simply interpolates the boundary condition in the limit of a1. Then, as the kernel
measure increases, the error on the boundary increases as before.
In the next section, it will be shown that the errors on the boundary in the test and trial functions are directly
related to the error in the solution of PDEs. That is, while O(h) in L2(g), the errors manifest as errors of O(h2)
in L2(Ω) and O(h) in H1(Ω), limiting the rate of convergence of the solution.
2.3.3 Error assessment of inconsistencies
As a point of departure in considering the error induced by these inconsistencies, we first examine the weighted
residual formulation, which is more a more general way to arrive at a weak formulation than a potential. The latter
point of view will be revisited.
Integrating the product of an arbitrary weight function vand the residual of (8a) over Ω we have:
(v, 2u+s)= 0.(18)
Integrating (18) by parts and employing divergence theorem one obtains
a(v, u)= (v, s)+ (v , n· ∇u).(19)
Per the usual procedures, employing (8b), v= 0 on g, and the boundary decomposition, we have the weak form
(W) in (9) which asks to find uH1
gsuch that for all vH1
0the following equation holds:
a(v, u)= (v, s)+ (v , h)h.
Provided uis sufficiently smooth, the above equation can be integrated by parts to obtain
(v, 2u+s)+(v, h − ∇u·n)h(v, u·n)g= 0 (20)
where
(v, u·n)g=Zg
vu·n.(21)
Employing the fact that v= 0 on g,u=gon g, and the arbitrary nature of vone obtains the strong form (8),
that is we have the following equivalence
(W)(S)
However, in meshfree methods it is difficult to achieve vh= 0 on gin the Galerkin discretization as discussed
previously. And, in fact, as shown in [12], the transformation method is actually consistent with a weak formulation
8
that only attests to strong enforcement of essential boundary conditions at nodal locations, rather than the entire
essential boundary in the true strong form.
Either way, to demonstrate one significant consequence of employing (9), consider the following relation found by
using Green’s first identity and the conditions in (8):
a(vh, u)=(vh,2u)+ (vh,n· ∇u)
= (vh, s)+ (vh, h)h+ (vh,n· ∇u)g.(22)
Subtracting (22) from (9) gives
a(vh, uhu)= (vh,n· ∇uh)g(23)
which is the relation given in [30], and demonstrates that if vh6= 0 on gGalerkin orthogonality is lost. It can be
easily shown that using this relation, the best approximation property no longer holds, i.e., the minimum error in
the norm induced by a(·,·) is not obtained for the Galerkin solution. One immediate consequence is that the patch
test will fail.
Now, as discussed in [30], the left hand side is bounded by a(uhu, uhu)1/2
. Since the discrepancy on the
boundary induced by the inadmissibility of test functions has been numerically observed as O(h), one should expect
O(h) error in the energy norm of the problem. This will be confirmed numerically in the next Subsection.
Remark 1 To further elucidate the failure of the patch test, consider the viewpoint of variational consistency
presented in [10]. Starting from (9), and following [10], it can be shown using (5) and (8), that the requirements for
obtaining an exact solution u[p]of order pusing the traditional weak formulation is
ahvh, u[p]i=−hvh,2u[p]i+hvh,n· ∇u[p]ih(24)
where a,·i,,·i, and ,·ihdenote the quadrature versions of a(·,·), (·,·), and (·,·)h, respectively. However,
using integration by parts, with sufficiently high order (e.g. machine precision) quadrature it is obvious that
ahvh, u[p]i≈ −(vh,2u[p])+ (vh,n· ∇u[p])6=(vh,2u[p])+ (vh,n· ∇u[p])h(25)
and a patch test will fail unless vh= 0 on g. That is, no matter how high order the quadrature (or even with
exact integration), one will not be able to pass the patch test.
2.3.4 Numerical assessment of the order of errors in boundary value problems
To examine the effect of these inconsistencies on the numerical solution to PDEs, and verify the assertions made
in the previous section, a few patch tests are first performed, with the solution obtained using the transformation
method.
Twenty by twenty Gaussian quadrature per background cell is used for domain integration over a two-dimensional
domain Ω, with twenty Gauss points on each cell boundary intersecting Ω for integration of boundary terms. Gauss
cells are coincident with the nodal spacing such that each cell is associated with four nodes. The reason that this
”overkill” quadrature is employed is to avoid the effect of numerical integration (which has a strong effect on solution
accuracy and convergence, cf. [10,14]) and isolate the issue of boundary condition enforcement. That is, the twenty
by twenty Gauss integration employed is sufficient to element errors due to quadrature [10], and any remaining error
should be due to any other variational crimes. For the test cases below, the only inconsistency present is the inability
to satisfy the requirements on test and trial functions in the weak form [30]. Cubic B-spline kernels are employed for
the kernel function in the RK approximation. Unless otherwise stated, these parameters will be employed throughout
this manuscript.
Consider the Poisson problem (8) on the domain ¯
Ω=[1,1] ×[1,1] with the pure essential boundary condition
g=Ω. First, let the prescribed body force and boundary conditions be consistent with the linear solution
u= 0.1x+ 0.3y:
u= 0.1x+ 0.3yon g,(26a)
s= 0 in Ω.(26b)
The problem is solved with linear basis, which can exactly represent the solution, and the ”overkill” quadrature
employed should, according to conventional wisdom, result in passing the patch test. The errors in the L2(Ω) norm
and H1(Ω) semi-norm are shown in Figure 4. First, it can be seen that the patch tests are indeed not passed,
which can be attributable to the errors in constructing the proper approximation spaces, since there are no other
9
variational crimes committed. It is also seen that through refinement of the discretization (decreasing h), the order
of error induced by the inconsistency in the boundary conditions on the test and trial functions manifest as O(h2)
and O(h), for the L2(Ω) norm and H1(Ω) semi-norm, respectively. That is, the errors reduce with refinement, at
a rate consistent with employing linear basis. One may thus expect that these errors will have no influence on the
convergence rates in the solution of PDEs with linear basis, which will be confirmed later.
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
log (L2 error)
2.0
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
log (semi-H1 error)
1.0
Figure 4: Norms of error transformation method in linear patch test of Poisson problem, rate of convergence indicated.
Next, consider a quadratic patch test with quadratic basis, which should according to conventional wisdom, also
result in a solution with machine precision error when high-order quadrature is employed. Here the following quadratic
solution is considered u= 0.1x+ 0.3y+ 0.8x2+ 1.2xy + 0.6y2. The following conditions result in this solution:
u= 0.1x+ 0.3y+ 0.8x2+ 1.2xy + 0.6y2on g,(27a)
s= 2.8 in Ω.(27b)
The error in the L2(Ω) norm and H1(Ω) semi-norm are shown in Figure 5; here, again it can be seen that the
inconsistent enforcement of boundary conditions result in errors O(h2) and O(h), respectively. That is, the errors
again decrease at a rate consistent with ”linear” accuracy, despite the fact that higher-order accurate basis functions
are employed. One may then expect that these errors will limit the order of convergence in the solution of PDEs,
which will be confirmed later.
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
log (L2 error)
2.0
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
log (semi-H1 error)
1.0
Figure 5: Norms of error for transformation method in quadratic patch test of Poisson problem, rate of convergence
indicated.
To conclude, the patch test results indicate that the error due to the inability to construct proper approximation
spaces manifest as errors of linear order in the solution of PDEs.
To examine the possible, and now expected, effect on convergence rates, consider (8) with the source term and
10
pure essential boundary g=Ω with domain ¯
Ω = [0,1] ×[0,1]:
g(x, 0) = sin(πx), g(x, 1) = g(0, y ) = g(1, y) = 0 on g,(28a)
s= 0 in Ω.(28b)
The exact solution of this problem is high order [32]:
u={cosh(πy)coth(π) sinh(πy)}sin(πx).(29)
Linear, quadratic, and cubic bases are employed with the transformation method, with uniform refinements of the
domain. Various normalized support sizes (denoted ”a” in the Figure legends) are employed, to examine the effect of
varying the measure of Φa(xxI), as it is well known that linear basis degenerates to linear finite elements as the
normalized measure aapproaches unity. Thus, larger values of aare expected to show more pronounced error due to
boundary condition enforcement, since finite elements have little to no difficulty in constructing proper approximation
spaces, or at least ones which do not induce significant solution errors.
Figure 6 shows the convergence for linear basis in the L2(Ω) norm and H1(Ω) semi-norm; it can be seen that the
optimal rates of two and one are essentially maintained, regardless of the kernel measure.
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
a = 1.5 : 1.06
a = 2.0 : 1.48
a = 2.5 : 1.17
a = 3.0 : 0.98
Optimal : 1
Figure 6: Convergence of transformation method with linear basis with various kernel measures a: rates indicated in
legend.
For quadratic basis, it can be seen in Figure 7 that these same linear rates are also generally obtained, yet the
optimal rates for quadratic basis should be three and two for the L2(Ω) norm and H1(Ω) semi-norm, respectively.
Therefore optimal rates are not obtained in this case, and rather, the solution exhibits linear accuracy rather than
quadratic.
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 2.5 : 2.62
a = 3.0 : 2.08
a = 3.5 : 1.97
a = 4.0 : 1.44
Suboptimal (linear) : 2
Optimal : 3
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 2.5 : 1.84
a = 3.0 : 1.15
a = 3.5 : 0.99
a = 4.0 : 0.35
Suboptimal (linear) : 1
Optimal : 2
Figure 7: Convergence of transformation method with quadratic basis with various kernel measures a: rates indicated
in legend.
11
For the case of cubic basis, shown in Figure 8, it can again be seen that the the rates obtained are far lower than
expected; the linear rates of two and one are again obtained in most cases for the L2(Ω) norm and H1(Ω) semi-norm,
respectively, when the optimal convergence rates associated with employing approximations with cubic completeness
in displacements are four and three, respectively. Again, the solution exhibits linear accuracy, rather than cubic.
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 3.5 : 1.98
a = 4.0 : 1.52
a = 4.5 : 1.94
a = 5.0 : 1.87
Suboptimal (linear) : 2
Optimal : 4
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 3.5 : 0.98
a = 4.0 : 0.51
a = 4.5 : 0.96
a = 5.0 : 0.91
Suboptimal (linear) : 1
Optimal : 3
Figure 8: Convergence of transformation method with cubic basis with various kernel measures a: rates indicated in
legend
Higher order bases were also tested but are not shown here for conciseness of presenting the present study. The
transformation method also provided only linear solution accuracy.
To conclude, the numerical results in this section indicate that the error due to the inability to satisfy the
requirements of the conventional weak form (9) is characterized as O(h2) error in the L2(Ω) norm and O(h) error in
the H1(Ω) semi-norm, limiting the rate of convergence for bases higher than linear.
It seems that through the popular choice of linear basis in meshfree approximations over the past two decades,
this observation has somehow been overlooked, or hardly reported in the literature. To the best of the authors’
knowledge, only [8] reports results with quadratic basis and strong enforcement of boundary conditions (using the
RK approximation with interpolation property), where the same trend was observed.
3 Consistent weak forms for meshfree methods
3.1 Consistent weak form I: A consistent weak formulation for inadmissible test func-
tions
A consistent weak formulation for test functions inadmissible in the conventional weak form can be derived by
considering the possibility of vh6= 0 on gin between nodes. First, consider the weighted residual of (8), as before:
(v, 2u+s)= 0.
Integrating (18) by parts and employing divergence theorem one obtains
a(v, u)= (v, s)+ (v , n· ∇u).
Now, by employing (8b) and allowing v6= 0 on g, a consistent weak form which we denote (W1
C) is arrived at,
which asks to find uH1
g, such that for all vH1, the following equation holds:
a(v, u)(v, n· ∇u)g= (v , s)+ (v, h)h(30)
where the requirement on vH1
0has been relaxed to simply vH1where H1=H1(Ω), which allows the employment
of (15) for the test function without committing a variational crime.
It is important to note, that when (30) is integrated by parts, it is straightforward to show the weak form (30)
attests to (8), and the equivalence of the weak form and the strong form is verified, that is, W1
CS:
(v, 2u+s)+(v, h − ∇u·n)h.(31)
Since vin the above is arbitrary and uH1
g, the strong form (8) is recovered.
12
The corresponding Galerkin approximation seeks uh∈ Sg,SgH1
gsuch that for all vh∈ S,S H1the following
holds
a(vh, uh)(vh,n· ∇uh)g= (vh, s)+ (vh, h)h(32)
where vhis constructed from (15) and uhis constructed from (16).
In this formulation, we have relaxed the condition on the test function, but still attempt to construct approximation
spaces that satisfy the usual conditions. That is, the present weak formulation (W1
C) can be considered a consistent
way to employ the condition vh= 0 on gstrongly at nodes.
So far, the inconsistency in the construction of the trial function is neglected, yet in the numerical examples in
Section 5 it is shown that this has little consequence on the solution accuracy.
Remark 2 Subtracting (22) from (30) gives
a(vh, uhu)= 0 (33)
and Galerkin orthogonality is restored (compare to (23)). If one recalls that the left hand side is bounded by
a(uhu, uhu)1/2
, this indicates that the limiting error on the boundary in (23) will be released and proper
convergence rates associated with the approximation space should be achieved.
Remark 3 From a potential point of view, it is easy to show (33) is equivalent to the minimization of the following
energy functional for the present problem:
Π(W1
C)uh=1
2auhu, uhu(34)
and the best approximation property is also restored. This relation also will be useful for comparison purposes later.
Remark 4 The consistent weighted residual procedure generalizes easily to various boundary value problems (see
Appendix A).
3.2 Consistent weak form II: A consistent weak formulation for inadmissible test and
trial functions with symmetry
The employment of (30) yields a non-symmetric stiffness matrix which is often undesirable. In addition, unless trial
functions can satisfy the essential boundary conditions exactly, we do not have W1
CS, and strictly speaking W1
C
is still not consistent with a meshfree discretization.
To address these two issues, consider a more general form of the weighted residual formulation with weights v
on Ω and vgon g:
(v,2u+s)+ (vg, u g)g= 0.(35)
Various weights can be chosen, however the choice of v=vand vg=n· ∇vyields a symmetric weak form which
will be shown as follows. Further impetus is provided by the fact that a flux term n· ∇uis the “work-conjugate”
to uin terms of the potential associated with (8) and yields consistent ”units” of the problem at hand. With this
choice, (35) is expressed as
(v, 2u+s)+ (n· ∇v, u g)g= 0.(36)
Integrating (36) by parts and employing the natural boundary condition (8b), one obtains a symmetric weak form
that we denote (W2
C), which asks to find to find uH1such that for all vH1, the following equation holds
a(v, u)(v, n· ∇u)g(n· ∇v , u)g= (v, s)+ (v, h)h(n· ∇v, g)g.(37)
The above allows the complete relaxations of simply requiring vH1and uH1, and now both (15) and (16) can
be employed without committing a variational crime.
Applying integration by parts to a(·,·) in (37) yields:
(v, 2u+s)+ (v, n· ∇uh)h+ (v , u g)g= 0 (38)
where it is immediately apparent that the strong form of the problem can be recovered, hence (W2
C)(S).
The weak from (W2
C) is the same one identified in reference [26], and can be also derived from a variational
viewpoint. Here, the key difference between this work and that in [26], is that the weak form is employed with (15)
13
and (16) as to rectify the deficiencies of the standard use of these approximations. We also note that employing (37)
alone does not guarantee stability [17].
The corresponding Galerkin approximation seeks uh∈ S such that for all vh∈ S,S H1the following holds
a(vh, uh)(vh,n· ∇uh)g(n· ∇vh, uh)g= (vh, s)+ (vh, h)h(n· ∇vh, g)g(39)
where vhis again constructed from (15) and uhis constructed from (16). It is easy to see that when a Bubnov-Galerkin
approximation is employed, (39) leads to a symmetric system matrix.
With the complete relaxation on test and trial functions, this weak formulation (W2
C) can be considered a consistent
way to employ both the conditions vh= 0 on gand uh=gon gstrongly at nodes.
Remark 5 Rather than satisfying Galerkin orthogonality, by employing (22), the Galerkin discretization of the
consistent weak form (W2
C) satisfies the following:
avh, uhu=n· ∇vh, uhgg+vh,n· ∇(uhg)g.(40)
Note that if uh=gon g, then the standard orthogonality relation is recovered.
Remark 6 The relation (40) leads to the insight that a Galerkin discretization of (W2
C) minimizes the error in the
norm induced by a(·,·) augmented by the the ”work” of the error on the essential boundary (compare to (34)):
Π(W2
C)uh=1
2auhu, uhu(uhu, n· ∇(uhu))g.(41)
That is, (W2
C) can be obtained by minimization of the above potential with respect to uh. This illuminates the
possibility of balancing errors on the domain and boundary, following [19], although the numerical examples in
Section 5 indicate that this is likely not necessary since optimal rates are obtained—that is, with (41), the order of
errors due to the imposition of conditions on the domain and boundary may already be balanced.
Remark 7 The potential associated with (39) can also be stated in a more conventional manner:
Π(W2
C)uh=1
2auh, uh(uh, s)(uh, h)h(uhg, n· ∇uh)g(42)
where it can be seen that the last term accounts for the work done by the error on the essential boundary. Thus,
considering the possibility of error on the boundary is one way to arrive at a consistent weak form. The other, is to
minimize the error in both the domain and boundary, in terms of appropriate work-conjugates, as in (41).
Remark 8 This weak form can also be generalized to other boundary value problems, for a discussion, refer to the
Appendix.
Remark 9 The employment of (W2
C) or (W1
C) is consistent with the variationally consistent framework proposed
in [10], which requires the weak form attest to the strong form. In contrast, the pure transformation method does
not.
In summary, two weak forms have been developed, which are consistent with the inability of an approximation to
meet the requirements of the conventional weak form. The first considers the fact that the weight function is possibly
non-zero on the essential boundary, but that the essential boundary conditions still hold strongly. This results in a
non-symmetric stiffness matrix, but is more consistent with meshfree approximations. This weak form attests to the
strong form, and is shown to restore Galerkin orthogonality and the best approximation property. The second weak
form relaxes the requirements on both the test and trial functions, and they only need to be constructed to possess
square-integrable derivatives. The particular form taken here results in a symmetric system, at least for the model
problem at hand (see the Appendix for a brief discussion). This weak form attests to the strong form, and is shown
to satisfy a different orthogonality relation, which illuminates that it minimizes the error in the domain in terms of
the energy norm, as well as the error on the boundary in terms of the field variable and it’s corresponding ”flux” (or
work-conjugate) term.
14
4 Numerical procedures
In this section, the matrix forms for the consistent weak forms are given and boundary condition enforcement
procedures are discussed. As a starting point, let us first define terms common to the weak formulations discussed:
let ddenote a column vector of {uI}Iη, ΨIand BIdenote the Ith shape function and the column vector of it’s
derivatives respectively, and let nrepresent the unit normal to gin column vector form. In two dimensions this
yields:
d=
d1
d2
.
.
.
dNP
,BI=ΨI,1
ΨI,2,n=n1
n2.(43)
The following final system of matrix equations is also common to all formulations:
Kd =f(44)
where the system size is Np×Np. The above system is left statically uncondensed purposefully, as special procedures
are needed to apply boundary conditions in meshfree methods. These techniques are discussed in Section 4.4.
4.1 Conventional weak formulation
Under the conventional weak formulation (9), the scalar entries of Kand fin (44) are computed as
KIJ =Z
BT
I(x)BJ(x) dΩ,(45a)
fI=Z
ΨI(x)sdΩ + Zh
ΨI(x)h.(45b)
4.2 Consistent weak form I (CFW I)
For the Consistent weak form I (32), the scalar entries of Kand fin (44) are computed as
KIJ =Z
BT
I(x)BJ(x) dΩ Zg
ΨI(x)nTBJ(x) dΓ,(46a)
fI=Z
ΨI(x)sdΩ + Zh
ΨI(x)h.(46b)
Comparing (46) to (45), it can be seen that only one new term is added to the stiffness matrix of the system. Later,
it will be seen that the addition of this one term results in a drastic increase in solution accuracy and is able to
restore optimal convergence rates. Indeed, the main problem with the inability to construct proper subspaces in the
conventional weak formulation is due to the term in (23), which this weak form corrects for.
4.3 Consistent weak form II (CFW II)
For the discretization of consistent weak form II (39), the scalar entries of Kand fin (44) are computed as
KIJ =Z
BT
I(x)BJ(x) dΩ Zg
BT
I(x)nΨJ(x) dΓ Zg
ΨI(x)nTBJ(x) dΓ,(47a)
fI=Z
ΨI(x)sdΩ + Zh
ΨI(x)hZg
BT
I(x)ng.(47b)
In the above, it can be seen that compared to (45), both the stiffness matrix and the force vector contain new terms.
For the stiffness matrix, the two additional terms are the transpose of each other, so that only one of these matrices
needs to be constructed for the analysis (or just the upper triangle of the entire system matrix). In addition, since
the original stiffness matrix is symmetric, the resulting system matrix will also be symmetric, and efficient solvers
can be employed with this method.
15
4.4 Enforcement of boundary conditions
Procedurally, due to the nature of the approximations involved, it is uncommon to employ the formal definitions
of test and trial approximations in (15) and (16) directly in the weak form for meshfree methods. Rather, the full
systems are formed with the RK approximation defined over all nodes (1) leading to (44), and boundary conditions
are applied after. That is to say, the system in (44) represents a statically uncondensed system and cannot be solved
directly.
Instead, two favorable possibilities to enforce boundary conditions on the uncondensed systems are recommended
here: (1) meshfree transformation procedures can be applied—the reader is referred to [13] for more details, where
a simple and convenient row-swap implementation of the transformation method is presented; or (2) straightforward
static condensation with direct enforcement of boundary conditions is possible (equivalent of course to using (15) and
(16) directly in the weak form), provided either singular kernels [12] or shape functions with interpolation property [8]
are introduced for nodes that lie on the essential boundary.
5 Numerical examples
For the following examples, the parameters of the RK approximation and the numerical integration method have
been discussed in Section 2.3.3 in detail, but are briefly recalled here: twenty-by-twenty Gaussian integration per
background cell is employed with cells aligned with uniformly distributed nodes. Cubic B-spline kernels are used in
the RK approximation, with varying nodal spacing denoted h, kernel measures normalized with respect to hdenoted
a, and order of bases denoted p.
Three main methods are compared in terms of the transformation method [12]:
The transformation method (denoted as T)
The transformation method with consistent weak form I (denoted as T+CWF I)
The transformation method with consistent weak form II (denoted as T+CWF II)
Later, the boundary singular kernel method [12] is employed to complete the study to demonstrate the method works
with other types of strong enforcement, with permutations denoted following the same convention:
The boundary singular kernel method (denoted as B)
The boundary singular kernel method with consistent weak form I (denoted as B+CWF I)
The boundary singular kernel method with consistent weak form II (denoted as B+CWF II)
The error in the L2(Ω) norm and the H1(Ω) semi-norm are assessed, computed using the same quadrature rules as
forming the system matrices.
5.1 Patch test for the 2D Poisson equation
Consider the Poisson problem (8) on the domain ¯
Ω = [1,1] ×[1,1] with the pure essential boundary condition
g=Ω. Two cases for the patch test are considered: linear and quadratic.
As previously discussed, the ”overkill” quadrature in the following numerical examples should result, by conven-
tional wisdom, in passing the patch tests. For an in-depth discussion on the Galerkin meshfree formulations and
patch tests see [10], where it was shown that the residual of the error in numerical integration drives the error in
patch tests. Thus, ”overkill” quadrature drives the residual to machine precision in the limit, resulting in the method
being variationally consistent (passing the patch test), to machine precision. However in [10] it was also discussed
that the weak form must also attest to the strong form, which is not the case for the pure transformation method.
5.1.1 Linear solution
Let the prescribed body force and boundary conditions be consistent with an exact linear solution u= 0.1x+ 0.3y
(see (26) for the conditions).
The error in the L2(Ω) and H1(Ω) semi-norm for the three versions of the transformation method with linear
basis are shown in Figure 9. It is seen that both the proposed T+CWF I and T+CWF II are able to pass the linear
patch test (with machine precision error). The transformation method fails to pass the patch test, and meanwhile,
shows error associated with linear accuracy as discussed previously.
16
-1.4 -1.2 -1 -0.8 -0.6
log (h)
-16
-14
-12
-10
-8
-6
-4
-2
log (L2 error)
2.00
T
T+CWF I
T+CWF II
-1.4 -1.2 -1 -0.8 -0.6
log (h)
-16
-14
-12
-10
-8
-6
-4
-2
log (Semi-H1 error)
1.00
T
T+CWF I
T+CWF II
Figure 9: Norms of error for various methods in linear patch test: rates for T indicated.
5.1.2 Quadratic solution
For the quadratic patch test, the following quadratic solution is considered: u= 0.1x+ 0.3y+ 0.8x2+ 1.2xy + 0.6y2
(see (27) for associated prescribed conditions).
Here quadratic bases is introduced into the RK approximations; the L2(Ω) and H1(Ω) semi-norms of error are
shown in Figure 10. And again it is seen that T+CWF I and T+CWF II are able to pass the patch test (with
machine-level error) while the transformation method does not. Again, the error due to the inconsistent week form
tends to manifest as linear, even though quadratic basis is employed.
-1.4 -1.2 -1 -0.8 -0.6
log (h)
-16
-14
-12
-10
-8
-6
-4
-2
log (L2 error)
2.00
T
T+CWF I
T+CWF II
-1.4 -1.2 -1 -0.8 -0.6
log (h)
-16
-14
-12
-10
-8
-6
-4
-2
log (Semi-H1 error)
1.00
T
T+CWF I
T+CWF II
Figure 10: Norms of error for various methods in quadratic patch test of Poisson problem: rates for T indicated.
For both tests, it can be noted that the inability, and ability to pass the patch test by these methods, respectively,
is consistent with mesfhree patch test results reported in [20], where test functions were identically zero on the
essential boundary. Additionally, the ability to pass the patch test by both T+CWF I and T+CWF II, and failure to
pass the patch test by the transformation method alone, is consistent with the orthogonality relations (23), (33), and
(40), where the resulting best approximation properties, or lack thereof, indicate which methods should or should
not pass the patch tests. Thus the results of the patch tests are consistent with the discussions in Section 2.
5.2 Poisson equation with high-order solution
Now consider the poisson problem (8), on ¯
Ω = [0,1] ×[0,1], with source term and the pure essential boundary
condition as the same as (28):
g(x, 0) = sin(πx), g(x, 1) = g(0, y ) = g(1, y) = 0 on g,
s= 0 in Ω.
17
The exact solution of this problem is high order:
u={cosh(πy)coth(π) sinh(πy)}sin(πx).
In this study, the effect of the three weak forms is examined in terms of convergence rates with respect to varying
the support sizes a, order of basis functions p, and nodal spacing h.
5.2.1 p-refinement and h-refinement
First consider linear, quadratic, cubic, and quartic bases (denoted with p= 1, p= 2, p= 3, and p= 4, respectively),
with normalized support sizes of a=p+ 1. h-refinement is performed for each of the basis, starting with an 11 ×11
uniform node distribution. The solution errors in the L2(Ω) norm and H1(Ω) semi-norm of the various bases are
plotted in Figure 11-12, showing that T+CWF I and T+CWF II can yield optimal convergence rates (p+1 in L2and
pin semi-H1), while the traditional weak form (T) only yields linear rates (2 in L2and 1 in semi-H1), regardless of
the order of basis. Therefore the present approach can yield h-refinement with pth order optimal rates of convergence.
In addition, it can be seen in Figures 11b 11c, 12b, and 12c, that by increasing p, for any given h(with the
exception of one case), more accuracy can be obtained, yielding the ability to also provide p-refinement. These
two features of the present approach are in stark contrast to the results in Figures 11a and 12a, where increasing
pdoes not give consistently more accurate results, and in fact moving from p= 1 to p= 2 provides only marginal
improvement in accuracy, while increasing pfrom two to three and three to four actually provides worse results.
Comparing to Tables 1, 2, and 3, it can be inferred that this is due to the additional error in the representation of
boundary conditions in the test and trial functions, decreasing from p= 1 to p= 2, and increasing from p= 2 to
p= 3.
Finally, it can be noted that both T+CWF I and T+CWF II can provide p-refinement and h-refinement with
pth order optimal rates with nearly the same levels of error, and one may select either based on need or preference
(T+CWF I has only one new term, but yields a non-symmetric system, while T+CWF II yields a symmetric system,
but has three additional terms).
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
Linear : 2.44
Quadratic : 2.08
Cubic : 1.52
Quartic : 1.91
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
Linear : 2.50
Quadratic : 3.16
Cubic : 4.25
Quartic : 5.05
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
Linear : 2.50
Quadratic : 3.17
Cubic : 4.28
Quartic : 5.03
(c) T+CWF II
Figure 11: Convergence with various bases in the L2norm: rates indicated in legend.
18
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
Linear : 1.48
Quadratic : 1.15
Cubic : 0.51
Quartic : 0.95
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
Linear : 1.50
Quadratic : 1.97
Cubic : 3.26
Quartic : 4.09
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
Linear : 1.50
Quadratic : 1.98
Cubic : 3.25
Quartic : 3.87
(c) T+CWF II
Figure 12: Convergence with various bases in the H1semi-norm: rates indicated in legend.
5.2.2 Dilation analysis
The effect of varying normalized support sizes in the proposed method is now examined, since as shown previously,
increased support sizes in the RK approximation can yield different behavior on the essential boundary of the domain
for both test and trial functions. In addition, the present test is to show that the previous results were not a special
case—window functions and their measure can have an effect on accuracy and convergence rates [25], and even super-
convergence can be obtained for special values of window functions [22,23]. Thus the current permutations on aand
pwill examine the robustness of the formulation under the variety of free parameters in the RK approximation. For
this study, the discretizations and solution technique for the previous example are employed, refining has before,
while varying aand p.
First, linear basis (p= 1) is tested. The errors in the L2(Ω) norm and H1(Ω) semi-norm are plotted in Figures 13
and 14 respectively, for T, T+CWF I, and T+CWF II. First it can be seen that optimal rates are obtained for all
cases of a, for all methods. Also, when comparing to the results for the transformation alone (T), much lower levels of
error can be obtained with the present approach: nearly an order of magnitude when ais sufficiently large. The error
also decreases monotonically with increasing a—this point will be revisited. Finally, it be seen that little difference
in the solution error is observed for T+CWF I and T+CWF II, as in the previous cases.
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 1.5 : 2.04
a = 2.0 : 2.09
a = 2.5 : 2.44
a = 3.0 : 1.97
Optimal : 2
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 1.5 : 2.05
a = 2.0 : 2.50
a = 2.5 : 2.54
a = 3.0 : 2.17
Optimal : 2
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 1.5 : 2.08
a = 2.0 : 2.50
a = 2.5 : 2.55
a = 3.0 : 2.32
Optimal : 2
(c) T+CWF II
Figure 13: Convergence for linear basis (p= 1) with various ain the L2norm: rates indicated in legend.
19
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
a = 1.5 : 1.06
a = 2.0 : 1.48
a = 2.5 : 1.17
a = 3.0 : 0.98
Optimal : 1
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
a = 1.5 : 1.07
a = 2.0 : 1.50
a = 2.5 : 1.51
a = 3.0 : 1.14
Optimal : 1
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
a = 1.5 : 1.07
a = 2.0 : 1.50
a = 2.5 : 1.51
a = 3.0 : 1.15
Optimal : 1
(c) T+CWF II
Figure 14: Convergence for linear basis (p= 1) with various ain the H1semi-norm: rates indicated in legend.
Next, quadratic (p= 2) basis is tested for various values of a; the same error measures are presented in Figure 15
and 16. Here it can be seen that the use of T+CWF I and T+CWF II provides a large improvement in performance
over T alone, regardless of the value of a. The proposed methods provide optimal convergence rates consistently,
and do not depend on the dilation parameter. Meanwhile, with T alone, consistently worse rates are obtained with
increasing the kernel measure a. Finally, from the figures, it is starkly apparent that the magnitude of error can be
reduced anywhere from one to two orders of magnitude by employing the proposed techniques.
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 2.5 : 2.62
a = 3.0 : 2.08
a = 3.5 : 1.97
a = 4.0 : 1.44
Suboptimal (linear) : 2
Optimal : 3
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 2.5 : 3.36
a = 3.0 : 3.16
a = 3.5 : 2.88
a = 4.0 : 2.90
Optimal : 3
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 2.5 : 3.36
a = 3.0 : 3.17
a = 3.5 : 2.88
a = 4.0 : 2.89
Optimal : 3
(c) T+CWF II
Figure 15: Convergence for quadratic basis (p= 2) with various ain the L2norm: rates indicated in legend.
20
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 2.5 : 1.84
a = 3.0 : 1.15
a = 3.5 : 0.99
a = 4.0 : 0.35
Suboptimal (linear) : 1
Optimal : 2
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 2.5 : 2.25
a = 3.0 : 1.97
a = 3.5 : 1.82
a = 4.0 : 1.81
Optimal : 2
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 2.5 : 2.25
a = 3.0 : 1.98
a = 3.5 : 1.82
a = 4.0 : 1.81
Optimal : 2
(c) T+CWF II
Figure 16: Convergence for quadratic basis (p= 2) with various ain the H1semi-norm: rates indicated in legend.
Finally, cubic (p= 3) basis is tested. The same error measures are presented in Figure 17 and 18 for all
cases. Again, the two proposed methods consistently provide optimal convergence rates regardless of the value of a.
However in this case, it seems that the actual value has little effect on solution accuracy. On the other hand, the
transformation method (T) provides only linear rates, as expected, while the value of aalso has little effect. Similar
to the last example, it is apparent from Figure 17 and 18 that these technique provide the ability to reduce the
solution error by several orders of magnitude, in this case, by three orders, or 99.9%.
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 3.5 : 1.98
a = 4.0 : 1.52
a = 4.5 : 1.94
a = 5.0 : 1.87
Suboptimal (linear) : 2
Optimal : 4
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 3.5 : 3.39
a = 4.0 : 4.25
a = 4.5 : 4.06
a = 5.0 : 4.21
Optimal : 4
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
a = 3.5 : 3.37
a = 4.0 : 4.28
a = 4.5 : 4.34
a = 5.0 : 4.19
Optimal : 4
(c) T+CWF II
Figure 17: Convergence for cubic basis (p= 3) with various ain the L2norm: rates indicated in legend.
21
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 3.5 : 0.98
a = 4.0 : 0.51
a = 4.5 : 0.96
a = 5.0 : 0.91
Suboptimal (linear) : 1
Optimal : 3
(a) T
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 3.5 : 2.96
a = 4.0 : 3.26
a = 4.5 : 3.21
a = 5.0 : 3.25
Optimal : 3
(b) T+CWF I
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
log (semi-H1 error)
a = 3.5 : 2.96
a = 4.0 : 3.25
a = 4.5 : 3.35
a = 5.0 : 3.09
Optimal : 3
(c) T+CWF II
Figure 18: Convergence for cubic basis (p= 3) with various ain the H1semi-norm: rates indicated in legend.
5.2.3 A new concept: a-refinement
From the previous study, it can be noted that increasing the support size tends to yield lower error. This seems
to run counter-intuitive as reported results in the meshfree community seem to indicate an ”optimal” dilation (e.g.,
see [25]); this contradiction motivates the current study.
Here, a fixed distribution of the nodal spacing h= 1/10 is employed, while varying the normalized support afor
different values of p. Figure 19 shows the error for linear basis, where it is seen that by increasing a, lower error
can be obtained with T+CWF I and T+CWF II. On the other hand, with T alone, the optimal value appears to be
a= 2.5, which likely strikes a balance between approximation accuracy, and error due to the inability to construct
proper spaces required of the weak form.
As shown in Figure 20, the trends are similar for quadratic basis. However this time, increasing aconsistently
yields larger errors for the transformation method. Meanwhile, for both T+CWF I and T+CWF II, the error is
generally monotonically reduced by increasing a.
Finally, the results for cubic basis are presented in Figure 21. Here it is seen that the kernel measure has little
effect on solution accuracy, for all three methods. However for the transformation method, increasing the kernel
measure monotonically increases the error. At least, the present method can obtain robust results for any selection
of ain cubic basis.
To conclude, with the transformation method alone, there is an optimal value of afor linear basis. For higher-
order approximations, increasing the kernel measure seems to always increase the solution error. For the proposed
method, increasing afor both linear and quadratic basis very consistently yields lower error. Meanwhile, for cubic
basis, the solution is relatively unaffected. In this work, we term the former effect, the ability to decrease the solution
error by increasing the kernel measure, a-refinement. Thus with the proposed method, users may have confidence in
consistent behavior of meshfree approximations in the Galerkin solution.
1.5 2.0 2.5 3.0 3.5
a
-5.0
-4.0
-3.0
log (L2 error)
T
T+CWF I
T+CWF II
1.5 2 2.5 3 3.5
a
-3
-2.5
-2
-1.5
-1
log (semi-H1 error)
T
T+CWF I
T+CWF II
Figure 19: Norms of error of various methods with linear basis and various kernel measures a.
22
2.5 3.0 3.5 4.0 4.5
a
-6
-5.5
-5
-4.5
-4
-3.5
-3
log (L2 error)
T
T+CWF I
T+CWF II
2.5 3.0 3.5 4.0 4.5
a
-4
-3.5
-3
-2.5
-2
-1.5
-1
log (semi-H1error)
T
T+CWF I
T+CWF II
Figure 20: Norms of error of various methods with quadratic basis and various kernel measures a.
3.5 4 4.5 5 5.5
a
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
-3
log (L2 error)
T
T+CWF I
T+CWF II
3.5 4 4.5 5 5.5
a
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
log (semi-H1 error)
T
T+CWF I
T+CWF II
Figure 21: Norms of error of various methods with cubic basis and various kernel measures a.
23
5.3 Boundary singular kernel method
The boundary singular kernel method is another strong type of boundary condition enforcement. The singular kernels
for the reproducing kernel shape functions are introduced for essential boundary nodes, which recovers the properties
(12)-(13). The imposition of boundary conditions in this method is therefore similar to the finite element method.
However, since (12)-(13) do not imply the weak Kronecker delta property, values imposed may actually deviate
between the nodes, just as in the transformation method.
Here we also consider the Poisson equation with high-order solution given in section 5.2: with the boundary
singular kernel method (B), boundary singular kernel method with consistent weak form one (B+CWF I), and
boundary singular kernel method with consistent weak form two (B+CWF II). h-refinement is performed as before,
varying p, with a=p+ 1 fixed.
Figures 22 and 23 show the errors in the L2(Ω) norm and H1(Ω) semi-norm, respectively. Here it can be seen
that for B alone, the convergence rates are far from optimal, as expected from previous results and the previous
discussions, and are in fact, linear. When CWFs are considered, both B+CWF I and B+CWF II can yield optimal
convergence rates. That is, they allow h-refinement with pth order rates in the boundary singular kernel method.
In addition, since accuracy can be increased monotonically with increasing p(again with one case as an exception),
both B+CWF I and B+CWF II offer the ability to perform p-refinement.
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
Linear : 1.97
Quadratic : 1.99
Cubic : 1.79
Quartic : 1.85
Suboptimal (linear) : 2
(a) B
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
Linear : 1.97
Quadratic : 3.01
Cubic : 3.87
Quartic : 4.61
(b) B+CWF I
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
log (L2 error)
Linear : 2.53
Quadratic : 3.39
Cubic : 4.40
Quartic : 7.09
(c) B+CWF II
Figure 22: Convergence with various bases in the L2norm for the boundary singular kernel method: rates indicated
in legend.
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
Linear : 1.57
Quadratic : 1.09
Cubic : 0.72
Quartic : 0.90
Suboptimal (linear) : 1
(a) B
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
Linear : 1.57
Quadratic :2.35
Cubic : 3.46
Quartic : 3.95
(b) B+CWF I
-1.4 -1.2 -1.0 -0.8 -0.6
log (h)
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
log (semi-H1 error)
Linear : 1.53
Quadratic : 2.35
Cubic : 3.32
Quartic : 5.81
(c) B+CWF II
Figure 23: Convergence with various bases in the H1semi-norm for the boundary singular kernel method: rates
indicated in legend.
6 Conclusion
In this work, it has first been shown that traditional strong enforcement of boundary conditions at nodal locations
in meshfree methods is inconsistent with the traditional weak formulation of the problem. That is, without the weak
Kronecker delta property, large, non-trivial deviations between the desired conditions on test and trial functions exist
24
between nodes. This was shown to result loss of Galerkin orthogonality, and an O(h) error in the L2(g) norm,
which in turn resulted in an O(h) error in the energy norm of the problem at hand. This error was also shown to
be independent of the order of approximation employed. Thus, when solving PDEs, it was expected that this error
would limit the rate of convergence in the numerical solution.
It was then demonstrated through patch tests, and convergence tests, that indeed this O(h) energy norm error
appeared in the solution, limiting the rate of convergence in meshfree methods to that of linear basis. Thus, this
inconsistency resulted in a barrier for meshfree approximations, to solutions with linear accuracy in the energy norm
of the problem.
To remedy this deficiency, two new weak forms were introduced. The first accounts for the inconsistency in
the test function construction. Here, the weak form relaxes the requirements on the test functions, to include the
approximations introduced in the Galerkin equation under the strong-form enforcement framework. This weak form
attests to the strong form of the problem at hand, and also was shown to restore Galerkin orthogonality and the
best approximation property. Only one new term is required in the matrix formulation, however this results in a
non-symmetric system matrix for self-adjoint systems.
The second weak form introduced relaxes the requirements on both the test and trial functions, to include both
approximations in the strong-form enforcement methods. This weak form also attests to the strong form, and results
in a symmetric system, which is favorable. Interestingly, this method results in an alternate orthogonality relation
related to the boundary conditions, and an alternate best approximation property. The latter feature demonstrates
that the method simultaneously minimizes the error in the energy norm, and the error on the boundary.
In numerical tests, it was first shown that the two proposed methods can restore the ability to pass the patch test
to machine precision. It was then demonstrated that pth-order optimal convergence rates under h-refinement could
be obtained, which is in stark contrast to the existing strong-type methods under the conventional weak formulation.
In addition, by increasing pfor a fixed h, it was shown that lower error can be obtained, thus providing the ability to
perform p-refinement for the first time under this framework. It was also shown that these results were independent
of the particular dilation achosen, and in fact, lower error can be obtained by increasing a, which was termed a-
refinement. Taken together, the proposed method provides the ability to perform p-refinement, h-refinement with pth
order rates, and a new capability called a-refinement.
Finally, it should be noted that in this work, high-order quadrature was employed, which is atypical of a practical
meshfree implementation. In future work, this aspect should be investigated: for instance, what is the lowest order
quadrature required to maintain these high-order properties? And, with methods such as variationally consistent
integration, which can greatly reduce the burden of quadrature, what would be the order required? It is noteworthy
that the present approach is compatible with the variationally consistent approach, in that the weak forms attest to
the strong form of the problem, which is in contrast to traditional strong enforcement of boundary conditions. Lastly,
this method was tested for the Poisson equation, but can be applied to other boundary value problems as well, as
described in the appendix.
Acknowledgments The authors greatly acknowledge the support of this work by the L. Robert and Mary L.
Kimball Early Career Professorship, and the College of Engineering at Penn State.
Appendix
Consider the following abstract boundary value problem governing a scalar u:
Lu +s= 0 in Ω (49a)
Bu =hon h(49b)
u=gon g(49c)
where Lis scalar differential operator acting in the domain Ω Rd,sis a source term, gis the prescribed values of u
on the essential boundary g,Bis a scalar boundary operator acting on the natural boundary h,gh=
and Ω = gh.
Consider the weighted residual of the boundary value problem:
(v, Lu +s)= 0.(50)
Manipulation yields a bilinear form a(·,·) which results from the integration by parts formula (v, Lu)= (v, B u)
a(v, u), and the following problem statement for (W1
C): find uHk
g,Hk
g={u|uHk(Ω), u =gon g}such that
for all vHkthe following equation holds:
a(v, u)(v, B u)g= (v, s)+ (v, h)h(51)
25
where Hkis an adequate Sobolev space. The above is a consistent weight residual of (49) as v= 0 on gis not
required to verify (49). Note that this procedure does not require the governing equation to emanate from a potential.
To take a concrete example, consider the equations for elasticity:
∇ · σ+b=0in Ω (52a)
σ·n=hon h(52b)
u=gon g(52c)
where uis the displacement, bis the body force, his the traction, gis the prescribed displacement, nis the unit
normal to the domain, σ=C:suis the Cauchy stress tensor; Cis the elasticity tensor and su= 1/2(∇⊗u+u⊗∇)
is the strain tensor.
The following form for (W1
C) can be obtained following the given procedures: find u∈ Sg,Sg={u|u
H1(Ω), ui=gion gi}such that for all wH1the following equation holds:
a(w,u)(w,n·σ(u))g= (w,b)+ (w,h)h(53)
where
a(w,u)=Z
sw:C:sudΩ,(54a)
(w,b)=Z
w·bdΩ,(54b)
(w,h)h=Zh
w·h,(54c)
(w,n·σ(u))g=Zg
w·(n·σ(u)) dΓ.(54d)
For the symmetric weak form of the abstract boundary value problem (49), consider a more general weighted
residual:
(v, Lu +s)+ (vg, u g)g= 0.(55)
Choosing v=vand vg=Bv one obtains the following formulation for (W2
C): find uHksuch that for all vHk
the following equation holds
a(v, u)(v, B u)g(Bv, u)g= (w, s)+ (v , h)h(Bv, g)g(56)
where Hkis again an adequate Sobolev space. The above verifies (49) without the use of v= 0 on gand
u=gon g. Note that if Lis non-self-adjoint a(·,·) is not symmetric, and the resulting Galerkin system matrix
will not be symmetric.
To take an example, consider the elasticity equations (52) again. The (W2
C) can be derived as: find uH1, such
that for all wH1the following equation holds:
a(w,u)(w,σ(u)·n)g(σ(w)·n,u)g= (w,b)+ (w,h)h(σ(w)·n,g)g.(57)
Again, this procedure does not require the governing equation to emanate from a potential, although from the
discussions in the manuscript, it seems that this is likely always possible to do so if the original governing equation
does.
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