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Preprint accepted in Journal of Computational Physics (originally submitted in September, 2019).

Download the journal version at https://doi.org/10.1016/j.jcp.2020.109358

A Hyperbolic Poisson Solver for Tetrahedral Grids

Hiroaki Nishikawa∗

National Institute of Aerospace, Hampton, VA 23666, USA

March 23, 2020

Abstract

This paper presents a robust and eﬃcient Poisson solver that can produce accurate solutions and gradi-

ents (e.g., heat ﬂux) on unstructured tetrahedral grids. The solver is constructed based on the hyperbolic

method for diﬀusion, where the Laplacian operator is discretized in the form of a hyperbolic system with

solution gradients introduced as additional unknown variables. A practical formula for deﬁning a reference

length is proposed, which is needed to properly scale the relaxation length associated with the hyper-

bolic formulation for scale-invariant computations. The resulting system of residual equations is eﬃciently

solved by a Jacobian-Free Newton-Krylov solver with an implicit defect-correction solver used as a variable-

preconditioner. Robustness and superior gradient accuracy are demonstrated for linear and nonlinear Poisson

equations through a series of numerical tests with unstructured tetrahedral grids.

1 Introduction

The Poisson equation:

div(νgrad u)=f(x, y, z),(1)

where uis a solution variable (e.g., temperature), νis a positive coeﬃcient (e.g., heat conductivity), and

fis a forcing term, arises across many disciplines of science and engineering: e.g., the pressure equation in

incompressible ﬂows [1], steady heat conduction [2], electrostatic problems [3], static-state elasticity problems [4],

geophysical electromagnetic modeling [5], and so on. Whether it is linear with νas a function of space (including

a constant) or nonlinear with νas a function of u, eﬃcient numerical methods are available for Cartesian grids

[6, 7, 8]. But current methods for unstructured grids are far from optimal in eﬃciency, robustness, and accuracy.

The demand for eﬃcient, robust, and accurate unstructured-grid solvers, however, only continues to grow as

computational methods are being applied to many practical problems with increasingly complex geometries.

Since there is no systematic ordering in an unstructured grid, e.g., no grid lines along the coordinate directions (or

any particular direction), special solution algorithms such as a Fourier transform or Thomas’ algorithm cannot be

applied. Therefore, unstructured-grid solvers need to rely on iterative methods. Classical iterative methods such

as the Jacobi and Gauss-Seidel relaxation schemes can be easily implemented but are known to signiﬁcantly

slow down for large-scale grids [9, 10]. Multigrid methods, which in principle can achieve grid-independent

convergence [10], are still not widely employed in practical solvers because their implementation is not simple and

requires further developments for fully unstructured grids. Moreover, robustness issues (e.g., solver divergence)

often arise in iterative methods especially for highly irregular grids. Such issues are often resolved by repairing

the grid but, in general, should be dealt with by improving discretizations and solvers because grids can and will

be highly irregular in adaptive-grid simulations [11]. For these reasons, the development of robust unstructured-

grid solvers remains an active area for research [12, 13, 14, 15]. In addition, it is well known that gradient

accuracy severely deteriorates on unstructured grids [16, 17, 18]. This problem is highly relevant to the Poisson

equation since many quantities governed by the Poisson equation carry physical importance in their gradients

such as heat ﬂux, pressure gradient, stresses, etc., and gradient prediction is often the primary target for

numerical simulations. Despite progress made over the past years [17, 19, 20], satisfactory techniques are not

available yet and a grid with some regularity needs to be generated for each practical simulation. In this

paper, we address these issues of iterative convergence and accuracy by the hyperbolic method for diﬀusion

and demonstrate that a practical Poisson solver can be constructed, which achieves robust convergence and

∗Associate Research Fellow (hiro@nianet.org), 100 Exploration Way, Hampton, VA 23666 USA,

1

accurate gradient prediction on fully irregular tetrahedral grids. Although not discussed in this paper, the

developed hyperbolic solver will be directly applicable to three-dimensional unsteady diﬀusion problems with

implicit time-stepping schemes [21].

The hyperbolic method for diﬀusion is a method for discretizing a second-order-derivative operator, e.g.,

the left hand side of Eq.(1), in the form of a hyperbolic system, introducing solution gradients as additional

variables [22]:

∂τu=∂xp+∂yq+∂zr−f,

∂τp=ν

Tr

(∂xu−p/ν),

∂τq=ν

Tr

(∂yu−q/ν),(2)

∂τr=ν

Tr

(∂zu−r/ν),

where τis a pseudo time variable, Tris a relaxation time, and p,q, and rare called the gradient variables. Note

that the above system is equivalent to the Poisson equation (1)with(p, q, r)=νgradu, for any nonzero Tr,

in the pseudo steady state or as soon as the pseudo time derivatives are dropped. Therefore, one can derive a

consistent discretization for the Poisson equation by discretizing the above system by a method for hyperbolic

systems (e.g., an upwind method) and then dropping the pseudo time derivatives. Although the hyperbolic

system (2) looks similar to the classical hyperbolic heat equations of Refs.[23, 24], it is fundamentally diﬀerent

in that Tris a free parameter and can be deﬁned to improve iterative convergence. This is the core idea of the

hyperbolic method considered here. The relaxation time is then deﬁned by

Tr=L2

r

ν,(3)

where Lris a relaxation length, and an optimal formula for Lris given by

Lr=1

2π,(4)

which was derived by maximizing the eﬀect of error propagation with a Fourier analysis on a unit square domain

[22, 25].

It should be noted also that the hyperbolic method considered here is not, unlike the classical approach

[23, 24, 26, 27], proposed a new physical model for diﬀusion and Poisson equations; it is merely a way of con-

structing a superior spatial discretization for the second-derivative diﬀusion operator expressed in a consistent

ﬁrst-order system form. As it has been demonstrated for many problems since Ref.[22], upwind discretizations

of the hyperbolized diﬀusion system (2) have unique and superior properties over conventional discretizations:

improved order of accuracy and higher quality in the solution gradients on unstructured grids due to the strong

coupling among the solution and gradient variables, and convergence acceleration by the elimination of second

derivatives. Improvements in iterative convergence, especially for implicit methods, come also from enabling

a ﬁrst-order viscous Jacobian, which is more accurate than a typical zeroth order Jacobian used in conven-

tional viscous solvers [28, 29, 30]. These advantages have been successfully demonstrated for linear/nonlinear

anisotropic diﬀusion equations [31], a tensor-coeﬃcient diﬀusion equation [32, 33], incompressible/compressible

Navier-Stokes equations [18, 34, 35, 36], an incompressible magnetohydrodynamics model [37], quasi-neutral

plasma models [32, 38], and the dispersion equation by Ricchiuto’s non-standard hyperbolic formulation [39].

There is, however, a long-standing issue of scale-invariant computations in three-dimensional problems, which

has prevented the method from being applied to practical problems and dimensional equations. For example, if

uis temperature in a heat conduction problem with ν= 1, f= 0, and speciﬁed boundary values (i.e., Dirichlet

conditions), the temperature distribution in a given domain must not depend on the unit used to deﬁne the grid

(e.g., meter, inch, foot, etc.). This property is referred to, in this paper, as the scale-invariance property. It is

very important to preserve this property in the discrete level because the grid unit should be totally arbitrary

(i.e., freely chosen by a user) and thus it should not greatly aﬀect the numerical solution and cause any serious

problem in numerical simulations. Figure 1 shows an example from Ref.[21], where numerical solutions of the

Laplace equation with Dirichlet conditions obtained by a scale-invariant method on the same triangular grid

with three diﬀerent units, i.e., meter, kilometer, and millimeter, are shown. Note that the solution variation

2

along the boundary (i.e., the same Dirichlet conditions) is kept the same for all grids in order to simulate a

situation where the same physical problem is solved with diﬀerent units. As shown in Figure 1(d), an implicit

solver based on a non-scale-invariant method is very sensitive to the grid unit and even diverges for the grid

deﬁned in millimeter. On the other hand, as in Figure 1(e), a solver based on a scale-invariant method converges

in the same way at the same number of iterations and produces the same numerical solution for all grid units

[21].

x

y

00.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75

(a) Grid[m].

x

y

00.0005 0.001

0

0.0005

0.001

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75

(b) Grid[km]

x

y

0200 400 600 800 1000

0

200

400

600

800

1000

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75

(c) Grid[mm]

Iteration

Max L1 norm of the residual components

Linear relaxation / Iteration

010 20 30 40 50 60

10-10

10-8

10-6

10-4

10-2

100

102

0

100

200

300

400

500

[km]

[m]

[mm]

[km]

[m]

[mm]

(d) Not scale-invariant.

Iteration

Max L1 norm of the residual components

Linear relaxation / Iteration

010 20 30 40 50 60

10-10

10-8

10-6

10-4

10-2

100

0

100

200

300

400

500

[km]

[m]

[mm]

[km]

[m]

[mm]

(e) Scale invariant.

Figure 1: Solution (e.g., temperature) contours obtained by a scale-invariant method in a square domain in three

diﬀerent units (top) and iterative convergence histories for a non-scale-invariant solver and a scale-invariant

solver (bottom) [21].

Many numerical schemes directly discretizing the second-order operator preserves the scale-invariance prop-

erty, but those based on a ﬁrst-order system formulation, e.g., the hyperbolic method considered here and related

methods such as the one in Ref.[26], do not automatically satisfy this property and can encounter diﬃculties

in iterative convergence although numerical schemes are consistent and maintain the design order of accuracy.

This problem was ﬁrst recognized when the hyperbolic method was applied to a dimensional diﬀusion equation

and addressed in Ref.[21] for two-dimensional problems. As discussed in Ref.[21], successful scale-invariant

computations require two items: (1) a reference length Lto scale Lrin a consistent manner with the grid unit

in the form:

Lr=L

2π,(5)

(or equivalently to nondimensionalize the hyperbolic system in a consistent manner) and (2) an optimal value of

Lr/L for fast convergence, which ensures, for example, that all Fourier modes propagate just like advection. The

3

ﬁrst item eliminates the dependence on the grid unit and the second ensures optimal performance of numerical

schemes. To take into account the two, we proposed in Ref.[21], as a practical technique, to deﬁne Lsuch

that Lr/L =1/(2π) is optimal in a rectangular domain and then generalize the resulting formula to arbitrary

domains. The proposed formula was demonstrated for various two-dimensional diﬀusion problems, but it was

not immediately applicable to three-dimensional problems. In this paper, we derive an optimal formula for a

three-dimensional rectangular domain, carefully extend it to a general domain, and demonstrate scale-invariance

and superior properties of the hyperbolic solver in three dimensions. The proposed formula thus paves the way

for the hyperbolic method to be applied for solving numerous practical applications.

As the hyperbolic method being a reformulation technique, the hyperbolized Poisson system can be dis-

cretized by any discretization method. In this paper, we employ the node-centered edge-based discretization,

which is one of the most eﬃcient discretization methods and has widely been used in practical unstructured-grid

ﬂuid-dynamics solvers [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. In particular, we consider the edge-based

discretization on fully unstructured tetrahedral grids. Among various types of elements (e.g., hexahedra, prisms,

etc.), we focus on tetrahedral elements for three reasons. First, the edge-based discretization is formally second-

order accurate on arbitrary tetrahedral grids while it deteriorates to ﬁrst-order with other types elements unless

the grid is regular (see Ref.[52] and Appendix B in Ref.[53]). To maintain accuracy for problems with complex

geometries, it is therefore necessary to develop a method for purely tetrahedral grids. Second, eﬃcient and accu-

rate practical computations will eventually involve grid adaptation, if not used routinely today, and tetrahedral

elements are known to be more ﬂexible and suitable for anisotropic grid adaptation [54, 55, 56, 57, 58, 59, 60].

Finally, although not pursued in this paper, a very economical third-order method can be constructed for the

edge-based method without increasing the number of quadrature points and also without high-order curved

surface mesh on tetrahedral grids [61, 62, 63, 64]. This special third-order property holds only on tetrahedral

grids.

To solve the resulting discrete equations, we ﬁrst consider an implicit defect-correction (IDC) solver and then

construct a Jacobian-Free Newton-Krylov (JFNK) solver with the IDC solver used as a variable-preconditioner

[13, 21]. The IDC solver is constructed with an approximate residual Jacobian based on the derivative of a

compact lower-order residual; the Jacobian of the second-order residual can be too large to store (and hard to

relax) and thus is not practical. Therefore, the IDC solver is constructed within the same framework for both

linear and nonlinear Poisson equations. JFNK solvers, which solve the exactly linearized system without storing

the true Jacobian, have been increasingly popular in recent years for its robustness especially on unstructured

grids [15, 65]. Its robustness and eﬃciency will be demonstrated here for solving linear and nonlinear Poisson

equations on unstructured tetrahedral grids. However, for the hyperbolic Poisson discretization, the JFNK

solver is not scale-invariant, unless carefully constructed, because it treats all variables and residual equations

with mixed units as a single vector of variables. A suitable modiﬁcation is proposed in this paper, with which

the scale-invariance property can be guaranteed with the JFNK solver.

The paper is organized as follows. In Section 2, we describe the hyperbolic Poisson system, discuss the

eigen-structure, and derive a dissipation matrix, which will be used to construct an upwind numerical ﬂux. In

Section 3, we describe the edge-based discretization of the hyperbolic Poisson system and boundary conditions.

In Section 4, we discuss how the discretization is aﬀected by the grid unit, derive an optimal reference length for

a rectangular domain, and generalize it to arbitrary domains. In Section 5, we describe two solvers used in the

study: IDC and JFNK solvers, and a modiﬁcation necessary in the JFNK solver for ensuring the scale-invariance

property. In Section 6, we present numerical results for model problems of heat-conduction in bounded domains,

which is one of our primary target applications. Finally, we conclude the paper with remarks.

2 Hyperbolic Formulation of Poisson Equation

Consider the hyperbolic formulation of the Poisson equation written in the preconditioned conservative form:

P−1∂τU+∂xF+∂yG+∂zH=S,(6)

where

P−1=⎡

⎢

⎢

⎣

10 0 0

0Tr/ν00

00Tr/ν0

00 0Tr/ν

⎤

⎥

⎥

⎦

,U=⎡

⎢

⎢

⎣

u

p

q

r

⎤

⎥

⎥

⎦

,F=⎡

⎢

⎢

⎣

−p

−u

0

0

⎤

⎥

⎥

⎦

,G=⎡

⎢

⎢

⎣

−q

0

−u

0

⎤

⎥

⎥

⎦

,H=⎡

⎢

⎢

⎣

−r

0

0

−u

⎤

⎥

⎥

⎦

,S=⎡

⎢

⎢

⎣

−f

−p/ν

−q/ν

−r/ν

⎤

⎥

⎥

⎦

.(7)

4

The gradient variables correspond to the ﬂuxes when the pseudo time derivative terms are dropped:

p=ν∂xu, q =ν∂yu, r =ν∂zu, (8)

and therefore the solution gradient can be obtained by

gradu=(∂xu, ∂yu, ∂zu)='p

ν,q

ν,r

ν(.(9)

This particular formulation is called the nonlinear formulation and suitable for nonlinear diﬀusion problems,

where νis a function of the solution (also useful when νis a function of space) [66, 67]. As shown below, this

formulation allows us to derive the ﬂux Jacobian without diﬀerentiating νfor nonlinear equations.

For the preconditioned conservative formulation, a matrix relevant to the construction of the dissipation

matrix for an upwind numerical ﬂux is given by (see, e.g., Ref.[68])

P−1|PAn|,(10)

where

An=∂Fn

∂U=∂(Fˆnx+Gˆny+Hˆnz)

∂U,(11)

and ˆ

n=(ˆnx,ˆny,ˆnz) is an arbitrary unit vector. Note that the ﬂux vector Fndoes not involve the coeﬃcient ν

and therefore there is no need to diﬀerentiate ν. To construct the matrix (10), we ﬁrst obtain

An=⎡

⎢

⎢

⎣

0−ˆnx−ˆny−ˆnz

−ˆnx000

−ˆny000

−ˆnz000

⎤

⎥

⎥

⎦

,(12)

and then

PAn=∂Fn

∂U=⎡

⎢

⎢

⎣

0−ˆnx−ˆny−ˆnz

−νˆnx/Tr000

−νˆny/Tr000

−νˆnz/Tr000

⎤

⎥

⎥

⎦

.(13)

The eigenvalues of PA nare given by

±)ν/Tr,0,0,(14)

which are used to deﬁne the diagonal matrix:

Λ=λ⎡

⎢

⎢

⎣

−1000

0100

0000

0000

⎤

⎥

⎥

⎦

,(15)

where

λ=)ν/Tr=ν/Lr.(16)

The corresponding right eigenvectors are given by

R=⎡

⎢

⎢

⎣

Lr/ν−Lr/ν00

ˆnxˆnx−ˆny−ˆnz

ˆnyˆnyˆnx0

ˆnzˆnz0ˆnx

⎤

⎥

⎥

⎦

.(17)

Finally, the dissipation matrix is obtained as

P−1|PAn|=P−1R|Λ|R−1=⎡

⎢

⎢

⎢

⎢

⎢

⎣

ν/Lr0 0 0

0Lrˆn2

x/νLrˆnxˆny/νLrˆnxˆnz/ν

0Lrˆnyˆnx/νLrˆn2

y/νLrˆnyˆnz/ν

0Lrˆnzˆnx/νLrˆnzˆny/νLrˆn2

z/ν

⎤

⎥

⎥

⎥

⎥

⎥

⎦

.(18)

5

3 Edge-Based Discretization

3.1 Residual

Given a tetrahedral grid, we store the solution values at nodes and discretize the system (6) at a node jas

VjP−1

j

dUj

dτ=−*

k∈{kj}

ΦjkAjk +SjVj,(19)

where Vjis the measure of the dual control volume around the node j,P−1

jis the preconditioning matrix

evaluated at jin case that νis not a constant, {kj}is a set of neighbor nodes of j,Ajk is the magnitude of

the directed area vector njk, which is a sum of the directed-areas corresponding to the dual-triangular faces

associated with all tetrahedral elements sharing the edge [j, k] (see Figure 2), and Φjk is a numerical ﬂux

evaluated by the following upwind ﬂux:

Φjk(UL,UR,ˆ

njk)= 1

2[Fn(UR)+Fn(UL)] −1

2P−1|PAn|(UR−UL),(20)

where the projected ﬂux Fnand the Jacobian Ancomputed with the unit directed-area vector: ˆ

njk =njk/|njk|.

Note that the volume Vjis precisely equal to 1/4 of the sum of the volumes of the tetrahedra sharing the node

j. The left and right states, ULand UR, are computed, for second-order accuracy, by the kappa reconstruction

scheme [69, 70]:

UL=Uj++1−κ

2(∂jkUj−∆Ujk)+ 1+κ

2∆Ujk,,(21)

UR=Uk−+1−κ

2(∂jkUk−∆Ujk)+ 1+κ

2∆Ujk,,(22)

where κis a real-valued parameter, and

∂jkUj=∇ULSQ

j·(xk−xj),∂

jkUk=∇ULSQ

k·(xk−xj),∆Ujk =1

2(Uk−Uj).(23)

The nodal gradients ∇ULSQ

jand ∇ULSQ

kare computed by a weighted least-squares (LSQ) method:

∇uLSQ

j=⎡

⎢

⎢

⎢

⎣

ˆ

∂xuj

ˆ

∂yuj

ˆ

∂zuj

⎤

⎥

⎥

⎥

⎦

=*

k∈{kj}

⎡

⎢

⎢

⎣

cx

jk

cy

jk

cz

jk

⎤

⎥

⎥

⎦

(uk−uj),(24)

where (cx

jk,c

y

jk,c

z

jk) are LSQ coeﬃcients (i.e., the coeﬃcients of uk−ujin the solution of a linear LSQ problem),

which are obtained by solving the overdetermined problem (e.g., by the QR factorization):

Ax =b,(25)

where

A=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

w1(x1−xj)w1(y1−yj)w1(z1−zj)

.

.

..

.

.

wk(xk−xj)wk(yk−yj)wk(zk−zj)

.

.

..

.

.

wnj(xnj−xj)wnj(ynj−yj)wnj(znj−zj)

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,x=⎡

⎢

⎣

∂xuj

∂yuj

∂zuj

⎤

⎥

⎦,b=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

w1(u1−uj)

.

.

.

wk(uk−uj)

.

.

.

wnj(unj−uj)

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,(26)

wk=1

dP

k

,d

k=-(xk−xj)2+(yk−yj)2+(zk−zj)2,(27)

where njis the number of neighbor nodes in {kj}and Pis a parameter, which can take zero (unweighted LSQ),

one (fully weighted LSQ), or any other real value but set to be 0.5 in this study for the hyperbolic Poisson

6

solver. The resulting scheme is called Scheme-I [25] and it is known to bes a very robust scheme. There is a

more eﬃcient scheme called Scheme-II, where the LSQ gradients of the primal solution variable uare replaced

by the gradient variables:

∂jkuj='p

ν,q

ν,r

ν(j·(xk−xj),∂

jkuk='p

ν,q

ν,r

ν(k·(xk−xj).(28)

This scheme does not require LSQ gradients for the primal variables and is typically more accurate. For Scheme-

II, the choice of the parameter κhas been found to be critical for robustness. A typical choice κ= 0 works ﬁne

for isotropic grids, but can lead to instability for anisotropic grids. This issue can be avoided by deﬁning κas

κ=.0ifARjk <10,

0.5ifARjk ≥10,

(29)

where ARjk is a local grid-aspect-ratio deﬁned at the edge [j, k] as

ARjk = max(ARj,ARk),ARj= max

T∈{Tj}ART,ARk= max

T∈{Tk}ART,(30)

where {Tj}and {Tk}denote the sets of tetrahedra around the nodes jand k, respectively, and ARTis the

cell-aspect-ratio deﬁned for a tetrahedron Tas the ratio of the maximum edge length to the minimum height

(i.e., 3 ×volume divided by the maximum triangular face area). As can be seen from Equations (21) and (22),

the contribution of the nodal gradients is reduced for κ>0 and eventually vanishes at κ= 1. Therefore, κ=0.5

has the eﬀect of reducing the contribution of ∇ULSQ

jby half in the reconstruction and it provides a favorable

eﬀect when the gradient is inaccurate and will negatively aﬀect the solver stability. Note, however, that κ=1

cannot be taken because it makes the dissipation term identically vanish and the solver can become unstable.

Not used in this work, but it is also possible to use ARjk for other purposes, e.g., to switch from Scheme-II

to Scheme-I when ARjk is exceptionally large, for increased robustness. Note that the value of κthus deﬁned

depends only on the geometry of the grid and therefore is ﬁxed for a given grid. A more sophisticated control

may be devised by the use of limiter functions [71] although it introduces nonlinearity in the algorithm even for

linear problems. In this study, we have found the above control of κvery eﬀective and do not consider limiter

functions.

Finally, we drop the pseudo time derivative term and deﬁne the residual equation at the node j:

Resj≡− *

k∈{kj}

ΦjkAjk +SjVj=0,(31)

which leads to the global system of 4Nresidual equations:

Res(U)=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

Res1(1)

Res1(2)

Res1(3)

Res1(4)

Res2(1)

Res2(2)

Res2(3)

Res2(4)

.

.

.

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

=0,(32)

to be solved for the vector of 4Nunknowns, U=(u1,p

1,q

1,r

1,u

2,p

2,q

2,r

2,···,u

N,p

N,q

N,r

N), where Nis the

total number of nodes in a given grid and Resj=(Resj(1),Res

j(2),Res

j(3),Res

j(4)). The system of residual

equations is eﬃciently solved by implicit methods, which will be discussed later in Section 5. It is possible

to keep the pseudo time derivative, if desired, possibly to increase solver robustness, but such has been found

unnecessary for the test problems considered in this paper and therefore it is completely dropped here. As a

result, the residual equation (31) becomes a consistent approximation to the Poisson equation expressed in the

ﬁrst-order system, i.e., the system (2) with the left hand side ignored.

Compared with a conventional Poisson solver that directly discretizes Equation (1), the hyperbolic method

requires the solution of a larger discrete system with 3Nadditional equations and unknowns. Therefore,

7

it requires more memory to store the additional quantities and also more computational time per iteration.

However, as already demonstrated for many problems in previous papers [22, 25, 34, 36], iterative methods

converge more rapidly for the hyperbolic discretization than for a conventional discretization with a speed-up

factor O(1/h), where his a typical mesh spacing, because of the elimination of second derivatives from target

governing equations. Moreover, it leads to a superior implicit solver with a more accurate residual Jacobian

(i.e., the derivative of a compact ﬁrst-order accurate residual) than that for a conventional scheme, where the

Jacobian is typically constructed with a compact zeroth-order accurate residual [28, 29, 30]. Also, as we will

demonstrate, the hyperbolic solver can produce accurate and useful numerical solutions (accurate gradients,

in particular) on unstructured grids when a conventional solver cannot. This is a decisive advantage of the

hyperbolic solver because then eﬃciency comparison does not even make sense as there is no point of obtaining

inaccurate solutions faster. It is critically important, in a practical point of view, to be able to produce useful

solutions and gradients on fully unstructured grids even with a larger memory requirement.

3.2 Boundary Conditions

Boundary conditions are imposed weakly through the right state that is sent to the numerical ﬂux at bound-

ary nodes. The residual equations are, therefore, deﬁned not only at interior nodes but also at boundary

nodes, and numerical solutions will be determined by solving them at all nodes. To guarantee second-order

accuracy, the boundary residual must be closed by boundary ﬂuxes with an accuracy-preserving boundary-

ﬂux quadrature formula, which depends on the type of elements adjacent to the boundary. A comprehensive

list of second-order quadrature formulas for quadrilateral/triangular elements in two dimensions and hexahe-

dral/tetrahedral/prismatic/pyramidal elements in three dimensions can be found in Appendix B in Ref.[53].

For tetrahedral grids, consider a dual face of a control volume around a boundary node jas shown in Figure

3. The residual at the node jneeds to be closed by the ﬂux contribution across the boundary dual face. The

boundary ﬂux contribution should be computed by the following second-order accurate formula [53, 72, 73]:

+6

8Φj(Uj,UR,ˆ

nB)+1

8Φ2(U2,UR,ˆ

nB)+1

8Φ3(U3,UR,ˆ

nB),|nB|,(33)

where nBdenotes the boundary element normal vector (pointing outward from the interior domain) and

ˆ

nB=nB/|nB|with |nB|being 1/3 of the triangle area. The ﬂux at each node is computed by the upwind ﬂux

(20) with the left state ULgiven by the state stored at the boundary node (e.g., UL=Ujat the node j) and

the right state URdeﬁned by the boundary condition. Note that URdepends on ULand therefore is diﬀerent

at nodes j, 2, and 3. For the Poisson equation, two conditions are considered: Dirichlet and Neumann conditions.

Dirichlet condition: The primal solution value is given: u=ub,whereubis a given value. This condi-

tion is implemented as

UR=⎡

⎢

⎢

⎣

uR

pR

qR

rR

⎤

⎥

⎥

⎦

=⎡

⎢

⎢

⎣

2ub−uL

pL

qL

rL

⎤

⎥

⎥

⎦

,(34)

where p,q, and rare evaluated by the left state since their values are not known, so that the average of uLand

uRmatches the given solution:

uL+uR

2=ub.(35)

Neumann condition: The gradient normal to the boundary surface is given: ∂u/∂n=pnb,wherenis

taken in the direction towards the interior domain and pnbis the given normal gradient. This condition is

implemented as

UR=⎡

⎢

⎢

⎢

⎢

⎢

⎣

uR

pR

qR

rR

⎤

⎥

⎥

⎥

⎥

⎥

⎦

=⎡

⎢

⎢

⎢

⎢

⎣

uL

pL+ 2(pnb−pn

L)ˆn′

x

qL+ 2(pnb−pn

L)ˆn′

y

rL+ 2(pnb−pn

L)ˆn′

z

⎤

⎥

⎥

⎥

⎥

⎦

,(36)

8

where the solution value is copied from the left state since it is not known and pn=(p, q, r)·ˆ

n′

Bwith ˆ

n′

B=

(ˆn′

x,ˆn′

y,ˆn′

z)=−ˆ

nB, so that the average of the normal derivatives matches the given value:

pn

L+pn

R

2=pnb.(37)

Note that the Neumann condition speciﬁes the normal derivative at a boundary, but there is no need, in the

hyperbolic method, to discretize the normal derivative at a boundary because gradients are available directly

at boundary nodes as variables and they are second-order accurate. In eﬀect, the Neumann condition can be

imposed in the same way as the Dirichlet condition via gradient variables, which is another useful feature of the

hyperbolic method.

4 Deﬁning Reference Length for Scale Invariance

4.1 Scale Dependence

As previously discussed in Ref.[21] for two-dimensional problems, the relaxation length Lrneeds to be scaled

by a suitable reference length in order to ensure the scale-invariance property: the numerical solution to the

Laplace equation (f= 0 and ν= 1) must not depend on the grid unit. In conventional methods, as one can

easily see, for example, from the classical ﬁnite-diﬀerence scheme on a Cartesian grid with a uniform spacing h,

ui−1,j,k −2ui,j,k +ui+1,j,k

h2+ui,j−1,k −2ui,j,k +ui,j +1,k

h2+ui,j,k−1−2ui,j,k +ui,j,k+1

h2=0,(38)

which is

ui,j,k =1

6[ui−1,j,k +ui+1,j,k +ui,j−1,k +ui,j+1,k +ui,j,k−1+ui,j,k+1],(39)

the discretization and thus the numerical solution do not depend on (x, y, z), and hence, it is scale invariant.

It is pointed out at this point that the scale-invariance property is deﬁned for the Laplace equation. For

a more general equation such as linear and nonlinear Poisson equations, the discretization is said to be scale

invariant if it has the scale-invariance property when applied to the Laplace equation. Scale-invariant discretiza-

tions will be scale invariant for linear and nonlinear Poisson equations if ν(x, y, z , u) and f(x, y, z) are deﬁned

consistently with the grid unit used as we will show later in Section 6 for cases where f=f(x, y, z) and ν=ν(u).

On an unstructured grid, many conventional discretizations of the Poisson equation will depend on (x, y, z )

but they will not depend on the grid unit. For example, consider the residual deﬁned by a conventional scheme

[53, 74]:

Resj=*

k∈{kj}+'∇uLSQ

j+∇uLSQ

k(·ˆ

njk +α

|ˆ

ejk ·ˆ

njk|Ljk

(uR−uL),Ajk =0,(40)

where ˆ

ejk =ejk/|ejk|,ejk =(xk−xj,y

k−yj,z

k−zj), Ljk =|ejk|=)(xk−xj)2+(yk−yj)2+(zk−zj)2,uL

and uRare computed by the kappa reconstruction scheme, and the damping parameter αis given an optimal

value of 4/3[53, 74]. This is a generalized version of the classical scheme called the Mathur-Murthy [75, 76]

or face-tangent scheme [29], and has been used in practical unstructured-grid codes [42, 77]. Observe that all

terms in the square bracket is of O(1/length) and therefore their balance is not aﬀected by the change in the

grid unit and the same numerical solution will be obtained for diﬀerent grid units. This makes sense since, for

example in the case of a steady heat conduction problem, a temperature distribution is uniquely determined

for a given shape of the domain and boundary conditions, and it should not depend on the scale of the domain.

However, in the hyperbolic method, the discretization is aﬀected by the coordinate scaling unless a special care

is taken. To see this, consider the ﬁrst component of the residual (31)withf= 0 and ν= 1:

*

k∈{kj}+(pL,q

L,r

L)·ˆ

njk +(pR,q

R,r

R)·ˆ

njk +1

Lr

(uR−uL),Ajk =0.(41)

The ﬁrst two terms in the square bracket are solution gradients (e.g., p=∂xusince ν= 1). Therefore, they

change their values when the grid unit is changed. For example, if uis temperature in Kelvin, [K], then

p= 300[K/m]→p= 1000/300[K/mm]. It means that the third term (i.e., dissipation) in the square bracket

9

must also change its magnitude in the same manner. Otherwise, the discrete equation will change by the amount

of dissipation, thus leading to changes in the solution and iterative convergence, which is exactly the case when

Lr=1/(2π). This is the scale-invariance problem originally pointed out in Ref.[21]. Note, however, that the

diﬀerence is only in the amount of dissipation and thus the discretization remains consistent and second-order

accurate. But second-order accurate solutions are not guaranteed to be obtained because the solver may get

unstable and diverge by the altered dissipation. To ensure the scale-invariance property, the relaxation length

must be scaled by a reference length, L:

Lr=L

2π,(42)

where Lneeds to be deﬁned for a given problem and we cannot set L=Ljk as in the conventional scheme

because then the hyperbolic discretization will lose its advantages and reduce to a conventional scheme [22].

This raises two questions: (1)How should we deﬁne Lfor a given problem?, and (2)Is Lr/L =1/(2π), which

was derived for a square domain, still optimal for other shapes of domains and what value is optimal if not?

The ﬁrst question is diﬃcult to answer for the Poisson equation, where there are no particular representative

length scales; or it is easy to answer since it can be any length, which however makes the second question more

diﬃcult because the choice of Laﬀects the optimal value of Lr/L.InRef.[21], we answered these questions, in

two dimensions, by deriving a formula for Lthat makes Lr/L =1/(2π) optimal for a rectangular domain and

then rearranging it in a form that can be applied to arbitrary domains. Below, we extend the approach to three

dimensions and develop a practical formula for L.

Remark: As mentioned in Introduction, many discretization methods for diﬀusion inherently possess the

scale-invariance property. However, those involving a characteristic length of O(1) in the discretization, e.g., a

method in Ref.[78] and a hybridizable discontinuous Galerkin method with a stabilization parameter of O(1)

[79], will need a similar technique to properly choose and/or scale a characteristic length for a given grid in

order for them to behave consistently (i.e., to give the same error and iterative convergence) for diﬀerent grid

units.

4.2 Rectangular Domain

Consider a rectangular domain: (x, y , z)∈[0,L

x]×[0,L

y]×[0,L

z]. Assume, for simplicity, a Cartesian grid

with a uniform spacing h=Lx/Nx=Ly/Ny=Lz/Nz,whereNx,Ny, and Nzare the numbers of cells in x-,

y−, and z-coordinate directions, respectively. Substitute a Fourier mode,

U0exp [i(βxx/h +βyy/h +βzz/h)],(43)

where U0=[u0,p

0,q

0,r

0] is a vector of amplitudes, βx,βy, and βzare frequencies in x,y, and zdirections,

respectively, and i=√−1, into the ﬁrst-order version of the residual (i.e., no linear reconstruction, and with

f= 0) to get

dU0

dτ=MU0,(44)

where Mis given for smooth components as

M=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

−νβ2

2hLr

iβx

h

iβy

h

iβz

h

iν2βx

hL2

r−ν

L2

r

00

iν2βy

hL2

r

0−ν

L2

r

0

iν2βz

hL2

r

00−ν

L2

r

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,(45)

whose eigenvalues are

−ν

L2

r

,−ν

L2

r

,−ν

2L2

r/1+Lrβ2

2h±1

Lr01−Lrβ2

h+β2L2

r

4h2(β2−16)1.(46)

10

Although slightly diﬀerent from those in Ref.[21] because of the nonlinear formulation employed in this paper,

it is very convenient that the expression inside the square root is exactly the same as in the two-dimensional

case. The relaxation length is determined so as to keep the last two eigenvalues complex conjugate for all

possible frequencies, so that Fourier modes will propagate just like advection instead of being purely damped

(see Ref.[21] for the derivation):

Lr≥2h

(β+ 4)β,(47)

which we impose with the smooth-smooth-smooth component (βx,β

y,β

z)=(π/Nx,π/N

y,π/N

z)withβ=

-β2

x+β2

y+β2

z=π

˜

N:

Lr=2h

(π/˜

N+ 4)π/˜

N,(48)

where

˜

N=1

-1/N 2

x+1/N 2

y+1/N 2

z

=1

h-1/L2

x+1/L2

y+1/L2

z

.(49)

It is further simpliﬁed for 1/˜

N≪1 as

Lr=1

2πh˜

N=1

2π

1

-1/L2

x+1/L2

y+1/L2

z

,(50)

from which we identify the optimal Lfor a rectangular domain, which is denoted by Lopt, as

Lopt =1

-1/L2

x+1/L2

y+1/L2

z

.(51)

If the domain of interest is indeed rectangular, one can employ this formula to compute Lopt and use it to scale

Lras Lr=Lopt/(2π). However, it cannot be directly applied to arbitrary domains. To generalize it to arbitrary

domains, we consider writing the formula in terms of computable quantities such as the volume and surface

area of the domain as discussed further in the next section.

4.3 Arbitrary Domain

For an arbitrary domain, it is not possible to derive Lopt in the same way because the Fourier analysis is not

applicable. In two dimensions, the optimal reference length Lopt, which can be obtained also from Equation (51)

by taking Lz→∞, can be fully expressed in terms of the area and the perimeter of a rectangular domain. The

resulting expression can be directly applied to arbitrary domains with the area and the perimeter computed from

a given computational grid; and it was indeed found to serve very well as a reference length to guarantee the

scale-invariance property and achieve fast iterative convergence [21]. In three dimensions, we can express Lopt

given in Equation (51) in terms of the volume V=LxLyLz, the total surface area S= 2(LxLy+LyLz+LzLx),

and the diagonal distance Diag2=L2

x+L2

y+L2

z:

Lopt =V

-S2/4−2V)Diag2+S

.(52)

This formula can be directly applied to arbitrary domains, but it requires the computation of the diagonal

distance (i.e., such as the maximum distance between two boundary points), which may not be as simple and

eﬃcient as one might hope for. As a practical option, we propose the following formula:

Diag2= max(˜

L2

x,˜

L2

y,˜

L2

z),(53)

where

˜

Lx=xmax −xmin,˜

Ly=ymax −ymin,˜

Lz=zmax −zmin,(54)

11

and xmin and xmax denote the minimum and maximum x-coordinates of the nodes in a given grid, and similarly

for others. As one might have noticed, the diagonal formula (53) does not reduce precisely to the diagonal

distance when the domain is rectangular. However, the diﬀerence is very minor and computational results

for a rectangular domain are almost the same whether the above approximation is used or not. Also, there

is another reason that we recommend the approximation (53). When the Lopt formula (52) is applied to an

arbitrary domain, there is no guarantee that the expression inside the square root, i.e., S2/4−2V)Diag2+S,

is positive. In fact, it can get negative for a sphere if we compute the diagonal distance, for example, by

Diag2=˜

L2

x+˜

L2

y+˜

L2

z, which is a reasonable formula. To see this, write the term inside the square root in

Equation (52) as

S2/4−2V)Diag2+S=S2+1/4−2

(S/V ))Diag2/S2+1/S,.(55)

This needs to be positive but can be negative if the domain has too small Sfor a ﬁxed V. The limiting case is

a sphere, which has the smallest surface area for a given volume. Consider a sphere of radius R, for which we

have V=4πR3/3 and S=4πR2. For Diag2=˜

L2

x+˜

L2

y+˜

L2

z= 12R2,weﬁnd

S2/4−2V)Diag2+S=S2+1/4−2R

3)12R2/(4πR2)2+1/(4πR2),(56)

=S2+3π−4√3+π

12π,(57)

≈−0.01294770009 ×S2,(58)

but for Equation (53), we have Diag2=4R2and thus

S2/4−2V)Diag2+S=S2+1/4−2R

3)4R2/(4πR2)2+1/(4πR2),(59)

=S2+3π−4√1+π

12π,(60)

≈0.03407020948 ×S2>0.(61)

Therefore, the formula (52) with the diagonal estimate (53) is expected to be valid for arbitrary domains.

To summarize, for a given grid, we ﬁrst compute

V=*

j∈{J}

Vj,S=*

E∈{Eb}

SE,(62)

where {J}is the set of all nodes in the grid, {Eb}is the set of all triangular faces of all the boundaries in the

grid, and SEis the area of the triangle E, then compute Lopt by the formula (52)withDiag2as in Equation

(53), and ﬁnally deﬁne the relaxation length Lras

Lr=Lopt

2π.(63)

Note that Lopt can be computed conveniently by using nodal coordinates, cell-volumes, cell-face areas avail-

able from a given computational grid. However, it is not a grid-dependent quantity but a domain-dependent

quantity. Of course, if it is known that the domain is rectangular, one can directly employ the optimal formula

given in Equation (51). The generalized formula (52) is proposed for more complex domains, but can also be

used for a rectangular domain without signiﬁcantly degrading the performance of the hyperbolic solver. It is

emphasized also that Lopt must be computed and Lrmust be deﬁned as above even when the grid coordinates

are already non-dimensionalized in order to eliminate the ambiguity in choosing the reference length used for

non-dimensionalization [21]. Finally, as in Ref.[21], we focus here on problems over bounded domains and leave

discussions on open-boundary problems to future work, which will be discussed in relation to ﬂuid-dynamic

applications. Note, however, that the proposed technique can be directly applied to open-boundary problems

of any kind. It is just not guaranteed to be optimal and may require some minor adjustments for the best per-

formance. Also, for ﬂuid-dynamic applications, one can alternatively nondimensionalize the problem of interest

for ﬂuid dynamic equations as typically done and then set Lto be the length scale used to nondimensionalize

the ﬂuid dynamic equations as it has been done for the hyperbolic Navier-Stokes method [18, 34, 35, 36].

12

5 Solvers

To eﬃciently solve the global system of residual equations, we employ implicit solvers: IDC and JFNK solvers.

The IDC solver is simple and eﬃcient, but encounters diﬃculties on irregular grids. Nevertheless, it can still

serve well as a variable-preconditioner and help the JFNK solver converge robustly and eﬃciently. The JFNK

solver is therefore the recommended solver for practical problems, but it requires a careful implementation

in order to ensure the scale-invariant convergence as discussed below. Note that these solvers are typically

employed for solving nonlinear discrete equations but used here also for linear discrete equations arising from

the discretization of a linear Poisson equation because the matrix of the linear discrete equations can be too

expensive to store.

5.1 Implicit Defect Correction (IDC) Solver

The IDC solver is an iterative method deﬁned by

Um+1 =Um+∆U,(64)

where U=(u1,p

1,q

1,r

1,u

2,p

2,q

2,r

2,···,u

N,p

N,q

N,r

N) is the global solution vector, mis the iteration

counter, and the correction ∆Uis computed by solving the linearized system:

∂Res

∂U∆U=−Res(Um),(65)

where the Jacobian ∂Res/∂Uis the exact derivative of the ﬁrst-order version of the residual Res (i.e., with

zero LSQ gradients but the Scheme-II reconstruction (28) retained). It is emphasized that the ﬁrst-order

Jacobian is an advantage of the hyperbolic solver and an improvement over conventional viscous schemes, for

which a Jacobian based on a zeroth-order scheme is common (because a ﬁrst-order version does not typically

exist)[28, 29, 30]. To solve the linear system, we employ a multi-color Gauss-Seidel relaxation scheme with a

speciﬁed tolerance of one order of magnitude residual reduction or for a speciﬁed maximum number (25 or 100) of

relaxations. As previously demonstrated for two-dimensional problems (and will be shown for three-dimensional

problems later), the IDC solver is very eﬃcient and scale invariant: convergence histories for diﬀerent grid units

match if the residual norms are scaled by the initial values. The same is true in three dimensions. Although we

have found that it is not as robust and eﬃcient as one would wish (e.g., the linear relaxation may diverge on

highly distorted grids), it still serves well as a variable-preconditioner for the JFNK solver as described in the

next section.

5.2 Jacobian-Free Newton-Krylov (JFNK) Solver

5.2.1 Standard implementation

The JFNK solver is constructed based on the algorithm presented in Ref.[13], with the generalized conjugate

residual (GCR) method: with x0=0,r0=−Res(Um), p0=˜

A−1r0, perform for i=0,1,2,··· ,i

max

αi=(Api)Tri

(Api)T(Api),xi+1 =xi+αipi,ri+1 =ri−αiApi,, (66)

where imax is a user-deﬁned integer, and, if not converged yet, compute pi+1 =˜

A−1ri+1, perform the orthogo-

nalization for k=0,1,2,···,i:

βi=(Api)T(Apk)

(Apk)T(Apk),pi+1 =pi+1 +βkpk,Api+1 =Api+1 −βkApk,(67)

and go back to the step (66). At convergence, we obtain the correction as ∆U=xkand update the solution

by Equation (64). The symbol ˜

A−1denotes the variable-preconditioner based on the multi-color Gauss-Seidel

relaxation, which approximately inverts the approximate Jacobian ˜

A=∂Res/∂Uwith the tolerance of 50%

residual reduction or for a speciﬁed maximum number 25 of relaxations. Furthermore, A=∂Res/∂Uis the

true Jacobian, but the computation and storage are avoided by computing the matrix-vector product, e.g., Apk,

by the Fr´echet derivative approximation:

Apk=Res(Um+ϵpk)−Res(Um)

ϵ,(68)

13

where

ϵ= max(1,|Um|)√10−16.(69)

5.2.2 Scale-invariant implementation

Unlike the IDC solver, the JFNK solver as described above is not scale-invariant because it treats all

variables/residuals with mixed units as a single vector of solutions/residuals. In our case, the variable udoes

not change its value for a change in the grid unit, but the variables p,q, and rdo change their values. Then,

the length (or the norm) of a vector and dot products will change in a manner that will change the behavior of

the solver. In order to make the solver scale-invariant, we need to unify the units of the solution variables and

residual components. It can be done conveniently with the reference length L, which is used to scale Lr. Deﬁne

the 4N×4Ndiagonal matrix:

Ds= diag

21,1

L,1

L,1

L,1,1

L,1

L,1

L,···3.(70)

Then, compute r0as

r0=DsRes(Um),(71)

and re-scale the Jacobian used in the preconditioner as

˜

A=∂(DsRes)

∂(D−1

sU),(72)

which means that each 4×4 Jacobian block, expressed here by ai,j, is scaled as

⎡

⎢

⎢

⎢

⎢

⎢

⎣

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

⎤

⎥

⎥

⎥

⎥

⎥

⎦

→⎡

⎢

⎢

⎢

⎢

⎢

⎣

a11 a12/L a13 /L a14/L

a21/L a22 /L2a23/L2a24 /L2

a31/L a32 /L2a33/L2a34 /L2

a41/L a42 /L2a43/L2a44 /L2

⎤

⎥

⎥

⎥

⎥

⎥

⎦

,(73)

and also that the solution xhas just been re-scaled as x→D−1

sx. Furthermore, the Fr´echet derivative needs

to be computed as

Apk=Ds

Res(Uk+ϵDspk)−Res(Um)

ϵ,(74)

where

ϵ= max(1,|D−1

sUm|)√10−16.(75)

As a result, the correction ∆Uis obtained as

∆U=Dsxk.(76)

Note that Lshould be uniquely deﬁned for a given problem, and Lopt serves the purpose very well. These

re-scalings will guarantee the dimensional consistency in xand make the algorithm scale-invariant.

6 Results

In this section, we present numerical results and demonstrate the scale-invariance property and advantages

of the hyperbolic Poisson solver for problems with various domain shapes. In the ﬁrst four problems, we solve

a linear Poisson equation with ν= 1:

div(grad u)=f(x, y, z),(77)

14

|ˆ

ejk ·ˆ

njk|AR

Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.

Grid1 512 2,058 0.19 0.83 1.0 1.4 3.9 97.3

Grid2 4,096 20,250 0.21 0.84 1.0 1.4 3.3 613.4

Grid3 32,768 178,746 0.20 0.85 1.0 1.3 3.3 41983.8

Grid4 110,592 622,938 0.22 0.85 1.0 1.3 3.0 160.2

Grid5 252,144 1,500,282 0.25 0.86 1.0 1.4 2.9 276.3

Table 1: Summary of grid statistics (independent of the grid unit) for cubic-domain grids. The skewness measure

|ˆ

ejk ·ˆ

njk|is deﬁned at edges, and the aspect ratio AR is deﬁned at cells.

where the forcing term fwill be determined for a chosen exact solution. The IDC solver is used with the

maximum of 100 relaxations and the tolerance of 0.1 for the ﬁrst two problems, and the maximum of 25

relaxations and the tolerance of 0.5 (i.e., the reduction of the linear-system residual by half) for all other

problems. These values are representative of those typically used in practical applications, so as to move onto

the next iteration without spending too much time with the linear solver. As a preconditioner in the JFNK

solver, the latter set of parameters (i.e., 25 and 0.5) are always used. The JFNK solver uses the maximum 10

Krylov vectors (i.e., imax = 10) and the tolerance of 0.1. For all problems, the initial solution is set to be 1 for

the variable u, and zero for all the gradient variables. The solvers are taken to be converged, unless otherwise

stated, when all the four residual components are reduced in the L1norm by six orders of magnitude from their

initial values. Iterative convergence histories will be shown for the maximum of the L1norms (over the four

equations) scaled by the corresponding initial norm over the four equations. Results will be compared for L=1

and L=Lopt with Lrgiven as in Equation (42). All computations were performed by a serial code; a study on

eﬀects of parallelization is left as future work.

6.1 Cubic domain

We begin with the linear Poisson equation (77) with the following forcing function:

f(x, y, z)=−π2(a2+b2+c2)sin(π(ax +by +cz)),(78)

which leads to the following exact solution:

u(x, y, z)=sin(π(ax +by +cz)),(79)

where a=2.2, b=2.3, and c=2.4, to be solved in a cubic domain with the weak Dirichlet condition speciﬁed

by the exact solution as described in Section 3.2.

We generated a series of ﬁve irregular tetrahedral grids for a cubic domain. As the baseline, the side length

1 is considered as measured in meter, [m]. To demonstrate that the same numerical solution is obtained by

the hyperbolic solver with the optimal reference length Lopt, we solve the same problem with grids expressed

in three diﬀerent units: [km], [m], and [mm]. The coarsest grids are shown in Figure 4 for diﬀerent units,

which are referred to as Grid1[km], Grid1[m], Grid1[mm], respectively, and similarly for ﬁner levels. Note that

the exact solution (79) is always computed with the baseline coordinates (i.e., in [m]), the exact gradients are

computed in the same way with the baseline coordinates but then divided by 0.001 for grids in [km] and 1000

for grids in [mm], and the forcing term (78) is computed again with the baseline coordinates but then will be

divided by 0.0012for grids in [km] and 10002for grids in [mm]. The same applies to all the test problems that

follow. Table 1 shows the grid statistics, which is independent of the grid unit. The skewness measure |ˆ

ejk ·ˆ

njk|

is computed at each edge, where ˆ

ejk is the unit vector of the edge between nodes jand k, and ˆ

njk is the unit

directed-area vector. A very small value indicates high skewness: the two vectors do not align with each other

and make an angle close to 90 degrees. The aspect ratio AR is deﬁned for each cell as the ratio of the longest

edge length to the smallest height computed for the largest triangular face area. As can be seen, these grids

are isotropic with low skewness and cell-aspect-ratio. Grid3 exhibits a very large maximum aspect ratio, but it

is an isolated eﬀect as the average is very low and close to those for other grids. The optimal reference length

15

|ˆ

ejk ·ˆ

njk|AR

Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.

Grid1 512 2,058 0.00033 0.014 1.0 518 2035 57,958

Grid2 4,096 20,250 0.00030 0.057 1.0 635 1826 441,588

Grid3 32,768 178,746 0.00027 0.038 1.0 572 1825 16,912,443

Grid4 110,592 622,938 0.00040 0.035 1.0 566 1692 81,377

Grid5 252,144 1,500,282 0.00034 0.033 1.0 568 1670 141,358

Table 2: Summary of grid statistics (independent of the grid unit) for high-aspect-ratio rectangular grids. The

skewness measure |ˆ

ejk ·ˆ

njk|is deﬁned at edges, and the aspect ratio AR is deﬁned at cells.

Lopt was computed, as described in Section 4, for each grid, but it is essentially the same for all levels of grids

in the same grid unit and diﬀers only by the magnitude for diﬀerent units: 0.5192793E-03 in [km], 0.5192793

in [m], and 0.5192793E+03 in [mm].

Convergence results for the IDC solver with L= 1 and L=Lopt are shown in Figure 5 for the ﬁnest

grid, Grid5[m], Grid5[km], Grid5[mm], where the solid lines indicate the residual histories and the dashed lines

indicate the number of linear relaxations per iteration. Note again that the residual norm is the maximum

among the four equations and normalized by the corresponding initial norm. As can be seen in Figure 5(a),the

solver behaves very diﬀerently, as expected, with L= 1: it converges reasonably fast for Grid5[m], signiﬁcantly

slows down for Grid5[km], and diverges for Grid5[mm]. This is the problem with L= 1 as reported for two-

dimensional problems in Ref.[21]. On the other hand, convergence histories perfectly match, as desired, with

L=Lopt as shown in Figure 5(b). Superior gradient accuracy by the hyperbolic solver is demonstrated in

Figure 6, where error convergence results for L=Lopt are shown for u,r=∂zu, and ∂zuLSQ. Here, the errors

in the gradient are multiplied by 0.001 for grids in [km] and 1000 for grids in [mm], so that their magnitude

match across the three units. Clearly, all the error norms match in magnitude, and the solution uand the

gradient rare second-order accurate while the LSQ gradient is only ﬁrst-order accurate. The same order of

accuracy in the solution and the gradient is one of the advantageous features of the hyperbolic method, and

it is demonstrated here to produce much more accurate gradients than the LSQ gradients computed from the

soluion. Although not shown, other gradients pand qare also obtained with second-order accuracy. Figure 7

provides a qualitative comparison of the z-derivative over the boundary surface for Grid2[m]. As expected, the

contours contain noise for the LSQ gradient (Figure 7(c)) whereas those for the gradient variable rare much

smoother (Figure 7(b)) and very close to those of the exact derivative (Figure 7(a)).

6.2 High-aspect-ratio rectangular domain

To investigate the impact of high skewness and large cell aspect ratio, we created a ﬂat domain from the

previous cubic domain by re-scaling the z-axis by a factor of 1/1000. The coarsest grids are shown in Figure 8,

again for the three diﬀerent grid units. The same forcing and exact solutions as in the previous case were used,

except that we set c= 200.4 in order to generate enough solution variation in the z-direction. As can be seen in

the grid statistics shown in Table 2, these grids are highly skewed with |ˆ

ejk ·ˆ

njk|≈0.035, which corresponds to

an angle of 89.8◦, and ﬂat with the aspect ratio of O(1000) on average. Lopt was, again, computed for each grid,

but it is essentially the same for all levels of grids in the same grid unit: 9.997313E-07 in [km], 9.997313E-04 in

[m], and 9.997313E-01 in [mm].

Convergence histories are shown for Grid5[km], Grid5[m], and Grid5[mm] in Figure 9. The solver is not

scale-invariant with L= 1 and signiﬁcantly slows down for Grid5[km] and Grid5[m]. On the other hand, as

shown in Figure 9(b), the solver converges for all grids in the scale-invariant manner for L=Lopt. However,

for this problem, the solver encounters diﬃculties in the linear relaxation, hitting the maximum of 100 at every

iteration, which can take up signiﬁcant computational time for ﬁne 3D grids. Then, the JFNK solver was

applied to see if it improves convergence and eﬃciency. Results are shown in Figures 9(c) and 9(d). Figure 9(c)

shows that the JFNK solver converged rapidly in seven iterations and it is scale-invariant as desired. Note that

the linear relaxation in the plots as indicated by “Relax” in the legend is the total number of preconditioner

relaxations performed per iteration. The number of Krylov vectors used per iteration is indicated by “‘KV” in

16

the legend. Figure 9(d) compares the IDC and JFNK solvers in terms of CPU time and clearly indicates that

the JFNK solver converges faster than the IDC solver. These results demonstrate that the JFNK solver can be

more eﬃcient and reliable than the IDC solver for highly stretched grids.

Error convergence results are shown in Figure 10, again, for u,r, and ∂zuLSQ. As before, the hyperbolic

solver achieves second-order accuracy for both the solution and gradients, and the LSQ gradient is ﬁrst-order

accurate. In Figure 11, contours of ∂zuare compared on Grid3[m]. It is observed again that the LSQ gradient

is very noisy on this grid while the gradient variable r=∂zuis very smooth and close to the exact contours.

It is important to note that the noise in the LSQ gradient does not vanish on ﬁner grids. This can be seen in

Figure 12, which shows the contours on Grid5[m]. Noise can be seen although smaller than that on Grid3[m].

6.3 Hemisphere cylinder

To explore other shapes of domains, we consider a domain around a hemisphere-cylinder enclosed by a large

hemisphere and a plane at the base. For eﬃciency, we consider only the half of the entire domain (y>0).

See Figure 13(a), which shows the domain without the outer half-hemisphere boundary. For this problem, we

consider solving the linear Poisson equation with the following forcing function:

f(x, y, z)=−A(a2+b2+c2)π2sin(aπx) cos(bπy)sin(cπz),(80)

and the exact solution:

u(x, y, z)=Asin(aπx) cos(bπy)sin(cπz),(81)

where A= 100, a=0.05, b=0.025, and c=0.5. The Dirichlet condition is applied at the hemisphere-cylinder

and the outer half-hemisphere; the Neumann condition is imposed elsewhere (i.e., two plane boundaries), where

the solution derivative normal to the boundary is zero. The problem is solved, again, in three diﬀerent units:

[km], [m], and [mm], and with four levels of grids. The coarsest grids are shown for the three diﬀerent units in

Figures 13(b), 13(c), and 13(d). As shown in Table 3, these grids are relatively skewed and high-aspect-ratio

near the hemisphere-cylinder surface with the minimum skewness measure being around 0.01 and the maximum

aspect ratio of O(100). Lopt was computed as 8.725970E-03 in [km], 8.725970 in [m], and 8.725970E+03 in

[mm].

|ˆ

ejk ·ˆ

njk|AR

Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.

Grid1 680 2,880 0.0097 0.54 1.0 2.2 27.2 206.2

Grid2 4,587 23,040 0.0083 0.53 1.0 2.2 26.5 240.0

Grid3 33,605 184,320 0.0076 0.53 1.0 2.2 26.4 261.7

Grid4 257,097 1,474,560 0.0073 0.53 1.0 2.2 26.4 274.0

Table 3: Summary of grid statistics (independent of the grid unit) for the hemisphere-cylinder grids. The

skewness measure |ˆ

ejk ·ˆ

njk|is deﬁned at edges, and the aspect ratio AR is deﬁned at cells.

For this problem, we consider both the IDC and JFNK solvers. Convergence histories are shown in Figure

14. As expected, the IDC solver is not scale-invariant for L= 1 as shown in Figure 14(a). On the other hand, as

shown in Figure 14(b), it is scale-invariant with L=Lopt. However, the linear relaxation hits the maximum of

25 every time as in the previous case, meaning that 25 relaxations is not enough to reduce the linear residual by

half. It is, nevertheles, suﬃcient as a preconditioner in the JFNK solver. As shown in Figure 14(c). the JFNK

solver converges rapidly in 9 iterations and it is scale-invariant. The 10th iteration in Grids[m] is considered

due to a round-oﬀeﬀect, where the residual norm was actually reduced by nearly six orders of magnitude in the

9th iteration (as can be seen in the ﬁgure) but not exactly. Figure 14(d) shows that the JFNK solver is faster

in CPU time than the IDC solver. Although the linear relaxation in the preconditioner hits the maximum of

25 and also it requires 10 Krylov vectors in the last six iterations, the JFNK solver converges very fast in terms

of CPU time.

Figure 15 shows error convergence results. As expected, second-order accuracy is observed for the solution

uand the gradient variable r=∂zu, but lower-order accuracy (if not ﬁrst-order) is observed for the LSQ

17

gradient ∂zuLSQ. Not shown, but results are similar in other gradient components. As can be seen in the

comparison given in Figure 16, large LSQ gradient errors occur near the outer hemisphere boundary. These

plots demonstrate, again, the superior gradient accuracy by the hyperbolic solver.

6.4 Curved tube

To further investigate the impact of Lopt for a more general domain, we consider a curved-tube domain as

shown in Figure 17. We solve, again, the linear Poisson equation with the Dirichlet condition. The same forcing

term and exact solution are used, as in the cubic domain case except a=1.5, b=1.5, and c=1.5. Four levels of

grids have been generated with the coarsest grid as shown in Figure 17 for three diﬀerent units: [km], [m], and

[mm]. Grid statistics is shown in Table 4, in terms of which these grids are similar to those in the previous case

with the minimum skewness measure of O(0.01) and the maximum aspect ratio of O(100). Lopt was computed

as 1.471851E-03 in [km], 1.471851 in [m], and 1.471851E+03 in [mm].

|ˆ

ejk ·ˆ

njk|AR

Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.

Grid1 6,480 30,240 0.015 0.40 1.0 1.7 24.0 187.5

Grid2 46,656 245,376 0.013 0.40 1.0 1.5 24.0 163.2

Grid3 151,632 832,032 0.013 0.40 1.0 1.5 24.0 155.0

Grid4 352,512 1,976,832 0.013 0.40 1.0 1.5 24.0 151.7

Table 4: Summary of grid statistics (independent of the grid unit) for the curved-tube grids. The skewness

measure |ˆ

ejk ·ˆ

njk|is deﬁned at edges, and the aspect ratio AR is deﬁned at cells.

As before, the IDC solver convergence wildly varies for diﬀerent grid units with L= 1 as shown in Figure

18(a). On the other hand, scale-invariant convergence is achieved with the optimal length L=Lopt as can be

clearly seen in Figure 18(b). Convergence is further improved by the JFNK solver as shown in Figure 18(c):

the JFNK solver is scale-invariant and converges with much less iterations than the IDC solver. It is faster also

in CPU time than the IDC solver as can be seen in Figure 18(d).

Figure 19 shows the error convergence results for u,r=∂zu, and the LSQ gradient ∂zuLSQ. Again, the

hyperbolic solver gives second-order accuracy not only for ubut also for the gradient r. The LSQ gradient

is signiﬁcantly less accurate although it seems to exhibit nearly second-order convergence for these grids. In

Figure 20, derivative contours are compared over the inner boundary in Grid2[m], where the grid is stretched

and relatively high-aspect-ratio in the surface normal direction. On this grid, the gradient ris already accurate

and close to the exact contours, but the LSQ gradient contours are distorted and far from the exact contours.

6.5 Nonlinear problem

To demonstrate the capability for nonlinear problems, we consider the following nonlinear Poisson equation:

div(ν(u) grad u)=f(x, y, z),(82)

where

ν(u)=1+u2,f(x, y, z)=2A3(cπ)2cos(cπx) cos(cπy)4{cos(cπx)}2+{cos(cπy)}25exp '3√2cπz(,(83)

with c=0.5 and A=0.1. The exact solution is given by

u(x, y, z)=Acos(cπx) cos(cπy)exp'√2cπz(.(84)

The nonlinear Poisson equation is solved in an incomplete-donut-shaped domain as in Figure 21.Atthetwo

annulus end-planes at x= 0 and y= 0, the Neumann condition is imposed with a zero normal derivative,

and the Dirichlet condition is imposed elsewhere. Three levels of tetrahedral grids were generated by a two-

dimensional irregular triangular grid and then subdividing prisms into tetrahedra; the coarsest grids are shown

18

|ˆ

ejk ·ˆ

njk|AR

Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.

Grid1 23,328 120,960 0.027 0.49 1.0 3.7 19.9 69.5

Grid2 176,256 981,504 0.011 0.40 1.0 3.8 39.6 165.3

Grid3 1,368,576 7,907,328 0.106 0.40 1.0 3.5 38.9 169.0

Table 5: Summary of grid statistics (independent of the grid unit) for the nonlinear problem. The skewness

measure |ˆ

ejk ·ˆ

njk|is deﬁned at edges, and the aspect ratio AR is deﬁned at cells.

for diﬀerent units in Figure 21. The irregularity of the two-dimensional section grid can be observed in Figure

21(d). These grids are relatively skewed and high-aspect-ratio as in the previous two cases. See Table 5 for grid

statistics. Lopt was computed as 2.685715E-03 in [km], 2.685715 in [m], and 2.685715E+03 in [mm].

For this problem, we compare the hyperbolic solver with a conventional solver, which is constructed based

on the alpha-damping diﬀusion scheme (40) modiﬁed for the nonlinear equation as

Resj=*

k∈{kj}+6ν(uj)∇uLSQ

j+ν(uk)∇uLSQ

k7·ˆ

njk +αν(ujk )

|ˆ

ejk ·ˆ

njk|Ljk

(uR−uL),Ajk −f(xj,y

j,z

j)Vj=0,(85)

where ujk =(uL+uR)/2withuLand uRcomputed by the kappa reconstruction (κ= 0). This is a scalar

residual equation at a node jfor the Poisson equation (82). The same IDC and JFNK solvers are used but they

are applied to the reduced number of residual equations (i.e., Nequations instead of 4N). As the conventional

discretization is scale-invariant with both the IDC and JFNK solvers, we solve the problem only on grids with

the unit [m]. It should be noted also that the Jacobian required in the IDC solver/preconditioner is constructed

by diﬀerentiating the damping term only (i.e., α

|ˆ

ejk·ˆ

njk|Ljk (uk−uj)), which by itself is an inconsistent scheme

on skewed grids, and such inconsistent Jacobians are known to degrade iterative convergence of the IDC solver

[28, 29, 30]. In this problem, the tolerance is reduced further and the solver is taken to be converged when the

residual is reduced by eight (not six) orders of magnitude. This is because we observed signiﬁcant diﬀerences in

the discretization errors for the conventional scheme with six orders of residual reduction; the tolerance of eight

orders of magnitude reduction has been found suﬃcient to obtain converged numerical solutions. Below, in the

ﬁgures, the conventional method will be referred to as Alpha4/3.

As in the linear cases, the IDC solver is highly scale-dependent with L= 1 as can be seen in Figure 22(a),

where convergence histories are shown for the ﬁnest grid. It is then made completely scale-invariant with

L=Lopt as shown in Figure 22(b). Also plotted here is the convergence history of the conventional solver.

The linear relaxation hits the maximum 25 at almost every iteration, failing to reduce the linear residual; and

thus the solver took more than 150 iterations to converge. As discussed earlier, there are two weak points in

the conventional solver: (1)the Jacobian is constructed as the derivative of an inconsistent scheme (i.e., the

damping term only), which is especially far from the true Jacobian for skewed grids, and (2)a higher condition

number of the linear system by the discretized second-derivative operator. The hyperbolic discretization oﬀers

advantages in both aspects. First, the Jacobian is constructed by the derivative of the ﬁrst-order accurate

(not inconsistent) scheme. It is therefore closer to the true Jacobian, and thus faster nonlinear convergence is

expected. Second, the condition number is O(1/h), where his a representative mesh spacing, for the hyperbolic

discretization, which is O(h) times smaller than O(1/h2) typical in conventional discretizations. The improved

condition number leads to faster convergence in the linear relaxation, and that is exactly what is observed in

the result shown in Figure 22(b).

More robust and eﬃcient convergence can be achieved for both hyperbolic and conventional methods by the

JFNK solver with the IDC solver used as a preconditioner as shown in Figures 22(c) and 22(d). Figure 22(c)

conﬁrms that the JFNK solver is again scale-invariant and converged with only 8 iterations. It converged rapidly

also for the conventional discretization, but took more Krylov vectors as well as preconditioner relaxations as

can be seen in Figure 22(d), which compares the IDC and JFNK solvers for the hyperbolic discretization and the

JFNK solver for the conventional discretization. Observe the larger numbers of Krylov vectors and relaxations

for the conventional discretization. Notice also that the JFNK solver is signiﬁcantly faster in CPU time than

the IDC solver.

It may be possible to further reduce the CPU time for the conventional solver, e.g., by smart programming,

19

but such is not even worth considering because the numerical solution obtained with the conventional method

is very inaccurate. Figure 23 shows the error convergence results, again for u,r=∂zu, and the LSQ gradients

∂zuLSQ , which are computed for both solutions (i.e., produced with the hyperbolic and conventional methods).

Second-order accuracy is observed for all, but the errors in the LSQ gradients are much larger than the error

in the gradient variable r. Also, the error in uis larger for the conventional method than for the hyperbolic

method. In Figure 24, contours of the derivative normal to the inner surface ∂nuare compared on all grids:

the exact ∂nu,∂nu=(p, q, r)·n′

B, and ∂nu=∂zuLSQ ·n′

Bfor the conventional method. As can be seen, the

contours of ∂nu=(p, q, r)·n′

Bis very smooth, accurate, and very close to the exact contours on all grids. On the

other hand, the contours of the normal derivative computed with the LSQ gradient in the conventional method

are very noisy and far from the exact contours. It seems to have improved on the ﬁnest grid, but it still contains

noise and is far from the exact contours. Such inaccuracy of gradients on unstructured tetrahedral grids is

very well known. Typically, unstructured tetrahedral grids are avoided for problems where gradient accuracy

is important, and prismatic/hexahedral grids are used instead (if possible). However, as mentioned earlier,

tetrahedral grids are more suitable for automatic grid generation and anisotropic adaptation. The numerical

results shown here indicate that the hyperbolic solver has a potential for making grid generation much easier

for complex geometries and allowing anisotropic grid adaptation methods to be applied to a wider range of

practical problems with purely tetrahedral grids.

6.6 Nonlinear problem over a more complex domain

To demonstrate the solver for a more complex geometry, we consider a spiral-shaped tube as shown in Figure

25 and solve the same nonlinear Poisson equation with c=0.06 and A=0.2 and the same boundary conditions.

Four levels of tetrahedral grids were generated with 23,040, 165,888, 1,253,376, and 9,732,096 nodes; Figure

25 shows the coarsest one. As can be seen from the grid statistics in Table 6, the cell aspect ratio is slightly

smaller than that of the grids used in the previous case. For this problem, we only compare the hyperbolic and

conventional methods with the JFNK solver on grids with a single grid unit. Lopt was computed as 2.563783 for

the ﬁnest grid, and it is very similar on other grids. Convergence results are shown for the ﬁnest grid in Figure

26. The solver converges very rapidly in 8 iterations for both discretization methods. However, it requires more

linear relaxations and Krylov vectors with the conventional method. As a result, the solver took less CPU time

with the hyperbolic discretization as can be seen in Figure 26(b). Figure 27 shows error convergence results for

uand its y-derivative. The error in uis slightly lower with the conventional method, but the gradient ∂yuis

more accurate with the hyperbolic method. Again, the hyperbolic method generates qualitatively more accurate

gradients as illustrated here for the surface normal gradient plotted over the inner boundary. See Figure 28.

|ˆ

ejk ·ˆ

njk|AR

Nodes Tetrahedra Min. Ave. Max. Min. Ave. Max.

Grid1 23,040 109,728 0.024 0.48 1.0 2.7 20.4 86.4

Grid2 165,888 881,280 0.029 0.49 1.0 2.5 19.0 82.4

Grid3 1,253,376 7,064,064 0.031 0.49 1.0 2.4 18.5 80.8

Grid4 9,732,096 56,567,808 0.019 0.49 1.0 2.1 18.3 84.2

Table 6: Summary of grid statistics (independent of the grid unit) for the second nonlinear test case. The

skewness measure |ˆ

ejk ·ˆ

njk|is deﬁned at edges, and the aspect ratio AR is deﬁned at cells.

7 Concluding Remarks

We have presented a hyperbolic Poisson solver for arbitrary tetrahedral grids, which is fundamentally more

eﬃcient and accurate than conventional methods, delivering second-order accuracy for both the solution and

the gradients on irregular tetrahedral grids. A practical formula has been proposed for estimating a reference

length in three dimensions, which is needed to scale the relaxation length associated with a hyperbolic formu-

lation of the Poisson equation for ensuring the scale-invariance property (i.e., the same solution is obtained

independently of the grid unit). The properly scaled hyperbolic Poisson system is discretized by the edge-based

20

method, and a robust Jacobian-Free Newton-Krylov solver has been developed to solve the resulting system of

residual equations. The developed hyperbolic Poisson solver has been shown to be more robust and accurate

than conventional solvers: (1) faster and more robust iterative convergence by the eﬀect of eliminating second

derivatives from the algorithm (i.e., reduced condition number) and by the use of a more accurate ﬁrst-order

Jacobian, and (2) superior gradient accuracy, which is not only one order higher than least-squares gradients

but also free from noise even on irregular tetrahedral grids. These advantages have been demonstrated for both

linear and nonlinear Poisson equations over various shapes of domains with unstructured tetrahedral grids. By

eﬀectively resolving the issue of scale-invariance for three-dimensional problems, the work presented has paved

the way for the hyperbolic method to be applied for solving numerous practical applications governed by linear

and nonlinear Poisson equations.

Future work includes computations with adaptive tetrahedral grids, applications to unsteady diﬀusion prob-

lems (e.g., heat transfer, elasticity, etc.) with the developed solver applied within each time step to solve

unsteady residual equations of an implicit time-stepping scheme [21], extensions to other discretization meth-

ods, e.g., cell-centered ﬁnite-volume methods, an extension to third-order accuracy based on the third-order

edge-based method [64], and the development of grid-independent multigrid solvers for the preconditioner as

well as for the nonlinear solver, in which the JFNK solver can serve as a robust smoother [10, 80]. Finally,

although the proposed reference-length formula has been shown to be useful for solving linear/nonlinear Poisson

equations over complex domains, it is theoretically optimal only for a rectangular domain and a truly optimal

formula for an arbitrary domain, in terms of both iterative convergence and accuracy, remains to be discovered.

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Acknowledgments

The author gratefully acknowledges support from Software CRADLE, part of Hexagon, and support from

the U.S. Army Research Oﬃce under the contract/grant number W911NF19-1-0429 with Dr. Matthew Munson

as the program manager.

24

j

k

c

m

cR

cL

(a) Dual face contributions from an adjacent tetrahedral

element to the edge [j, k].

j

k

(b) Total dual face at the edge [j, k].

Figure 2: Dual face contribution at the edge [j, k]. A numerical ﬂux is evaluated at the midpoint of the edge

m, indicated by an open circle. The centroid of the tetrahedral element is denoted by c, and the centroids of

the two adjacent triangles are denoted by cLand cR.

j, b

2

3

nB

||

||| |||

Figure 3: A dual-face (shaded area) of a control volume around a boundary node j. The boundary

dual-face normal vector nBis pointing outward from the interior domain.

25

(a) Grid1[km] (b) Grid1[m] (c) Grid1[mm]

Figure 4: The coarsest grids used for the cubic domain test case with diﬀerent grid units.

(a) L=1. (b) L=Lopt .

Figure 5: Comparison of convergence histories for Grid5 in the cubic domain test case.

26

-5 -4.5 -4

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Figure 6: Error convergence results for u,r=∂zu, and LSQ =∂zuLSQ in the cubic domain test case.

(a) Exact ∂zu(b) r=∂zu(c) ∂zuLSQ

Figure 7: Comparison of ∂zucontours for Grid2[m] in the cubic domain test case.

27

(a) Grid1[km] (b) Grid1[m] (c) Grid1[mm]

Figure 8: The coarsest grids used for the high-aspect-ratio rectangular domain test case with diﬀerent grid

units.

(a) IDC: L=1. (b) IDC: L=Lopt .

(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.

Figure 9: Comparison of convergence histories for Grid5 in the high-aspect-ratio rectangular domain test case.

28

-6 -5.5 -5

-4

-3

-2

-1

0

1

2

3

Figure 10: Error convergence results for u,r=∂zu, and LSQ =∂zuLS Q in the high-aspect-ratio rectan-

gular domain test case.

(a) Exact ∂zu(b) r=∂zu(c) ∂zuLSQ

Figure 11: Comparison of ∂zucontours for Grid3[m] in the high-aspect-ratio rectangular domain test case.

(a) Exact ∂zu(b) r=∂zu(c) ∂zuLSQ

Figure 12: Comparison of ∂zucontours for Grid5[m] in the high-aspect-ratio rectangular domain test case.

29

(a) Grid1. (b) Grid1[km]

(c) Grid1[m] (d) Grid1[mm]

Figure 13: The coarsest grids used for the hemisphere-cylinder domain test case with diﬀerent grid units.

30

(a) IDC: L=1. (b) IDC: L=Lopt .

(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.

Figure 14: Comparison of convergence histories for Grid4 in the hemisphere-cylinder domain test case.

31

-4 -3.5 -3

-1

-0.5

0

0.5

1

1.5

Figure 15: Error convergence results for u,r=∂zu, and LSQ =∂zuLS Q in the hemisphere-cylinder

domain test case.

(a) Exact ∂xu(b) p=∂xu(c) ∂xuLSQ

Figure 16: Comparison of ∂xucontours for Grid4[m] in the hemisphere-cylinder domain test case.

32

(a) Grid1[km] (b) Grid1[m] (c) Grid1[mm]

Figure 17: The coarsest grids used for the curved-tube domain test case with diﬀerent grid units.

(a) IDC: L=1. (b) IDC: L=Lopt .

(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.

Figure 18: Comparison of convergence histories for Grid4 in the curved-tube domain test case.

33

-4 -3.5 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Figure 19: Error convergence results for u,r=∂zu, and LSQ =∂zuLSQ in the curved-tube domain test

case.

(a) Exact ∂zu(b) r=∂zu(c) ∂zuLSQ

Figure 20: Comparison of ∂zucontours for Grid2[m] in the curved-tube domain test case. The outer boundary

is not shown.

34

(a) Grid1[km] (b) Grid1[m]

(c) Grid1[mm](d) Zoomed-in view.

Figure 21: The coarsest grids used for the nonlinear test case with diﬀerent grid units.

35

(a) IDC: L=1. (b) IDC: L=Lopt .

(c) JFNK: L=Lopt. (d) IDC and JFNK: L=Lopt.

Figure 22: Comparison of convergence histories for Grid3[m] in the nonlinear test case.

36

-4 -3.8 -3.6 -3.4

-5

-4

-3

-2

-1

Figure 23: Error convergence results for u,r=∂zu, and LSQ =∂zuLSQ in the nonlinear test case.

37

(a) Exact (Grid1). (b) Hyperbolic (Grid1). (c) Alpha4/3 (Grid1).

(d) Exact (Grid2). (e) Hyperbolic (Grid2). (f ) Alpha4/3 (Grid2).

(g) Exact (Grid3). (h) Hyperbolic (Grid3). (i) Alpha4/3 (Grid3).

Figure 24: Comparison of contours of the solution gradient normal to the inner boundary surface ∂nufor

Grid3[m] in the nonlinear test case. The outer boundary is not shown. The normal derivative is computed with

∂nu=(p, q, r )·n′

Bfor the hyperbolic method and ∂nu=∇uLS Q ·n′

Bfor the conventional method (Alpha4/3).

38

Figure 25: The coarsest grid used for the second nonlinear test case.

39

(a) JFNK solver: maximum residual norm versus itera-

tion.

(b) JFNK solver: maximum residual norm versus CPU

time.

Figure 26: Comparison of convergence histories for the hyperbolic and conventional methods on the ﬁnest grid

in the second nonlinear test case.

-1 -0.5 0

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

Figure 27: Error convergence results for u,q=∂yuand

∂yuLSQ in the second nonlinear test case.

(a) Exact (Grid4). (b) Hyperbolic (Grid4). (c) Alpha4/3 (Grid4).

Figure 28: Comparison of contours of the solution gradient normal to the inner boundary surface ∂nuon

the ﬁnest grid in the second nonlinear test case. The outer boundary is not shown. The normal derivative

is computed with ∂nu=(p, q , r)·n′

Bfor the hyperbolic method and ∂nu=∇uLS Q ·n′

Bfor the conventional

method (Alpha4/3).

40