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Journal of Computational and Applied Mathematics 275 (2015) 262–271
Contents lists available at ScienceDirect
Journal of Computational and Applied
Mathematics
journal homepage: www.elsevier.com/locate/cam
Accuracy analysis of an adaptive mesh refinement method
using benchmarks of 2-D steady incompressible lid-driven
cavity flows and coarser meshes
Zhenquan Li ∗, Robert Wood
School of Computing and Mathematics, Charles Sturt University, Thurgoona, NSW 2640, Australia
article info
Article history:
Received 12 July 2014
Received in revised form 21 July 2014
Keywords:
Adaptive mesh refinement
Finite volume method
Lid-driven cavity flow
abstract
Lid-driven cavity flows have been widely investigated and accurate results have been
achieved as benchmarks for testing the accuracy of computational methods. This paper
verifies the accuracy of an adaptive mesh refinement method numerically using 2-D steady
incompressible lid-driven cavity flows and coarser meshes. The accuracy is shown by
verifying that the centres of vortices given in the benchmarks are located in the refined
grids of refined meshes for Reynolds numbers 100, 1000 and 2500 using coarser meshes.
The adaptive mesh refinement method performs mesh refinement based on the numerical
solutions of Navier–Stokes equations solved by a finite volume method with a well known
SIMPLE algorithm for pressure–velocity coupling. The accuracy of the refined meshes is
shown by comparing the profiles of horizontal and vertical components of velocity fields
with the corresponding components of the benchmarks together and drawing closed
streamlines. The adaptive mesh refinement method verified in this paper can be applied to
find the accurate numerical solutions of any mathematical models containing continuity
equations for incompressible fluid, steady state fluid flows or mass and heat transfer.
©2014 Elsevier B.V. All rights reserved.
1. Introduction
Meshing is the process of breaking up a physical domain into finite smaller sub-domains (called elements, cells or grids) in
order to evaluate the numerical solutions of differential equations. Adaptive mesh refinement is a computational technique
to improve the accuracy of numerical solutions of differential equations by starting the calculations on a coarse initial mesh
and then refining this mesh where less accuracy may occur locally.
There is an extensive number of publications on adaptive mesh refinements and their applications. Some refinement
methods use a refinement criterion which is based on local truncation errors (e.g. [1]). Other common methods include the
so-called h-refinement (e.g. [2]), p-refinement (e.g. [3]) or r-refinement (e.g. [4]), with different combinations of these also
possible (e.g. [5]). The overall aim of these adaptive algorithms is to allow a balance to be obtained between accuracy and
computational efficiency in solving differential equations.
We introduced adaptive mesh refinement methods from a different point of view for 2-D velocity fields [6] and for
3-D fields [7] based on a theorem in qualitative theory of differential equations (Theorem 1.14, page 18, Ye et al. [8]). The
theorem indicates that a divergence free vector field has no limit cycles or one sided limit cycles, that is, the trajectories
(or streamlines) of divergence free vector fields are closed curves in bounded domains (singular points are streamlines) that
∗Corresponding author. Tel.: +61 260519604; fax: +61 60519604.
E-mail addresses: jali@csu.edu.au,zhenquanl@gmail.com (Z. Li), rwood@csu.edu.au (R. Wood).
http://dx.doi.org/10.1016/j.cam.2014.07.025
0377-0427/©2014 Elsevier B.V. All rights reserved.
Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271 263
Table 1
Jacobian matrices and corresponding expressions of f(C̸= 0).
Case Jacobian f
1r10
0r2(0̸= r1̸= r2̸= 0)C
y1+b1
r1y2+b2
r2
2r10
0 0(r1̸= 0)C
y1+b1
r1
3r10
0r1(r1̸= 0)C
y1+b1
r12
4µ λ
−λ µ(µ ̸= 0, λ ̸= 0)C
y1+µb1−λb2
µ2+λ22
+y2+λb1+µb2
µ2+λ22
have also been shown by benchmarks (e.g. [9]). The adaptive mesh refinement methods adaptively refine meshes based
on the information of evaluated numerical velocity fields to obtain refined meshes on which the linear interpolation of the
numerical velocity fields approximates divergence free vector fields. Using numerical velocity fields obtained by taking the
vectors of the analytical velocity fields at nodes of meshes, examples show the accuracy of the methods include: locating
the singular points and asymptotic lines for 2-D [6]; the singular points and asymptotic plane for 3-D [7]; and drawing
closed streamlines [10,11] using numerical velocity fields evaluated on the refined meshes with a pre-specified number of
refinements of the initial meshes. We showed that the adaptive mesh refinement method for 2-D velocity fields provides
accurate estimates for the singular points of 2-D steady incompressible lid-driven cavity flows using the numerical velocity
fields [12]. The numerical velocity fields are obtained by solving the Navier–Stokes equations with the boundary conditions
numerically using a second order colocated finite volume method (GSFV) with a splitting method for time discretization [13].
We applied the adaptive mesh refinement method to the initial meshes and the numerical velocity fields, and take the
centres of refined grids in the vortex regions as the estimates of the singular points. The comparison of the estimates with
the benchmarks shows that the estimates for the singular points are accurate. Identification of accurate locations of singular
points and asymptotic lines (planes), and drawing closed streamlines are some of the accuracy measures for computational
methods. Therefore the adaptive mesh refinement methods are accurate when we apply them to refine meshes and then
evaluate numerical vector fields on refined meshes.
Li [14] considers 2-D lid-driven cavity flows using finer meshes 99 ×99 for Re =1000, 121 ×121 for Re =2500 and
139 ×139 for Re =5000. The results show that the different sizes of vortices (primary, secondary, tertiary and quaternary
vortices) require different densities of mesh nodes in the separated-flow regions for similar relative errors of centre locations.
The same conclusion is shown in [15]. We conclude that mesh refinement is necessary if we want accurate centre locations
of vortices with similar relative errors. An investigation starting from coarser initial meshes and demonstrating that the
centres of vortices are contained in the refined grids of refined meshes is valuable when we consider the computational
time is an important issue.
This paper establishes the accuracy of the same adaptive mesh refinement method for 2-D proposed by Li [6] using 2-D
lid-driven cavity flows and a different finite volume method from GSFV. A comparison of the accuracy between GSFV [13]
and a finite volume method with SIMPLE algorithm [16] has been done [17]. The comparison shows that our implementation
for the finite volume method with SIMPLE algorithm provides more accurate outputs. We use coarser initial meshes with
sizes 45×45 for Re =100, 65×65 for Re =1000 and 85×85 for Re =2500. We report the results from the refined meshes
created by applying the adaptive mesh refinement method once using the numerical solutions of Navier–Stokes equations
obtained from the finite volume method with SIMPLE algorithm. We show that the centres of vortices (except one case) are
contained in refined grids of refined meshes so further refinements will provide more accurate locations of the centres.
2. Algorithm of adaptive mesh refinement and finite volume method
This section summarizes the adaptive mesh refinement method proposed by Li [6] based on Theorem 1.14 of [8] and the
finite volume method [16].
Assume that Vl=AX +Bis a vector field obtained by linearly interpolating the vectors at the three vertices of a triangle,
where
A=a11 a12
a21 a22
is a matrix of constants,
B=b′
1
b′
2
is a vector of constants, and X=(x1,x2)T. The vector Vlis unique if the area of the triangle is not zero [18]. Mass conservation
for a steady flow or an incompressible fluid requires that
∇ · Vl=trace(A)=0.(1)
264 Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271
Let fbe a scalar function depending only on spatial variables. We assume that fVlis divergence free and then calculate
the expressions of f. Li [6] derives the expressions of ffor the four different Jacobian forms of the coefficient matrix Aas
shown in Table 1. Variables y1and y2in Table 1 are the components of (y1,y2)T=V−1Xwhere Vsatisfies AV =VJ and
Jis one of the Jacobian matrices in Table 1. Vectors Vland fVlproduce same streamlines if f̸= 0,∞(refer to Section 2.2
of [10]). Therefore, if fVlis divergence free, Vlproduces divergence free streamlines. The introduction of the adaptive mesh
refinement is to achieve refined meshes on which Vlproduce streamlines that are not divergence free in a set of grids with
controllable small Lebesgue measure. Scalar functions freduce the number of refined grids in refined meshes. The functions
fare calculated by solving differential equations [19]. The conditions (MC) (MC is the abbreviation of mass conservation)
are the functions fin Table 1 not equalling zero or infinity at any point on the triangular domains when fVlis divergence
free on these triangular domains.
We describe the algorithm of adaptive mesh refinement for quadrilateral mesh in this paper. The algorithm is also
applicable to triangular mesh but it is an easier case than the quadrilateral mesh. The following grid refinement algorithm
describes how to use the conditions (MC) to refine a quadrilateral grid in a given mesh. To avoid an infinite refinement of
the mesh, we choose a pre-specified threshold number of refinements Tbased on the accuracy requirements. The algorithm
of grid refinement is:
Step 1 Subdivide a quadrilateral grid into two triangles and check if Vlsatisfies Eq. (1) on both triangles. If yes, no refinement
for the grid is required. If no, go to Step 2;
Step 2 Apply the conditions (MC) to both of the triangles. If the conditions (MC) are satisfied on both triangles, there is no
need to subdivide the grid. Otherwise, we subdivide the grid into a number of small grids such that the lengths of all
sides of the small grids are truly reduced (e.g. connecting the mid-points of opposite sides of a quadrilateral by line
segments produces four small quadrilaterals and the lengths of the sides of the four small quadrilaterals are truly
reduced).
We have checked results for two different subdivisions of a quadrilateral into two triangles and did not detect any
differences. The algorithm above can be applied to any of these subdivisions.
The algorithm of adaptive mesh refinement is:
Step 1 Evaluate the numerical velocity field for a given initial mesh;
Step 2 Refine the grids of the mesh one by one using the above algorithm of grid refinement;
Step 3 Take the refined mesh as initial mesh and go to Step 1 until a satisfactory numerical velocity field is obtained or the
threshold number Tis reached.
Section 3uses the finite volume method with SIMPLE algorithm for pressure–velocity coupling to evaluate numerical
velocity fields [16].
We have verified that the accuracy of the above adaptive mesh refinement method in identifying asymptotic line for T
from 1 to 5, and identifying singular points and drawing closed streamlines for Tfrom 1 to 4 by two analytical velocity field
examples [6]. We have also verified that the larger the value of T, the more accurate are the location of the asymptotic line,
the coordinates of the singular points and the closed streamlines [6].
In this paper, we subdivide a quadrilateral grid by connecting the mid-points of two opposite sides of a quadrilateral. The
abbreviations BR, BL and TL refer to bottom right, bottom left and top left corners of the cavity, respectively. The number
following these abbreviations refers to the vortices that appear in the flow, which are numbered according to size (for
example, BR1 refers to bottom right secondary vortex).
3. Accuracy analysis by comparisons with benchmarks
We take the results for Re =100 [20,21] and the results for Re =1000 and 2500 [9] as the benchmarks for comparisons.
We consider the accuracy of the adaptive mesh refinement method in the following two aspects:
•the variations of the refined meshes according to the comparison of the profiles of horizontal and vertical components
of the numerical velocity fields with the corresponding benchmarks;
•inclusion of the centres of vortices identified in the benchmarks in the refined mesh.
3.1. Variation of refined meshes
We consider the refined meshes for 2-D lid-driven cavity flows for different mesh sizes and Reynolds number Re =100,
1000 and 2500, respectively. Because the benchmarks of the profiles for horizontal component (u) of the velocity fields at
x=0.5 and vertical component (v) of the velocity fields at y=0.5 are available, we show the accuracy of the numerical
velocity fields by comparing the corresponding profiles obtained from the numerical velocity fields with the corresponding
benchmarks. We also show the streamlines of Vlgenerated by Matlab built-in function streamlines, and refined meshes. A
grid is called a refined grid if a cross is drawn inside.
One of the possible comparisons is the adaptive mesh refinement which refines everywhere that the solution gradient is
large [22, 293–299]. The refinement criteria enforce
∥∇u(k)∥ ≤ ϵ∥uh∥1
Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271 265
Fig. 1. Refined meshes and vorticity fields in pairs.
everywhere in the mesh, where ∥·∥is the L2norm, ∥·∥1is the H1norm, ϵis the discretization tolerance, uhis
finite-dimensional approximation for u, and kin ∥∇u(k)∥is the number of subdomains. Fig. 1 (Fig. 5.7 of [22]) shows the
266 Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271
Fig. 2. uprofile for Re =100.
Fig. 3. Streamlines for Re =100.
refined meshes (left) and vorticity fields (right) in pairs at different values of the refinement parameter ϵ: (a) ϵ=10−3,
(b) ϵ=10−4, (c) ϵ=10−5, and (d) ϵ=10−6for lid-driven cavity flow at Re =1000. Even though there might be some
relations between the refined meshes and the vorticity field as ϵdecreases, no one provides any information on the pattern
of the flow field such as locations of the centres of vortices.
3.1.1. Re =100
We show the figures for Re =100 generated from an initial mesh with 45 ×45 uniform grids.
From Fig. 2, the profile of u(the solid line) of the numerical velocity field at x=0.5 shows a slight difference with the
corresponding benchmark (the dash dot line) [20]. The profile of vof the numerical velocity field at y=0.5 shows the
similar difference (not shown). However, the profiles of uand vreflect the local accuracy of the numerical velocity field.
The streamlines shown in Fig. 3 provide the global accuracy of the numerical velocity field depending on the accuracy of
streamline generation method.
Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271 267
Fig. 4. Refined mesh for Re =100 with mesh size 45 ×45.
Fig. 5. uprofile for Re =1000.
The streamlines shown in the graph of Fig. 3 are not closed (spiral lines) so we conclude that the velocity field Vlis not
divergence free or the corresponding fin Table 1 does not satisfy the condition (MC) on some grids in the regions.
Fig. 4 shows the refined mesh and the centres of vortices (dots) given by the benchmarks. The refined grid in the
primary regions contains the centre of primary vortex, and the refined grid on the bottom left side contains the centre
of the bottom left secondary vortex (BL1), and the refined grid on the bottom right side contains the centre of the bottom
right secondary vortex (BR1). Further mesh refinement is needed for more accurate locations of centres of the secondary
vortices.
3.1.2. Re =1000
We show the figures for Re =1000 generated from an initial mesh with 65 ×65 uniform grids.
From the graph of Fig. 5, the profile of u(the solid line) of the numerical velocity field at x=0.5 shows a slight difference
with the corresponding benchmark (the dash dot line). The profile of vof the numerical velocity field at y=0.5 fits the
268 Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271
Fig. 6. Streamlines for Re =1000.
Fig. 7. Refined mesh for Re =1000 with mesh size 65 ×65.
benchmark well (not shown). Fig. 5 indicates that the evaluated velocity field has small errors along x=0.5. The streamlines
shown in Fig. 6 provide the global accuracy of the numerical velocity field if we ignore the errors of streamline generation
method.
The streamlines shown in Fig. 6 are not closed so we conclude that the velocity field Vlis not divergence free or the
corresponding fin Table 1 does not satisfy the condition (MC) on some grids in the regions.
Fig. 7 shows the refined mesh and the centres of vortices (dots) given by the benchmarks. The refined grid in the primary
regions contains the centre of primary vortex, and the isolated refined grid on the bottom left side contains the centre of the
bottom left secondary vortex (BL1), and the isolated refined grid on the bottom right side contains the centre of the bottom
right secondary vortex (BR1). Even though the centres of tertiary vortices (BL2 and BR2) are included in the refined grids, we
cannot identify the refined grids that containing these centres in the refined mesh if no centres (dots) from the benchmarks
are shown in the figure. Further mesh refinement is needed for more information on this matter.
Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271 269
Fig. 8. uprofile for Re =2500.
Fig. 9. Streamlines for Re =2500.
3.1.3. Re =2500
We show the figures for Re =2500 generated from an initial mesh with 85 ×85 uniform grids.
From the graph of Fig. 8, the difference between the profile of u(the solid line) of the numerical velocity field at x=0.5
and the corresponding benchmark (the dash dot line) is small. The profile of vof the numerical velocity field at y=0.5 fits
the corresponding benchmark well (not shown). The streamlines in the primary vortex region shown in Fig. 9 are not closed.
We conclude that the velocity field Vlis not divergence free or the corresponding fin Table 1 does not satisfy the condition
(MC) on some grids in the region.
Fig. 10 shows the refined mesh and the centres of vortices (dots) given by the benchmarks. The refined grid in the primary
regions contains the centre of primary vortex, and the isolated refined grid on the bottom left side contains the centre of the
bottom left secondary vortex (BL1), and the isolated refined grid on the bottom right side contains the centre of the bottom
270 Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271
Fig. 10. Refined mesh for Re =2500 with mesh size 85 ×85.
Table 2
Locations of the centre of vortices.
Vortex type Reynolds numbers
Re =100 (45 ×45) Re =1000 (65 ×65) Re =2500 (85 ×85)
Primary vortex (0.6160, 0.7380) (0.5325, 0.5668) (0.5207, 0.5452)
(0.6189, 0.7400) (0.5300, 0.5650) (0.5200, 0.5433)
BR1 (0.8692, 0.0112) (0.8634, 0.1128) (0.8299, 0.0914)
(0.9424, 0.0610) (0.8627, 0.1137) (0.8350, 0.0917)
BL1 (0.0774, 0.1250) (0.0842, 0.0774) (0.0849, 0.1106)
(0.0332, 0.0352) (0.0833, 0.0783) (0.0850, 0.1100)
BR2 – – –
– (0.9917, 0.0067) (0.9900, 0.0100)
BL2 – (0.0081, 0.0081) (0.0096, 0.0092)
– (0.0050, 0.0050) (0.0067, 0.0067)
TL1 – – –
– – (0.0433, 0.8900)
right secondary vortex (BR1). Even though the centres of tertiary vortices are included in the refined grids, we cannot identify
the refined grids that containing these centres in the refined mesh. The centre of the top left secondary vortex (TL1) in Fig. 10
is not contained in a refined grid. Further mesh refinement is needed for more information on these issues.
3.1.4. Vortex centre locations
This subsection shows the comparison of the centres of vortices between the benchmarks and the corresponding esti-
mates obtained in this paper.
Table 2 presents coordinates of centres of vortices calculated from the linearly interpolated velocity fields Vland the
corresponding benchmarks (the coordinates on the second line in each row) for Re =100 [21], 1000 and 2500 [9].
The differences between estimated coordinates and the coordinates from the benchmarks in Table 2 are accurate up to
three decimal places for the centres of primary and secondary vortices, and bottom left tertiary vortex (BL2) of Re =1000
and 2500 but the centres of bottom right tertiary vortices (BR2) are not found for these cases, and the centre of top left
secondary vortex (TL1) for Re =2500 is also not found. This indicates that different regions require different densities of
nodes for the similar information and further refinement is needed if more accurate results are required. Table 3 provides
the information on relative errors of the estimated centres of vortices. The relative errors of BL1 for Re =100, and the
relative errors of BL2 for Re =1000 and 2500 are bigger than 40% so these numbers also indicate that more refinements are
necessary.
Z. Li, R. Wood / Journal of Computational and Applied Mathematics 275 (2015) 262–271 271
Table 3
Relative errors of the estimates of vortex centres.
Vortex type Reynolds numbers
Re =100 (45 ×45) Re =1000 (65 ×65) Re =2500 (85 ×85)
Primary vortex 0.0037 0.0040 0.0027
BR1 0.0937 0.0013 0.0061
BL1 2.0685 0.0111 0.0044
BL2 – 0.6200 0.4041
4. Discussions
We applied the adaptive mesh refinement method once to the initial meshes based on the information of numerical solu-
tions of 2-D lid-driven cavity flows using finite volume method with SIMPLE algorithm. The mesh sizes of the initial meshes
are coarser than those used in literature but the refined meshes include the refined grids containing centres of all vortices
identified in benchmarks except for TL1 for Re =2500. Further refinements may locate the centre of TL1 for Re =2500.
If more accurate centre locations of the vortices are required, we repeat the algorithm of adaptive mesh refinement until a
satisfactory result is obtained.
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