doi: 10.3934/jcd.2021020
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A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

1. 

T-5 Applied Mathematics and Plasma Physics Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

2. 

Dipartimento di Ingegneria Civile, Edile e Ambientale - ICEA, Università di Padova, 35131 Padova, Italy

* Corresponding author: G. Manzini

Received  March 2021 Revised  July 2021 Early access December 2021

Fund Project: Dr. G. Manzini was supported by the LDRD-ER program of Los Alamos National Laboratory under project number 20180428ER. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001)

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $ {\mathbb{P}}_{{1}}-{\mathbb{P}}_{{0}} $ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.

Citation: Gianmarco Manzini, Annamaria Mazzia. A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem. Journal of Computational Dynamics, doi: 10.3934/jcd.2021020
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References:
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Figure 1.  Degrees of freedom of each component of the virtual element vector-valued fields of $ {\bf{V}}^{h}_{k}( {\rm{E}}) $ (left) and the scalar polynomial fields of $ Q^{ h}_{\underline{k}}( {\rm{E}}) $ (right) of an hexagonal element for the accuracy degrees $ k = 1,2,3 $ and $ \underline{k} = k-1 $. Vertex values and edge polynomial moments are marked by a circular bullet; cell polynomial moments are marked by a square bullet
Figure 2.  Base meshes (top row) and first refinement meshes (bottom row) of the three mesh families used in the general convergence tests: $ \mathcal{M}{1} $: randomly quadrilateral meshes; $ \mathcal{M}{2} $: general polygonal meshes; $ \mathcal{M}{3} $: concave element meshes
Figure 3.  Base meshes (top row) and first refinement meshes (bottom row) of the three mesh families used to investigate convergence and stability of the lowest-order scheme: $ \mathcal{M}{4} $: diagonal triangle meshes; $ \mathcal{M}{5} $: criss-cross triangle meshes; $ \mathcal{M}{6} $: square meshes
Figure 4.  Convergence curves for $ k = 1,\ldots,6 $ and $ \underline{k} = k-1 $ versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). Blue lines with circles represent the error curves using the enhanced virtual element space (14), and, accordingly, the right-hand is approximated by using the projection operator $ \Pi^{0, {\rm{E}}}_{{k}} $. The mesh families used in each calculations are shown in the left corner of each panel. The expected convergence slopes and rates are shown by the triangles and corresponding numerical labels
Figure 5.  Convergence curves for $ k = 1,\ldots,6 $ and $ \underline{k} = k-1 $ versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). Blue lines with circles represent the error curves using the virtual element space (13), and, accordingly, the right-hand is approximated by using the projection operator $ \Pi^{0, {\rm{E}}}_{{ \bar{k}}} $ with $ \bar{k} = max(0,k-2) $. A loss of accuracy for $ k = 2 $ in the $ L^2 $-norm error curves is visible. The mesh families used in each calculations are shown in the left corner of each panel. The expected convergence slopes and rates are shown by the triangles and corresponding numeric labels
Figure 6.  Values of the inf-sup constant $ \beta $ versus the mesh size parameter $ h $. The lines with circles represent the values of $ \beta $ with $ k = 1 $ and $ \underline{k} = 0 $. The mesh families used in each calculations are shown in the bottom-left corner of each panel
Figure 7.  Values of the inf-sup constant $ \beta $ versus the mesh size parameter $ h $. Blue lines with circles represent the values of $ \beta $ with $ k = 2 $ and $ \underline{k} = 0 $. Red lines with circles represent the values of $ \beta $ when $ k = 2 $ and $ \underline{k} = 1 $. Black, green and magenta lines with triangles are associate to $ k = 3 $ and $ \underline{k} = 0 $, $ \underline{k} = 1 $ and $ \underline{k} = 2 $, respectively. The mesh families used in each calculations are shown in the bottom-left corner of each panel
Figure 8.  Convergence curves for $ k = 2,\ldots,6 $, and $ \underline{k} = k-1 $ versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). The results with $ k = 1 $ are not reported because there is no convergence. The lines with circles represent the error curves using the enhanced virtual element space (14). The right-hand side is approximated by using the projection operator $ \Pi^{0, {\rm{E}}}_{{k}} $, i.e., the "enhanced" definition of the virtual element space given in (14). The mesh families used in each calculations are shown in the left corner of each panel. The expected convergence slopes and rates are shown by the triangles and corresponding numerical labels
Figure 9.  Convergence curves for $ k = 2 $, $ \underline{k} = 0 $, versus the mesh size parameter $ h $ for the velocity approximation measured using the energy norm (55) (top panels) and the $ L^2 $-norm (56) (mid panels), and for the pressure approximation measured using the $ L^2 $-norm (57) (bottom panels). Blue lines with circles represent the error curves for the formulation using the enhanced virtual element space (14) with the right-hand side approximated by using the projection operator $ \Pi^{0}_{{2}} $. Red lines with circles represent the error curve with the right-hand side approximated by using the projection operator $ \Pi^{0}_{{0}} $. The mesh families used in each calculations are shown in the left corner of each panel. The convergence slopes and rates are shown by the triangles and corresponding numeric labels
Table 1.  Diameter $ h $ of each grid of the six mesh families $ \mathcal{M}{1} $-$ \mathcal{M}{6} $
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $ $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
1 $ 3.72 \cdot 10^{-1} $ $ 4.26 \cdot 10^{-1} $ $ 3.81 \cdot 10^{-1} $ $ 7.07 \cdot 10^{-1} $ $ 5.00 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $
2 $ 1.99 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.91 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.77 \cdot 10^{-1} $
3 $ 1.01 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 9.54 \cdot 10^{-2} $ $ 1.77 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 8.84 \cdot 10^{-2} $
4 $ 5.17 \cdot 10^{-2} $ $ 6.21 \cdot 10^{-2} $ $ 4.77 \cdot 10^{-2} $ $ 8.84 \cdot 10^{-2} $ $ 6.25 \cdot 10^{-1} $ $ 4.42 \cdot 10^{-2} $
5 $ 2.61 \cdot 10^{-2} $ $ 3.41 \cdot 10^{-2} $ $ 2.38 \cdot 10^{-2} $ $ 4.42 \cdot 10^{-2} $ $ 3.12 \cdot 10^{-2} $ $ 2.21 \cdot 10^{-2} $
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $ $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
1 $ 3.72 \cdot 10^{-1} $ $ 4.26 \cdot 10^{-1} $ $ 3.81 \cdot 10^{-1} $ $ 7.07 \cdot 10^{-1} $ $ 5.00 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $
2 $ 1.99 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.91 \cdot 10^{-1} $ $ 3.53 \cdot 10^{-1} $ $ 2.50 \cdot 10^{-1} $ $ 1.77 \cdot 10^{-1} $
3 $ 1.01 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 9.54 \cdot 10^{-2} $ $ 1.77 \cdot 10^{-1} $ $ 1.25 \cdot 10^{-1} $ $ 8.84 \cdot 10^{-2} $
4 $ 5.17 \cdot 10^{-2} $ $ 6.21 \cdot 10^{-2} $ $ 4.77 \cdot 10^{-2} $ $ 8.84 \cdot 10^{-2} $ $ 6.25 \cdot 10^{-1} $ $ 4.42 \cdot 10^{-2} $
5 $ 2.61 \cdot 10^{-2} $ $ 3.41 \cdot 10^{-2} $ $ 2.38 \cdot 10^{-2} $ $ 4.42 \cdot 10^{-2} $ $ 3.12 \cdot 10^{-2} $ $ 2.21 \cdot 10^{-2} $
Table 2.  Number of elements $ N_{el} $ and vertices $ N $ of each grid of the three mesh families $ \mathcal{M}{1} $$ \mathcal{M}{3} $
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 16 25 22 46 16 73
2 64 81 84 171 64 305
3 256 289 312 628 256 1249
4 1024 1089 1202 2406 1024 5057
5 4096 4225 4772 9547 4096 20353
Level $ \mathcal{M}{1} $ $ \mathcal{M}{2} $ $ \mathcal{M}{3} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 16 25 22 46 16 73
2 64 81 84 171 64 305
3 256 289 312 628 256 1249
4 1024 1089 1202 2406 1024 5057
5 4096 4225 4772 9547 4096 20353
Table 3.  Number of elements $ N_{el} $ and vertices $ N $ of each grid of the three mesh families $ \mathcal{M}{4} $-$ \mathcal{M}{6} $
Level $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 8 9 16 13 16 25
2 32 25 64 41 64 81
3 128 81 256 145 256 289
4 512 289 1024 545 1024 1089
5 2048 1089 4096 2113 4096 4225
Level $ \mathcal{M}{4} $ $ \mathcal{M}{5} $ $ \mathcal{M}{6} $
$ N_{el} $ $ N $ $ N_{el} $ $ N $ $ N_{el} $ $ N $
1 8 9 16 13 16 25
2 32 25 64 41 64 81
3 128 81 256 145 256 289
4 512 289 1024 545 1024 1089
5 2048 1089 4096 2113 4096 4225
Table 4.  Size, rank and kernel's dimension of matrix B when $ k = 1 $ for the mesh family $ \mathcal{M}{4} $
$ \mathcal{M}{4} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 2\times 8 $ 2 6
2 $ 18\times 32 $ 18 14
3 $ 98\times 128 $ 98 30
4 $ 450\times 512 $ 450 62
5 $ 1922\times 2048 $ 1922 126
6 $ 7938\times 8192 $ 7938 254
$ \mathcal{M}{4} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 2\times 8 $ 2 6
2 $ 18\times 32 $ 18 14
3 $ 98\times 128 $ 98 30
4 $ 450\times 512 $ 450 62
5 $ 1922\times 2048 $ 1922 126
6 $ 7938\times 8192 $ 7938 254
Table 5.  Size, rank and kernel's dimension of matrix B when $ k = 1 $ for the mesh family $ \mathcal{M}{5} $
$ \mathcal{M}{5} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 10\times 16 $ 10 6
2 $ 50\times 64 $ 46 18
3 $ 226\times 256 $ 190 66
4 $ 962\times 1024 $ 766 258
5 $ 3970\times 4096 $ 3070 1026
6 $ 16130\times 16384 $ 12286 4098
$ \mathcal{M}{5} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 10\times 16 $ 10 6
2 $ 50\times 64 $ 46 18
3 $ 226\times 256 $ 190 66
4 $ 962\times 1024 $ 766 258
5 $ 3970\times 4096 $ 3070 1026
6 $ 16130\times 16384 $ 12286 4098
Table 6.  Size, rank and kernel's dimension of matrix B when $ k = 1 $ for the mesh family $ \mathcal{M}{6} $
$ \mathcal{M}{6} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 18\times 16 $ 14 2
2 $ 98\times 64 $ 62 2
3 $ 450\times 256 $ 254 2
4 $ 1922\times 1024 $ 1022 2
5 $ 7938\times 4096 $ 4094 2
6 $ 32258\times 16384 $ 16382 2
$ \mathcal{M}{6} $
Level size($ B $) $ rank(B) $ $ kernel(B) $
1 $ 18\times 16 $ 14 2
2 $ 98\times 64 $ 62 2
3 $ 450\times 256 $ 254 2
4 $ 1922\times 1024 $ 1022 2
5 $ 7938\times 4096 $ 4094 2
6 $ 32258\times 16384 $ 16382 2
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