# American Institute of Mathematical Sciences

## Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique

 1 Abdelmalek Essaadi University, Faculty of Sciences and Techniques of Tangier, Laboratory of Mathematics and Applications, Department of Mathematics, BP. 416, Tangier 90000, Morocco 2 Moulay Ismail University of Meknes Faculty of Sciences and Techniques of Errachidia Department of Mathematics MSISI Laboratory, AM2CSI Group BP. 509, Boutalamine Errachidia 57000, Morocco 3 Moulay Ismail University of Meknes, Faculty of Sciences of Meknes, Department of Mathematics and Informatics, BP. 11201 Zitoune Meknes, Morocco

* Corresponding author: Samir Khallouq

Received  March 2019 Revised  June 2019 Published  September 2020

We present in this paper a new algorithm combining a finite volume method with an improved Schur complement technique to solve $2D$ anisotropic diffusion problems on general meshes. After having proved the convergence of the finite volume method, we have given a description of the proposed algorithm in the case of two nonoverlapping subdomains. Several numerical tests are achieved which illustrate the theoretical results of convergence of the finite volume method and show the advantages of the proposed algorithm.

Citation: Hassan Belhadj, Samir Khallouq, Mohamed Rhoudaf. Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020260
##### References:
 [1] L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Acad. Sci. Paris. Ser. I, 346 (2008), 1007-1012.  doi: 10.1016/j.crma.2008.07.015.  Google Scholar [2] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general $2D$ meshes, Num. Meth. PDE., 23 (2007), 145-195.  doi: 10.1002/num.20170.  Google Scholar [3] H. Belhadj, M. Fihri, S. Khallouq and N. Nagid, Optimal number of schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 21-34.  doi: 10.3934/dcdss.2018002.  Google Scholar [4] P. E. Bjørstad and O. B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), 1097-1120.  doi: 10.1137/0723075.  Google Scholar [5] F. Boyer, F. Hubert and S. Krell, Non-overlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes, IMA J. Numer. Anal., 30 (2010), 1062-1100.  doi: 10.1093/imanum/drp001.  Google Scholar [6] S. C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math., 83 (1999), 187-203.  doi: 10.1007/s002110050446.  Google Scholar [7] T. F. Chan, E. Weinan and J. Sun, Domain decomposition interface preconditioners for fourth-order elliptic problems, Appl. Numer. Math., 8 (1991), 317-331.  doi: 10.1016/0168-9274(91)90072-8.  Google Scholar [8] Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM:M2AN, 33 (1999), 493-516.  doi: 10.1051/m2an:1999149.  Google Scholar [9] K. Domelevo and P. Omnès, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM:M2AN, 39 (2005), 1203-1249.  doi: 10.1051/m2an:2005047.  Google Scholar [10] R. Eymard, T. Gallouët and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math., 82 (1999), 91-116.  doi: 10.1007/s002110050412.  Google Scholar [11] R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.  doi: 10.1093/imanum/drn084.  Google Scholar [12] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, 7 (2000), 713–1020.  Google Scholar [13] R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in ISTE. Finite Volumes for Complex Applications V (eds. R. Eymard and J.-M. Hérard), Wiley, 5 (2008), 659–692.  Google Scholar [14] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), 481-499.  doi: 10.1006/jcph.2000.6466.  Google Scholar [15] L. Mansfield, On the conjugate gradient solution of the Schur complement system obtained from domain decomposition, SIAM J. Numer. Anal., 27 (1990), 1612-1620.  doi: 10.1137/0727094.  Google Scholar [16] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999.   Google Scholar

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##### References:
 [1] L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Acad. Sci. Paris. Ser. I, 346 (2008), 1007-1012.  doi: 10.1016/j.crma.2008.07.015.  Google Scholar [2] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general $2D$ meshes, Num. Meth. PDE., 23 (2007), 145-195.  doi: 10.1002/num.20170.  Google Scholar [3] H. Belhadj, M. Fihri, S. Khallouq and N. Nagid, Optimal number of schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 21-34.  doi: 10.3934/dcdss.2018002.  Google Scholar [4] P. E. Bjørstad and O. B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), 1097-1120.  doi: 10.1137/0723075.  Google Scholar [5] F. Boyer, F. Hubert and S. Krell, Non-overlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes, IMA J. Numer. Anal., 30 (2010), 1062-1100.  doi: 10.1093/imanum/drp001.  Google Scholar [6] S. C. Brenner, The condition number of the Schur complement in domain decomposition, Numer. Math., 83 (1999), 187-203.  doi: 10.1007/s002110050446.  Google Scholar [7] T. F. Chan, E. Weinan and J. Sun, Domain decomposition interface preconditioners for fourth-order elliptic problems, Appl. Numer. Math., 8 (1991), 317-331.  doi: 10.1016/0168-9274(91)90072-8.  Google Scholar [8] Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM:M2AN, 33 (1999), 493-516.  doi: 10.1051/m2an:1999149.  Google Scholar [9] K. Domelevo and P. Omnès, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM:M2AN, 39 (2005), 1203-1249.  doi: 10.1051/m2an:2005047.  Google Scholar [10] R. Eymard, T. Gallouët and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math., 82 (1999), 91-116.  doi: 10.1007/s002110050412.  Google Scholar [11] R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.  doi: 10.1093/imanum/drn084.  Google Scholar [12] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, 7 (2000), 713–1020.  Google Scholar [13] R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in ISTE. Finite Volumes for Complex Applications V (eds. R. Eymard and J.-M. Hérard), Wiley, 5 (2008), 659–692.  Google Scholar [14] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), 481-499.  doi: 10.1006/jcph.2000.6466.  Google Scholar [15] L. Mansfield, On the conjugate gradient solution of the Schur complement system obtained from domain decomposition, SIAM J. Numer. Anal., 27 (1990), 1612-1620.  doi: 10.1137/0727094.  Google Scholar [16] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999.   Google Scholar
DDFV mesh
Diamond cell $\mathcal{D}_{\sigma,\sigma^{*}}$
A DDFV mesh $\mathcal{T}$ of the whole domain $\Omega$
The compatible meshes $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, of the whole domain $\Omega$, corresponding to the DDFV mesh $\mathcal{T}$ of figure 3
Two independent DDFV meshes $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ for both subdomains $\Omega_{i}$
The compatible meshes corresponding to two independent DDFV meshes $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ of figure 5
Primal mesh $Mesh_{1}$ (left) and $Mesh_{2}$ (right)
Analytic solution (left) and DDFV solution (right) for the test case $1$ and $h = 0.0118$
Analytic solution (left) and DDFV solution (right) for the test case $2$ and $h = 0.0207$
Primal mesh $Mesh_{2,1}$ (left) and $Mesh_{1,3}$ (right)
CPU time to solve the test case $2$ for different numbers of diamonds $\mathcal{D}$
$L^{\infty}(\Omega)$ and $L^{2}(\Omega)$ errors by DDFV, for test case $1$ and for different values of $h$
 Mesh Nbr of $\mathcal{D}$ $h$ $e_{h,L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h,L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1}$ $488$ $0.1184$ $0.0037$ - $5.3348$E-$04$ - $Mesh_{2}$ $1912$ $0.0645$ $8.5460$E-$04$ $2.4127$ $1.3259$E-$04$ $2.2920$ $Mesh_{3}$ $7568$ $0.0364$ $2.5019$E-$04$ $2.1472$ $3.3329$E-$05$ $2.4136$ $Mesh_{4}$ $30112$ $0.0207$ $8.2489$E-$05$ $1.9658$ $8.3355$E-$06$ $2.4554$ Average $2.1752$ $2.3870$
 Mesh Nbr of $\mathcal{D}$ $h$ $e_{h,L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h,L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1}$ $488$ $0.1184$ $0.0037$ - $5.3348$E-$04$ - $Mesh_{2}$ $1912$ $0.0645$ $8.5460$E-$04$ $2.4127$ $1.3259$E-$04$ $2.2920$ $Mesh_{3}$ $7568$ $0.0364$ $2.5019$E-$04$ $2.1472$ $3.3329$E-$05$ $2.4136$ $Mesh_{4}$ $30112$ $0.0207$ $8.2489$E-$05$ $1.9658$ $8.3355$E-$06$ $2.4554$ Average $2.1752$ $2.3870$
$L^{\infty}(\Omega)$ and $L^{2}(\Omega)$ errors by DDFV, for test case $2$ and for different values of $h$
 Mesh Nbr of $\mathcal{D}$ $h$ $e_{h,L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h,L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1}$ $488$ $0.1184$ $0.2402$ - $0.0452$ - $Mesh_{2}$ $1912$ $0.0645$ $0.0423$ $2.8592$ $0.0106$ $2.3876$ $Mesh_{3}$ $7568$ $0.0364$ $0.0132$ $2.0356$ $0.0028$ $2.3269$ $Mesh_{4}$ $30112$ $0.0207$ $0.0048$ $1.7922$ $7.3969$E-$04$ $2.3584$ Average $2.2290$ $2.3576$
 Mesh Nbr of $\mathcal{D}$ $h$ $e_{h,L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h,L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1}$ $488$ $0.1184$ $0.2402$ - $0.0452$ - $Mesh_{2}$ $1912$ $0.0645$ $0.0423$ $2.8592$ $0.0106$ $2.3876$ $Mesh_{3}$ $7568$ $0.0364$ $0.0132$ $2.0356$ $0.0028$ $2.3269$ $Mesh_{4}$ $30112$ $0.0207$ $0.0048$ $1.7922$ $7.3969$E-$04$ $2.3584$ Average $2.2290$ $2.3576$
$L^{\infty}(\Omega)$ and $L^{2}(\Omega)$ errors by DDFV-SC, for the test case $1$ and for different mesh $Mesh_{i,i}$ with $i = 1,\ldots,4$
 Mesh Nbr of $\mathcal{D}$ in $\Omega_{1}\cup \Omega_{2}$ $h$ $e_{h_{1,2},L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h_{1,2},L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1,1}$ $546$ $0.1191$ $0.0020$ $-$ $5.4853$E-$04$ $-$ $Mesh_{2,2}$ $2124$ $0.0626$ $5.6512$E-$04$ $1.9650$ $1.3729$E-$04$ $2.1535$ $Mesh_{3,3}$ $8376$ $0.0354$ $2.0197$E-$04$ $1.8050$ $3.4357$E-$05$ $2.4301$ $Mesh_{4,4}$ $33264$ $0.0200$ $7.0126$E-$05$ $1.8527$ $8.5752$E-$06$ $2.4308$ Average $1.8742$ $2.3381$
 Mesh Nbr of $\mathcal{D}$ in $\Omega_{1}\cup \Omega_{2}$ $h$ $e_{h_{1,2},L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h_{1,2},L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1,1}$ $546$ $0.1191$ $0.0020$ $-$ $5.4853$E-$04$ $-$ $Mesh_{2,2}$ $2124$ $0.0626$ $5.6512$E-$04$ $1.9650$ $1.3729$E-$04$ $2.1535$ $Mesh_{3,3}$ $8376$ $0.0354$ $2.0197$E-$04$ $1.8050$ $3.4357$E-$05$ $2.4301$ $Mesh_{4,4}$ $33264$ $0.0200$ $7.0126$E-$05$ $1.8527$ $8.5752$E-$06$ $2.4308$ Average $1.8742$ $2.3381$
$L^{\infty}(\Omega)$ and $L^{2}(\Omega)$ errors by DDFV-SC, for the test case $2$ and for different mesh $Mesh_{i,i}$ with $i = 1,\ldots,4$
 Mesh Nbr of $\mathcal{D}$ in $\Omega_{1}\cup \Omega_{2}$ $h$ $e_{h_{1,2},L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h_{1,2},L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1,1}$ $546$ $0.1191$ $0.3694$ $-$ $0.0980$ $-$ $Mesh_{2,2}$ $2124$ $0.0626$ $0.0733$ $2.5145$ $0.0152$ $2.8975$ $Mesh_{3,3}$ $8376$ $0.0354$ $0.0191$ $2.3592$ $0.0036$ $2.5267$ $Mesh_{4,4}$ $33264$ $0.0200$ $0.0053$ $2.2452$ $8.6492$E-$04$ $2.4976$ Average $2.3730$ $2.6406$
 Mesh Nbr of $\mathcal{D}$ in $\Omega_{1}\cup \Omega_{2}$ $h$ $e_{h_{1,2},L^{\infty}}$ $EOC_{L^{\infty}}$ $e_{h_{1,2},L^{2}}$ $EOC_{L^{2}}$ $Mesh_{1,1}$ $546$ $0.1191$ $0.3694$ $-$ $0.0980$ $-$ $Mesh_{2,2}$ $2124$ $0.0626$ $0.0733$ $2.5145$ $0.0152$ $2.8975$ $Mesh_{3,3}$ $8376$ $0.0354$ $0.0191$ $2.3592$ $0.0036$ $2.5267$ $Mesh_{4,4}$ $33264$ $0.0200$ $0.0053$ $2.2452$ $8.6492$E-$04$ $2.4976$ Average $2.3730$ $2.6406$
$L^{\infty}(\Omega)$ and $L^{2}(\Omega)$ errors by DDFV-SC, for the test case $3$ and for different mesh $Mesh_{i,j}$ with $i,j = 1,\ldots,4$
 Mesh Nbr of $\mathcal{D}$ in $\Omega_{1}$ Nbr of $\mathcal{D}$ in $\Omega_{2}$ $h_{1}$ $h_{2}$ $e_{h_{1,2},L^{\infty}}$ $e_{h_{1,2},L^{2}}$ $Mesh_{1,1}$ $273$ $273$ $0.1191$ $0.1187$ $0.0238$ $0.0068$ $Mesh_{2,1}$ $1062$ $283$ $0.0626$ $0.1187$ $0.0195$ $0.0053$ $Mesh_{2,2}$ $1062$ $1062$ $0.0626$ $0.0621$ $0.0129$ $0.0035$ $Mesh_{3,2}$ $4188$ $1082$ $0.0354$ $0.0621$ $0.0103$ $0.0028$ $Mesh_{3,3}$ $4188$ $4188$ $0.0354$ $0.0347$ $0.0066$ $0.0018$ $Mesh_{4,3}$ $16632$ $4228$ $0.0200$ $0.0347$ $0.0053$ $0.0014$ $Mesh_{4,4}$ $16632$ $16632$ $0.0200$ $0.0191$ $0.0033$ $9.1142$E-$04$
 Mesh Nbr of $\mathcal{D}$ in $\Omega_{1}$ Nbr of $\mathcal{D}$ in $\Omega_{2}$ $h_{1}$ $h_{2}$ $e_{h_{1,2},L^{\infty}}$ $e_{h_{1,2},L^{2}}$ $Mesh_{1,1}$ $273$ $273$ $0.1191$ $0.1187$ $0.0238$ $0.0068$ $Mesh_{2,1}$ $1062$ $283$ $0.0626$ $0.1187$ $0.0195$ $0.0053$ $Mesh_{2,2}$ $1062$ $1062$ $0.0626$ $0.0621$ $0.0129$ $0.0035$ $Mesh_{3,2}$ $4188$ $1082$ $0.0354$ $0.0621$ $0.0103$ $0.0028$ $Mesh_{3,3}$ $4188$ $4188$ $0.0354$ $0.0347$ $0.0066$ $0.0018$ $Mesh_{4,3}$ $16632$ $4228$ $0.0200$ $0.0347$ $0.0053$ $0.0014$ $Mesh_{4,4}$ $16632$ $16632$ $0.0200$ $0.0191$ $0.0033$ $9.1142$E-$04$
CPU time to solve the test case $1$ for different numbers of diamonds $\mathcal{D}$
 Nbr of $\mathcal{D}$ in $\Omega$ CPU time for DDFV (in seconds) CPU time for DDFV-SC (in seconds) $8376$ $18.149494$ $24.458821$ $20860$ $94.936105$ $59.901873$ $33264$ $418.468599$ $229.306277$ $82920$ $2899.437600$ $1285.708950$ $132576$ $18312.136003$ $3947.530428$ $330960$ $126004.456142$ $7801.259200$
 Nbr of $\mathcal{D}$ in $\Omega$ CPU time for DDFV (in seconds) CPU time for DDFV-SC (in seconds) $8376$ $18.149494$ $24.458821$ $20860$ $94.936105$ $59.901873$ $33264$ $418.468599$ $229.306277$ $82920$ $2899.437600$ $1285.708950$ $132576$ $18312.136003$ $3947.530428$ $330960$ $126004.456142$ $7801.259200$
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