Neumann-Neumann methods for a DG discretization on geometrically nonconforming substructures
A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming....
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| Vydáno v: | Numerical methods for partial differential equations Ročník 28; číslo 4; s. 1194 - 1226 |
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| ISSN: | 0749-159X, 1098-2426 |
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| Abstract | A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ωi, a conforming finite element space associated to a triangulation
\documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} {\mathcal{T}}_{h_i}(\Omega_i)\end{align*}\end{document}
is introduced. To handle the nonmatching meshes across ∂Ωi, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ∂Ωi, a condition number estimate
\documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} C(1 + \max_i\log \frac{H_i}{h_i})^2\end{align*}\end{document}
is established with C independent of hi, Hi, hi/hj, and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012 |
|---|---|
| AbstractList | A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ωi, a conforming finite element space associated to a triangulation
\documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} {\mathcal{T}}_{h_i}(\Omega_i)\end{align*}\end{document}
is introduced. To handle the nonmatching meshes across ∂Ωi, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ∂Ωi, a condition number estimate
\documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} C(1 + \max_i\log \frac{H_i}{h_i})^2\end{align*}\end{document}
is established with C independent of hi, Hi, hi/hj, and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012 A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ω i of size O ( H i ). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ω i , a conforming finite element space associated to a triangulation \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} {\mathcal{T}}_{h_i}(\Omega_i)\end{align*}\end{document} is introduced. To handle the nonmatching meshes across ∂ Ω i , a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ∂ Ω i , a condition number estimate \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} C(1 + \max_i\log \frac{H_i}{h_i})^2\end{align*}\end{document} is established with C independent of h i , H i , h i / h j , and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012 |
| Author | Sarkis, Marcus Galvis, Juan Dryja, Maksymilian |
| Author_xml | – sequence: 1 givenname: Maksymilian surname: Dryja fullname: Dryja, Maksymilian organization: Department of Mathematics, Warsaw University, Warsaw 02-097, Poland – sequence: 2 givenname: Juan surname: Galvis fullname: Galvis, Juan organization: Department of Mathematics, Texas A&M University, College Station, Texas 3368 – sequence: 3 givenname: Marcus surname: Sarkis fullname: Sarkis, Marcus email: msarkis@wpi.edu organization: Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, Rio de Janeiro, Brazil |
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| Cites_doi | 10.1142/S0218202508002619 10.1137/0731086 10.1007/s10915-009-9293-1 10.1137/070697859 10.1002/num.20359 10.1090/S0025-5718-03-01484-4 10.2478/cmam-2003-0007 10.1137/S1064827502410657 10.1002/cpa.3160480203 10.2478/cmam-2001-0024 10.1002/cnm.1640090307 10.1002/nla.544 10.1090/S0025-5718-96-00757-0 10.1137/S0036142900378480 10.1007/s002110200392 10.1016/j.jco.2006.10.003 10.1051/m2an:2007006 10.1137/060667372 10.1016/j.jco.2007.02.003 10.1137/050634736 10.1137/S1064827596305593 10.1002/num.20293 10.1007/s002110050292 10.1007/s00211-005-0641-2 10.1007/b137868 10.1007/s002110050172 10.1137/S0036142901384162 10.1090/conm/180/01963 10.1002/nla.504 10.1137/0719052 10.1051/m2an:2008012 |
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| References | M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Commun Pure Appl Math 48 ( 1995), 121-155. I. Herrera, New formulation of iterative substructuring methods without Lagrange multipliers: Neumann-Neumann and FETI, Numer Methods Partial Differential Equations 24 ( 2008), 845-878. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal 39 ( 2001/02), 1749-1779. (electronic). A. Toselli and O. Widlund, Domain decomposition methods-algorithms and theory, Vol. 34, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2005. J. Mandel and M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients, Math Comp 65 ( 1996), 1387-1401. J. Mandel, Balancing domain decomposition, Commun Numer Methods Eng 9 ( 1993), 233-241. S. C. Brenner and K. Wang, Two-level additive Schwarz preconditioners for C0 interior penalty methods, Numer Math 102 ( 2005), 231-255. M. Dryja, B. F. Smith, and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J Numer Anal 31 ( 1994), 1662-1694. V. A. Dobrev, R. D. Lazarov, P. S. Vassilevski, and L. T. Zikatanov, Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations, Numer Linear Algebra Appl 13 ( 2006), 753-770. M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements, Numer Math 77 ( 1997), 383-406. G. Kanschat, Preconditioning methods for local discontinuous Galerkin discretizations, SIAM J Sci Comput 25 ( 2003), 815-831. (electronic). J. Xu and Y. Zhu, Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients, Math Models Methods Appl Sci 18 ( 2008), 77-105. I. G. Graham and M. J. Hagger, Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients, SIAM J Sci Comput 20 ( 1999), 2041-2066. (electronic). J. K. Kraus and S. K. Tomar, Multilevel preconditioning of two-dimensional elliptic problems discretized by a class of discontinuous Galerkin methods, SIAM J Sci Comput 30 ( 2008), 684-706. B. A. de Dios and L. Zikatanov, Uniformly convergent iterative methods for discontinuous Galerkin discretizations, J Sci Comput 40 ( 2009), 4-36. R. D. Lazarov, S. Z. Tomov, and P. S. Vassilevski, Interior penalty discontinuous approximations of elliptic problems, Comput Methods Appl Math 1 ( 2001), 367-382. J. K. Kraus and S. K. Tomar, A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems, Numer Linear Algebra Appl 15 ( 2008), 417-438. C. Lasser and A. Toselli, An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems, Math Comput 72 ( 2003), 1215-1238. (electronic). P. F. Antonietti and B. Ayuso, Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems, M2AN Math Model Numer Anal 42 ( 2008), 443-469. P. F. Antonietti and B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case, M2AN Math Model Numer Anal 41 ( 2007), 21-54. M. Dryja, M. V. Sarkis, and O. B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer Math 72 ( 1996), 313-348. M. Dryja, J. Galvis, and M. Sarkis, BDDC methods for discontinuous Galerkin discretization of elliptic problems, J Complex 23 ( 2007), 715-739. X. Feng and O. A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J Numer Anal 39 ( 2001), 1343-1365. (electronic). J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numer Math 95 ( 2003), 527-550. R. D. Lazarov and S. D. Margenov, CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems, J Complex 23 ( 2007), 498-515. H. H. Kim, M. Dryja, and O. B. Widlund, A BDCC method for mortar discretizations using a transformation of basis, SIAM J Numer Anal 47 ( 2008), 136-157. I. Herrera and R. A. Yates, Unified multipliers-free theory of dual-primal domain decomposition methods, Numer Methods Partial Differential Equations 25 ( 2009), 552-581. E. Burman and P. Zunino, A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems, SIAM J Numer Anal 44 ( 2006), 1612-1638. (electronic). D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J Numer Anal 19 ( 1982), 742-760. M. Dryja, On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients, Comput Methods Appl Math 3 ( 2003), 76-85. (electronic); dedicated to Raytcho Lazarov. 1993; 9 2009; 25 2009; 40 1982; 19 2006; 13 2008; 18 1996; 72 2008 2008; 15 1999; 20 1994 2008; 30 2002 2003; 72 2001/02; 39 2003; 95 1997; 77 1995; 48 2006; 44 2005; 102 2008; 47 2003; 3 2003; 25 2008; 24 2001; 39 2001; 1 2007; 41 2008; 42 2007; 23 1996; 65 2005; 34 1994; 31 e_1_2_12_3_2 e_1_2_12_5_2 e_1_2_12_19_2 e_1_2_12_18_2 e_1_2_12_17_2 e_1_2_12_15_2 Dryja M. (e_1_2_12_2_2) 2003; 3 Rivière B. (e_1_2_12_4_2) 2008 e_1_2_12_20_2 e_1_2_12_21_2 e_1_2_12_22_2 e_1_2_12_23_2 e_1_2_12_24_2 e_1_2_12_25_2 e_1_2_12_26_2 e_1_2_12_27_2 e_1_2_12_28_2 e_1_2_12_29_2 Dryja M. (e_1_2_12_34_2) 2008 e_1_2_12_30_2 e_1_2_12_31_2 e_1_2_12_32_2 Feng X. (e_1_2_12_16_2) 2002 e_1_2_12_33_2 e_1_2_12_35_2 e_1_2_12_14_2 e_1_2_12_13_2 e_1_2_12_12_2 e_1_2_12_11_2 e_1_2_12_7_2 e_1_2_12_10_2 e_1_2_12_6_2 e_1_2_12_9_2 e_1_2_12_8_2 |
| References_xml | – reference: J. Xu and Y. Zhu, Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients, Math Models Methods Appl Sci 18 ( 2008), 77-105. – reference: E. Burman and P. Zunino, A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems, SIAM J Numer Anal 44 ( 2006), 1612-1638. (electronic). – reference: J. Mandel, Balancing domain decomposition, Commun Numer Methods Eng 9 ( 1993), 233-241. – reference: H. H. Kim, M. Dryja, and O. B. Widlund, A BDCC method for mortar discretizations using a transformation of basis, SIAM J Numer Anal 47 ( 2008), 136-157. – reference: I. Herrera, New formulation of iterative substructuring methods without Lagrange multipliers: Neumann-Neumann and FETI, Numer Methods Partial Differential Equations 24 ( 2008), 845-878. – reference: M. Dryja, B. F. Smith, and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J Numer Anal 31 ( 1994), 1662-1694. – reference: S. C. Brenner and K. Wang, Two-level additive Schwarz preconditioners for C0 interior penalty methods, Numer Math 102 ( 2005), 231-255. – reference: D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J Numer Anal 19 ( 1982), 742-760. – reference: I. G. Graham and M. J. Hagger, Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients, SIAM J Sci Comput 20 ( 1999), 2041-2066. (electronic). – reference: V. A. Dobrev, R. D. Lazarov, P. S. Vassilevski, and L. T. Zikatanov, Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations, Numer Linear Algebra Appl 13 ( 2006), 753-770. – reference: J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numer Math 95 ( 2003), 527-550. – reference: M. Dryja, J. Galvis, and M. Sarkis, BDDC methods for discontinuous Galerkin discretization of elliptic problems, J Complex 23 ( 2007), 715-739. – reference: A. Toselli and O. Widlund, Domain decomposition methods-algorithms and theory, Vol. 34, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2005. – reference: M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Commun Pure Appl Math 48 ( 1995), 121-155. – reference: C. Lasser and A. Toselli, An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems, Math Comput 72 ( 2003), 1215-1238. (electronic). – reference: J. Mandel and M. Brezina, Balancing domain decomposition for problems with large jumps in coefficients, Math Comp 65 ( 1996), 1387-1401. – reference: P. F. Antonietti and B. Ayuso, Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems, M2AN Math Model Numer Anal 42 ( 2008), 443-469. – reference: J. K. Kraus and S. K. Tomar, Multilevel preconditioning of two-dimensional elliptic problems discretized by a class of discontinuous Galerkin methods, SIAM J Sci Comput 30 ( 2008), 684-706. – reference: X. Feng and O. A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J Numer Anal 39 ( 2001), 1343-1365. (electronic). – reference: M. Dryja, M. V. Sarkis, and O. B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer Math 72 ( 1996), 313-348. – reference: I. Herrera and R. A. Yates, Unified multipliers-free theory of dual-primal domain decomposition methods, Numer Methods Partial Differential Equations 25 ( 2009), 552-581. – reference: G. Kanschat, Preconditioning methods for local discontinuous Galerkin discretizations, SIAM J Sci Comput 25 ( 2003), 815-831. (electronic). – reference: D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal 39 ( 2001/02), 1749-1779. (electronic). – reference: M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements, Numer Math 77 ( 1997), 383-406. – reference: P. F. Antonietti and B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case, M2AN Math Model Numer Anal 41 ( 2007), 21-54. – reference: J. K. Kraus and S. K. Tomar, A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems, Numer Linear Algebra Appl 15 ( 2008), 417-438. – reference: M. Dryja, On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients, Comput Methods Appl Math 3 ( 2003), 76-85. (electronic); dedicated to Raytcho Lazarov. – reference: R. D. Lazarov and S. D. Margenov, CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems, J Complex 23 ( 2007), 498-515. – reference: R. D. Lazarov, S. Z. Tomov, and P. S. Vassilevski, Interior penalty discontinuous approximations of elliptic problems, Comput Methods Appl Math 1 ( 2001), 367-382. – reference: B. A. de Dios and L. 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| SubjectTerms | discontinuous Galerkin method elliptic problems with discontinuous coefficients finite element method interior penalty discretization Neumann-Neumann algorithms nonconforming decomposition preconditioners Schwarz methods |
| Title | Neumann-Neumann methods for a DG discretization on geometrically nonconforming substructures |
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