Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients
SUMMARYIn this paper, we present a multigrid V‐cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix–Raviart discretization of second‐order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We s...
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| Vydané v: | Numerical linear algebra with applications Ročník 21; číslo 1; s. 24 - 38 |
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01.01.2014
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| Abstract | SUMMARYIn this paper, we present a multigrid V‐cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix–Raviart discretization of second‐order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rates of the (multiplicative) two‐grid and multigrid V‐cycle algorithms will deteriorate rapidly because of large jumps in coefficient. However, the preconditioned systems have only a fixed number of small eigenvalues depending on the large jump in coefficient, and the effective condition numbers are independent of the coefficient and bounded logarithmically with respect to the mesh size. As a result, the two‐grid or multigrid preconditioned conjugate gradient algorithm converges nearly uniformly. We also comment on some major differences of the convergence theory between the nonconforming case and the standard conforming case. Numerical experiments support the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd. |
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| AbstractList | SUMMARYIn this paper, we present a multigrid V‐cycle preconditioner for the linear system arising from piecewise linear nonconforming Crouzeix–Raviart discretization of second‐order elliptic problems with jump coefficients. The preconditioner uses standard conforming subspaces as coarse spaces. We showed that the convergence rates of the (multiplicative) two‐grid and multigrid V‐cycle algorithms will deteriorate rapidly because of large jumps in coefficient. However, the preconditioned systems have only a fixed number of small eigenvalues depending on the large jump in coefficient, and the effective condition numbers are independent of the coefficient and bounded logarithmically with respect to the mesh size. As a result, the two‐grid or multigrid preconditioned conjugate gradient algorithm converges nearly uniformly. We also comment on some major differences of the convergence theory between the nonconforming case and the standard conforming case. Numerical experiments support the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd. |
| Author | Zhu, Yunrong |
| Author_xml | – sequence: 1 givenname: Yunrong surname: Zhu fullname: Zhu, Yunrong email: Correspondence to: Yunrong Zhu, Physical Sciences 318 P. O. Box 8085, Idaho State University, Pocatello, ID 83209, USA., zhuyunr@isu.edu organization: Department of Mathematics, Idaho State University, Pocatello, ID, USA |
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| References_xml | – reference: Hackbusch W. Iterative Solution of Large Sparse Systems of Equations, Vol. 95 of Applied Mathematical Sciences. Springer-Verlag: New York, 1994. xxii+429 pp. ISBN: 0-387-94064-2. – reference: Brenner SC. A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method. SIAM Journal on Numerical Analysis 1992; 29:647-678. – reference: Braess D, Verfürth R. Multigrid methods for nonconforming finite element methods. SIAM Journal on Numerical Analysis 1990; 27:979-986. – reference: Oswald P. Preconditioners for nonconforming discretizations. Mathematics of Computation 1996; 65(215):923-941. – reference: Wang F, Chen J, Huang P. A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients. Science China Mathematics 2012; 55:1513-1526. – reference: Brenner SC. Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM Journal on Numerical Analysis 2003; 41(1):306-324. – reference: Ayuso de Dios B, Zikatanov L. Uniformly convergent iterative methods for discontinuous Galerkin discretizations. Journal of Scientific Computing 2009; 40(1):4-36. – reference: Axelsson O. Iteration number for the conjugate gradient method. Mathematics and Computers in Simulation 2003; 61(3-6):421-435. MODELLING 2001 (Pilsen). – reference: Xu J, Zhu Y. Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Mathematical Models and Methods in Applied Science 2008; 18(1):77-105. – reference: Bramble JH. Multigrid Methods, Vol. 294 of Pitman Research Notes in Mathematical Sciences. Longman Scientific & Technical: Essex, England, 1993. – reference: Zhu Y. Domain decomposition preconditioners for elliptic equations with jump coefficients. Numerical Linear Algebra with Applications 2008; 15(2-3):271-289. – reference: Hoppe RHW, Wohlmuth B. Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems. SIAM Journal on Numerical Analysis 1997; 34(4):1658-1681. – reference: Brenner SC. Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second-order elliptic boundary value problems. Mathematics of Computation 2004; 73(247):1041-1066. (electronic). – reference: Oswald P. Optimality of multilevel preconditioning for nonconforming P1 finite elements. Numerische Mathematik 2008; 111(2):267-291. – reference: Axelsson O. Iterative Solution Methods. Cambridge University Press: Cambridge, 1994. – reference: Chen Z. Equivalence between and multigrid algorithms for nonconforming and mixed methods for second- order elliptic problems. East-West Journal of Numerical Mathematics 1996; 4:1-33. – reference: Bramble JH, Pasciak JE, Xu J. The analysis of multigrid algorithms with non-nested spaces or noninherited quadratic forms. Mathematics of Computation 1991; 56:1-34. – reference: Vassilevski PS, Wang J. An application of the abstract multilevel theory to nonconforming finite element methods. SIAM Journal on Numerical Analysis 1995; 32(1):235-248. – reference: Toselli A, Widlund O. Domain Decomposition Methods: Algorithms and Theory, Springer Series in Computational Mathematics, Springer-Verlag: Berlin, 2005. – reference: Galvis J, Efendiev Y. Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Modeling & Simulation 2010; 8(4):1461-1483. – reference: Falgout RD, Vassilevski PS, Zikatanov LT. On two-grid convergence estimates. Numerical Linear Algebra with Applications 2005; 12(5-6):471-494. – reference: Oswald P. On hierarchical basis multilevel method with nonconforming P1 elements. Numerische Mathematik 1992; 62:189-212. – reference: Xu J. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured meshes. Computing 1996; 56:215-235. – reference: Brenner SC. An optimal order multigrid for P1 nonconforming finite elements. Mathematics of Computation 1989; 52:1-15. – reference: Golub GH, Van Loan CF. Matrix Computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press: Baltimore, MD, 1996. – reference: Arnold DN, Brezzi F. Mixed and nonconforming finite element methods: implementation, postporcessing and error estimates. RAIRO Modélisation Mathématique Et Analyse Numérique 1985; 19:7-32. – reference: Dolejší V, Feistauer M, Felcman J. On the discrete Friedrichs inequality for nonconforming finite elements. Numerical Functional Analysis and Optimization 1999; 20(5-6):437-447. – reference: Zikatanov L. Two-sided bounds on the convergence rate of two-level methods. Numerical Linear Algebra with Applications 2008; 15(5):439-454. – reference: Brenner SC. Two-level additive Schwarz preconditioners for nonconforming finite element methods. Mathematics of Computation 1996; 65:897-921. – reference: Briggs WL, Henson VE, McCormick SF. A Multigrid Tutorial, 2nd edn. Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, 2000. – year: 2011 – volume: 56 start-page: 215 year: 1996 end-page: 235 article-title: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured meshes publication-title: Computing – volume: 40 start-page: 4 issue: 1 year: 2009 end-page: 36 article-title: Uniformly convergent iterative methods for discontinuous Galerkin discretizations publication-title: Journal of Scientific Computing – volume: 15 start-page: 271 issue: 2–3 year: 2008 end-page: 289 article-title: Domain decomposition preconditioners for elliptic equations with jump coefficients publication-title: Numerical Linear Algebra with Applications – volume: 27 start-page: 979 year: 1990 end-page: 986 article-title: Multigrid methods for nonconforming finite element methods publication-title: SIAM Journal on Numerical Analysis – volume: 65 start-page: 923 issue: 215 year: 1996 end-page: 941 article-title: Preconditioners for nonconforming discretizations publication-title: Mathematics of Computation – year: 2005 – volume: 8 start-page: 1461 issue: 4 year: 2010 end-page: 1483 article-title: Domain decomposition preconditioners for multiscale flows in high‐contrast media publication-title: Multiscale Modeling & Simulation – volume: 19 start-page: 7 year: 1985 end-page: 32 article-title: Mixed and nonconforming finite element methods: implementation, postporcessing and error estimates publication-title: RAIRO Modélisation Mathématique Et Analyse Numérique – volume: 20 start-page: 437 issue: 5–6 year: 1999 end-page: 447 article-title: On the discrete Friedrichs inequality for nonconforming finite elements publication-title: Numerical Functional Analysis and Optimization – year: 1989 – volume: 73 start-page: 1041‐1066 issue: 247 year: 2004 article-title: Convergence of nonconforming V‐cycle and F‐cycle multigrid algorithms for second‐order elliptic boundary value problems publication-title: Mathematics of Computation – year: 2000 – year: 1996 – volume: 32 start-page: 235 issue: 1 year: 1995 end-page: 248 article-title: An application of the abstract multilevel theory to nonconforming finite element methods publication-title: SIAM Journal on Numerical Analysis – volume: 18 start-page: 77 issue: 1 year: 2008 end-page: 105 article-title: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients publication-title: Mathematical Models and Methods in Applied Science – volume: 29 start-page: 647 year: 1992 end-page: 678 article-title: A multigrid algorithm for the lowest‐order Raviart–Thomas mixed triangular finite element method publication-title: SIAM Journal on Numerical Analysis – volume: 4 start-page: 1 year: 1996 end-page: 33 article-title: Equivalence between and multigrid algorithms for nonconforming and mixed methods for second‐ order elliptic problems publication-title: East‐West Journal of Numerical Mathematics – volume: 12 start-page: 471 issue: 5–6 year: 2005 end-page: 494 article-title: On two‐grid convergence estimates publication-title: Numerical Linear Algebra with Applications – year: 1994 – volume: 55 start-page: 1513 year: 2012 end-page: 1526 article-title: A multilevel preconditioner for the C‐R FEM for elliptic problems with discontinuous coefficients publication-title: Science China Mathematics – year: 2010 – start-page: 93 year: 1993 end-page: 109 – volume: 56 start-page: 1 year: 1991 end-page: 34 article-title: The analysis of multigrid algorithms with non‐nested spaces or noninherited quadratic forms publication-title: Mathematics of Computation – volume: 52 start-page: 1 year: 1989 end-page: 15 article-title: An optimal order multigrid for P1 nonconforming finite elements publication-title: Mathematics of Computation – volume: 41 start-page: 306 issue: 1 year: 2003 end-page: 324 article-title: Poincaré–Friedrichs inequalities for piecewise functions publication-title: SIAM Journal on Numerical Analysis – volume: 95 year: 1994 – volume: 111 start-page: 267 issue: 2 year: 2008 end-page: 291 article-title: Optimality of multilevel preconditioning for nonconforming P1 finite elements publication-title: Numerische Mathematik – volume: 61 start-page: 421 issue: 3–6 year: 2003 end-page: 435 article-title: Iteration number for the conjugate gradient method publication-title: Mathematics and Computers in Simulation – volume: 294 year: 1993 – volume: 34 start-page: 1658 issue: 4 year: 1997 end-page: 1681 article-title: Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems publication-title: SIAM Journal on Numerical Analysis – volume: 65 start-page: 897 year: 1996 end-page: 921 article-title: Two‐level additive Schwarz preconditioners for nonconforming finite element methods publication-title: Mathematics of Computation – volume: 15 start-page: 439 issue: 5 year: 2008 end-page: 454 article-title: Two‐sided bounds on the convergence rate of two‐level methods publication-title: Numerical Linear Algebra with Applications – start-page: 119 year: 1993 end-page: 124 – volume: 62 start-page: 189 year: 1992 end-page: 212 article-title: On hierarchical basis multilevel method with nonconforming P1 elements publication-title: Numerische Mathematik |
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| SubjectTerms | conjugate gradient Crouzeix-Raviart effective condition number jump coefficients multigrid preconditioner two-grid |
| Title | Analysis of a multigrid preconditioner for Crouzeix-Raviart discretization of elliptic partial differential equation with jump coefficients |
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