The analysis of PMHSS-multigrid methods for elliptic problems with smooth complex coefficients

In this paper, we investigate a class of multigrid methods with HSS smoothers applied to second order elliptic problems whose dominant coefficient is complex valued. These smoothers are based on Hermitian/skew-Hermitian splitting (HSS), modified HSS (MHSS) and preconditioned MHSS (PMHSS) iterations. It is shown that the abstract theory for multigrid method can be extended and applied to elliptic problems with complex coefficient. Under some assumption conditions, we prove that multigrid method with PMHSS smoother is convergent and optimal. Numerical results are reported to confirm the theoretical analysis.

[1]  Gene H. Golub,et al.  Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices , 2007, Math. Comput..

[2]  Gene H. Golub,et al.  Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices , 2006, SIAM J. Sci. Comput..

[3]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[4]  C. Douglas.  Multi-Grid Algorithms with Applications to Elliptic Boundary Value Problems , 1984 .

[5]  Hamilton-Jacobi Equations,et al.  Multigrid Methods for , 2011 .

[6]  Gene H. Golub,et al.  Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems , 2004, Numerische Mathematik.

[7]  Susanne C. Brenner,et al.  Convergence of the multigrid V-cycle algorithm for second-order boundary value problems without full elliptic regularity , 2002, Math. Comput..

[8]  Jinchao Xu,et al.  The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems , 1988 .

[9]  James H. Bramble,et al.  The analysis of multigrid methods , 2000 .

[10]  Zhong-Zhi Bai,et al.  Optimal parameters in the HSS‐like methods for saddle‐point problems , 2009, Numer. Linear Algebra Appl..

[11]  Fang Chen,et al.  Modified HSS iteration methods for a class of complex symmetric linear systems , 2010, Computing.

[12]  Jan Mandel,et al.  Multigrid convergence for nonsymmetric, indefinite variational problems and one smoothing step , 1986 .

[13]  Joseph E. Pasciak,et al.  THE ANALYSIS OF SMOOTHERS FOR MULTIGRID ALGORITHMS , 1992 .

[14]  Randolph E. Bank,et al.  A Comparison of Two Multilevel Iterative Methods for Nonsymmetric and Indefinite Elliptic Finite Element Equations , 1981 .

[15]  T. Pagel.  HUMBOLDT-UNIVERSITY OF BERLIN Faculty of Agriculture and Horticulture , 2009 .

[16]  Fang Chen,et al.  On preconditioned MHSS iteration methods for complex symmetric linear systems , 2011, Numerical Algorithms.

[17]  J. Pasciak,et al.  Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems , 1994 .

[18]  Junping Wang.  Convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems , 1993 .

[19]  H. Yserentant.  Old and new convergence proofs for multigrid methods , 1993, Acta Numerica.

[20]  Michele Benzi,et al.  New multigrid smoothers for the Oseen problem , 2010, Numer. Linear Algebra Appl..

[21]  Xuejun Xu,et al.  Local Multilevel Methods for Adaptive Finite Element Methods for Nonsymmetric and Indefinite Elliptic Boundary Value Problems , 2010, SIAM J. Numer. Anal..

[22]  Zhengda Huang,et al.  Convergence analysis of HSS-multigrid methods for second-order nonselfadjoint elliptic problems , 2013 .

[23]  Gene H. Golub,et al.  Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[24]  Gene H. Golub,et al.  Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems , 2005, SIAM J. Sci. Comput..

[25]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[26]  J. Pasciak,et al.  Multigrid Convergence for Second Order Elliptic Problems with Smooth Complex Coefficients , 2008 .

[27]  Gene H. Golub,et al.  A Preconditioner for Generalized Saddle Point Problems , 2004, SIAM J. Matrix Anal. Appl..