Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

AbstractWe present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincaré–Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.

[1]  J. Pasciak,et al.  Convergence estimates for product iterative methods with applications to domain decomposition , 1991 .

[2]  P. Grisvard Singularities in Boundary Value Problems , 1992 .

[3]  G. A. Baker Finite element methods for elliptic equations using nonconforming elements , 1977 .

[4]  Xiaobing Feng,et al.  Two-Level Additive Schwarz Methods for a Discontinuous Galerkin Approximation of Second Order Elliptic Problems , 2001, SIAM J. Numer. Anal..

[5]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[6]  J. Tinsley Oden,et al.  A Conservative DGM for Convection-Diffusion and Navier-Stokes Problems , 2000 .

[7]  Ohannes A. Karakashian,et al.  A Nonconforming Finite Element Method for the Stationary Navier--Stokes Equations , 1998 .

[8]  Ohannes A. Karakashian,et al.  Piecewise solenoidal vector fields and the Stokes problem , 1990 .

[9]  Susanne C. Brenner,et al.  Poincaré-Friedrichs Inequalities for Piecewise H1 Functions , 2003, SIAM J. Numer. Anal..

[10]  M. Wheeler An Elliptic Collocation-Finite Element Method with Interior Penalties , 1978 .

[11]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[12]  Susanne C. Brenner,et al.  A two-level additive Schwarz preconditioner for nonconforming plate elements , 1996 .

[13]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[14]  Talal Rahman,et al.  An additive average Schwarz method for the plate bending problem , 2002, J. Num. Math..

[15]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[16]  E. Süli,et al.  Discontinuous hp-finite element methods for advection-diffusion problems , 2000 .

[17]  D. Arnold An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

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

[19]  Paul Houston,et al.  Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems , 2001, SIAM J. Numer. Anal..

[20]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for nonconforming finite element methods , 1996, Math. Comput..

[21]  P. Oswald,et al.  Remarks on the Abstract Theory of Additive and Multiplicative Schwarz Algorithms , 1995 .

[22]  J. Douglas,et al.  Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods , 1976 .