An Iterative Substructuring Method for Raviart-Thomas Vector Fields in Three Dimensions

The iterative substructuring methods, also known as Schur complement methods, form one of two important families of domain decomposition algorithms. They are based on a partitioning of a given region, on which the partial differential equation is defined, into nonoverlapping substructures. The preconditioners of these conjugate gradient methods are then given in terms of local problems, defined on individual substructures and pairs of substructures, and, in addition, a global problem of low dimension. An iterative method of this kind is introduced for the lowest order Raviart--Thomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure. The theoretical bounds are confirmed by a series of numerical experiments.

[1]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[2]  Barry F. Smith,et al.  Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions , 1994 .

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

[4]  Andrea Toselli,et al.  Overlapping Schwarz methods for Maxwell's equations in three dimensions , 1997, Numerische Mathematik.

[5]  Alberto Valli,et al.  An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations , 1999, Math. Comput..

[6]  Barbara I. Wohlmuth,et al.  An iterative substructuring method for Maxwell's equations in two dimensions , 2001, Math. Comput..

[7]  Jinchao Xu,et al.  Some Estimates for a Weighted L 2 Projection , 1991 .

[8]  Douglas N. Arnold,et al.  Multigrid in H (div) and H (curl) , 2000, Numerische Mathematik.

[9]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[10]  Andrea Toselli,et al.  Overlapping and Multilevel Schwarz Methods for Vector Valued Elliptic Problems in Three Dimensions , 2000 .

[11]  R. S. Falk,et al.  PRECONDITIONING IN H (div) AND APPLICATIONS , 1997 .

[12]  Susanne C. Brenner.  A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method , 1992 .

[13]  Tarek P. Mathew,et al.  Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: Algorithms and numerical results , 1993 .

[14]  Tarek P. Mathew,et al.  Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part II: Convergence theory , 1993 .

[15]  Olof B. Widlund,et al.  Exotic coarse spaces for Schwarz methods for lower order and spectral finite elements , 1995 .

[16]  M. Sarkis.  Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements , 1997 .

[17]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[18]  J. Wang,et al.  Analysis of the Schwarz algorithm for mixed finite elements methods , 1992 .

[19]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .