On preconditioners for mortar discretization of elliptic problems

We consider elliptic problems with discontinuous coefficients defined on a union of two polygonal subdomains. The problems are discretized by the finite element method on non-matching triangulation across the interface. The discrete problems are described by the mortar technique in the space with constraints (the mortar condition) and in the space without constraints using Lagrange multipliers. To solve the discrete problems Preconditioned conjugate gradient iterations are used with Neumann–Dirichlet and Neumann–Neumann preconditioners in the first case, and dual Neumann–Dirichlet and dual Neumann–Neumann (or FETI, the finite element tearing and interconnecting) in the second case. An analysis of convergence of all four of these preconditioners is given. Numerical comparison of their performance on non-matching grids is presented. The general observation is that all preconditioners considered are very robust for the cases with the discontinuity ratio of 1000 across the interface. Copyright © 2002 John Wiley & Sons, Ltd.

[1]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[2]  O. Widlund,et al.  FETI and Neumann--Neumann Iterative Substructuring Methods: Connections and New Results , 1999 .

[3]  Maksymilian Dryja,et al.  A capacitance matrix method for Dirichlet problem on polygon region , 1982 .

[4]  Barbara I. Wohlmuth,et al.  Discretization Methods and Iterative Solvers Based on Domain Decomposition , 2001, Lecture Notes in Computational Science and Engineering.

[5]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[6]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[7]  P. Peisker,et al.  On the numerical solution of the first biharmonic equation , 1988 .

[8]  Olof B. Widlund,et al.  A FETI - DP Method for a Mortar Discretization of Elliptic Problems , 2002 .

[9]  Marian Brezina,et al.  Balancing domain decomposition for problems with large jumps in coefficients , 1996, Math. Comput..

[10]  J. Mandel,et al.  Convergence of a substructuring method with Lagrange multipliers , 1994 .

[11]  Olivier Pironneau,et al.  Substructuring preconditioners for the $Q_1$ mortar element method , 1995 .

[12]  C. Farhat,et al.  A method of finite element tearing and interconnecting and its parallel solution algorithm , 1991 .

[13]  L. Greengard,et al.  On the numerical solution of the biharmonic equation in the plane , 1992 .

[14]  Olof B. Widlund,et al.  Iterative Substructuring Preconditioners For Mortar Element Methods In Two Dimensions , 1997 .

[15]  O. Widlund,et al.  Schwarz Methods of Neumann-Neumann Type for Three-Dimensional Elliptic Finite Element Problems , 1993 .

[16]  Wolfgang Dahmen,et al.  A Multigrid Algorithm for the Mortar Finite Element Method , 1999, SIAM J. Numer. Anal..