A Numerical Study of FETI Algorithms for Mortar Finite Element Methods

The finite element tearing and interconnecting (FETI) method is an iterative substructuring method using Lagrange multipliers to enforce the continuity of the finite element solution across the subdomain interface. Mortar finite elements are nonconforming finite elements that allow for a geometrically nonconforming decomposition of the computational domain into subregions and, at the same time, for the optimal coupling of different variational approximations in different subregions. We present a numerical study of FETI algorithms for elliptic self-adjoint equations discretized by mortar finite elements. Several preconditioners which have been successful for the case of conforming finite elements are considered. We compare the performance of our algorithms when applied to classical mortar elements and to a new family of biorthogonal mortar elements, and we discuss the differences between enforcing mortar conditions instead of continuity conditions for the case of matching nodes across the interface. Our experiments are carried out for both two and three dimensional problems, and include a study of the relative costs of applying different preconditioners for mortar elements.

[1]  Andrea Toselli,et al.  A FETI Domain Decomposition Method for Edge Element Approximations in Two Dimensions with Discontinuous Coefficients , 2001, SIAM J. Numer. Anal..

[2]  C. Farhat,et al.  Optimal convergence properties of the FETI domain decomposition method , 1994 .

[3]  R. Tezaur.  Analysis Of Lagrange Multiplier Based Domain Decomposition , 1998 .

[4]  Charbel Farhat,et al.  A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems , 2000, Numerische Mathematik.

[5]  Daniel Rixen,et al.  Preconditioning the FETI Method for Problems with Intra- and Inter-Subdomain Coefficient Jumps , 1997 .

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

[7]  Andrea Toselli,et al.  A FETI Preconditioner for Two Dimensional Edge Element Approximations of Maxwell's Equations on Nonmatching Grids , 2001, SIAM J. Sci. Comput..

[8]  D. Rixen,et al.  A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems , 1999 .

[9]  Barbara I. Wohlmuth,et al.  A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier , 2000, SIAM J. Numer. Anal..

[10]  Charbel Farhat,et al.  A non Overlapping Domain Decomposition Method for the Exterior Helmholtz Problem , 1997 .

[11]  I. Wohlmuth.  MULTIGRID METHODS FOR SADDLEPOINT PROBLEMSARISING FROM MORTAR FINITE ELEMENT DISCRETIZATIONSBARBARA , 1999 .

[12]  Antonini Macedo,et al.  Two Level Finite Element Method and its Application to the Helmholtz Equation , 1997 .

[13]  C. Farhat,et al.  A Scalable Substructuring Method By Lagrange Multipliers For Plate Bending Problems , 1996 .

[14]  Xiao-Chuan Cai,et al.  Overlapping Non-Matching Grid Mortar Element Methods for Elliptic Problems ; CU-CS-842-97 , 1997 .

[15]  Yuri V. Vassilevski,et al.  Analysis and parallel implementation of adaptive mortar element methods , 1998 .

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

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

[18]  Padmanabhan Seshaiyer,et al.  Uniform hp convergence results for the mortar finite element method , 2000, Math. Comput..

[19]  Charbel Farhat,et al.  Residual-Free Bubbles for the Helmholtz Equation , 1996 .

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

[22]  Barbara I. Wohlmuth.  A MULTIGRID METHOD FOR SADDLE POINT PROBLEMS ARISING FROM MORTAR FINITE ELEMENT DISCRETIZATIONS , 2000 .

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

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

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

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

[27]  Talal Rahman,et al.  Additive Schwarz Methods for Elliptic Mortar Finite Element Problems , 2003, Numerische Mathematik.

[28]  Yvon Maday,et al.  The mortar element method for three dimensional finite elements , 1997 .

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

[30]  Marina Vidrascu,et al.  Three-dimensional domain decomposition methods with nonmatching grids and unstructured coarse solver , 1994 .

[31]  C. Farhat,et al.  The two-level FETI method for static and dynamic plate problems Part I: An optimal iterative solver for biharmonic systems , 1998 .

[32]  D. Rixen,et al.  FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method , 2001 .

[33]  Dan Stefanica,et al.  Domain Decomposition Methods for Mortar Finite Elements , 2000 .

[34]  C. Farhat,et al.  The two-level FETI method. Part II: Extension to shell problems, parallel implementation and performance results , 1998 .

[35]  A. Toselli,et al.  A FETI Domain Decomposition Method for Maxwell''s Equations with Discontinuous Coefficients in Two Dimensions , 1999 .

[36]  A. Buffa,et al.  A Sliding Mesh-Mortar Method for Two Dimensional Eddy Currents Model for Electric Engines , 1999 .

[37]  Charbel Farhat,et al.  Implicit parallel processing in structural mechanics , 1994 .

[38]  C. Farhat,et al.  A scalable dual-primal domain decomposition method , 2000, Numer. Linear Algebra Appl..

[39]  A. Klawonn,et al.  The FETI method for mortar nite elements , 1998 .

[40]  J. Mandel.  Balancing domain decomposition , 1993 .

[41]  Y. Maday,et al.  Two different approaches for matching nonconforming grids: The Mortar Element method and the Feti Method , 1997 .

[42]  Catherine Lacour,et al.  Analyse et resolution numerique de methodes de sous-domaines non conformes pour des problemes de plaques , 1997 .

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

[44]  Yvon Maday,et al.  The mortar method for the Maxwell's equations in 3D , 1999 .

[45]  O. Widlund,et al.  A Domain Decomposition Method With Lagrange Multipliers For Linear Elasticity , 1999 .

[46]  Olof B. Widlund,et al.  A Domain Decomposition Method with Lagrange Multipliers and Inexact Solvers for Linear Elasticity , 2000, SIAM J. Sci. Comput..

[47]  Dietrich Braess,et al.  A Multigrid Method for Nonconforming FE-Discretisations with Application to Non-Matching Grids , 1999, Computing.

[48]  C. Farhat,et al.  A scalable Lagrange multiplier based domain decomposition method for time‐dependent problems , 1995 .