A mortar element method for elliptic problems with discontinuous coefficients

This paper proposes a mortar finite element method for solving the two-dimensional second-order elliptic problem with jumps in coefficients across the interface between two subregions. Non-matching finite element grids are allowed on the interface, so independent triangulations can be used in different subregions. Explicitly realizable mortar conditions are introduced to couple the individual discretizations. The same optimal L 2 -norm and energy-norm error estimates as for regular problems are achieved when the interface is of arbitrary shape but smooth, though the regularity of the true solution is low in the whole physical domain.

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

[2]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

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

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

[5]  L. Marcinkowski.  The mortar element method for quasilinear elliptic boundary value problems , 2022 .

[6]  Zhilin Li.  The immersed interface method using a finite element formulation , 1998 .

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

[8]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[9]  Yvon Maday,et al.  Coupling finite element and spectral methods: first results , 1990 .

[10]  Ragnar Winther,et al.  A Preconditioned Iterative Method for Saddlepoint Problems , 1992, SIAM J. Matrix Anal. Appl..

[11]  James H. Bramble,et al.  A finite element method for interface problems in domains with smooth boundaries and interfaces , 1996, Adv. Comput. Math..

[12]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[13]  Jun Zou,et al.  An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems , 2001, SIAM J. Matrix Anal. Appl..

[14]  Qiang Du,et al.  A Gradient Method Approach to Optimization-Based Multidisciplinary Simulations and Nonoverlapping Domain Decomposition Algorithms , 2000, SIAM J. Numer. Anal..

[15]  Y. Achdou.  The mortar element method for convection diffusion problems , 1995 .

[16]  R. Scott.  Interpolated Boundary Conditions in the Finite Element Method , 1975 .

[17]  J. Zou,et al.  Finite element methods and their convergence for elliptic and parabolic interface problems , 1998 .

[18]  M. Gunzburger,et al.  Least-Squares Finite Element Approximations to Solutions of Interface Problems , 1998 .

[19]  Zhiming Chen,et al.  Finite Element Methods with Matching and Nonmatching Meshes for Maxwell Equations with Discontinuous Coefficients , 2000, SIAM J. Numer. Anal..