Finite Element Methods with Matching and Nonmatching Meshes for Maxwell Equations with Discontinuous Coefficients

We investigate the finite element methods for solving time-dependent Maxwell equations with discontinuous coefficients in general three-dimensional Lipschitz polyhedral domains. Both matching and nonmatching finite element meshes on the interfaces are considered, and optimal error estimates for both cases are obtained. The analysis of the latter case is based on an abstract framework for nested saddle point problems, along with a characterization of the trace space for H(curl;D), a new extension theorem for H(curl;D) functions in any Lipschitz domain D, and a novel compactness argument for deriving discrete inf-sup conditions.

[1]  François Dubois,et al.  Discrete vector potential representation of a divergence-free vector field in three dimensional domains: numerical analysis of a model problem , 1990 .

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

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

[4]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

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

[6]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[7]  J. Zou,et al.  Fully Discrete Nite Element Approaches for Time-dependent Maxwell Equations , 1996 .

[8]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[9]  Pierre Degond,et al.  An analysis of the Darwin model of approximation to Maxwell’s equations , 1992 .

[10]  Pierre Degond,et al.  On a finite-element method for solving the three-dimensional Maxwell equations , 1993 .

[11]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

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

[13]  Satya N. Atluri,et al.  A hybrid finite element method for stokes flow: Part II—Stability and convergence studies , 1983 .

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

[15]  J. Nédélec.  A new family of mixed finite elements in ℝ3 , 1986 .

[16]  Roy A. Nicolaides,et al.  Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions , 1998, Math. Comput..

[17]  Jun Zou,et al.  Finite element convergence for the Darwin model to Maxwell's equations , 1997 .

[18]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[19]  Peter Monk,et al.  Analysis of a finite element method for Maxwell's equations , 1992 .

[20]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[21]  R. Nicolaides.  Direct discretization of planar div-curl problems , 1992 .

[22]  C. Makridakis,et al.  Time-discrete finite element schemes for Maxwell's equations , 1995 .

[23]  Jun Zou,et al.  Fully discrete finite element approaches for time-dependent Maxwell's equations , 1999, Numerische Mathematik.

[24]  A. Bendali,et al.  Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. I - The continuous problem. II - The discrete problem , 1984 .

[25]  P. Werner,et al.  A local compactness theorem for Maxwell's equations , 1980 .

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

[27]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[28]  Junping Wang,et al.  Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions , 1996, Math. Comput..

[29]  John Ambrosiano,et al.  A finite element formulation of the Darwin PIC model for use on unstructured grids , 1995 .

[30]  C. W. Nielson,et al.  A multidimensional quasineutral plasma simulation model , 1978 .