Mixed Finite Elements, Strong Symmetry and Mass Lumping for Elastic Waves

We present here the continuation of our work on mixed finite elements for wave propagation problems. In a previous report, we constructed and analysed a new family of quadrangular (2D) or cubic (3D) mixed finite elements, for the approximation of the scalar anisotropic wave equation. This work is extended here to the elastic wave equation, including in the case of an anisotropic medium. These new elements present the specificity to enforce the symmetry of the stress tensor in a str ong way and lead to explicit schemes (via mass lumping), after time discretization. The convergence analysis of these mixed finite elements is not straightforward: neither the standard abstract theory nor the theory we developed for the scalar case can be applied. That is why we introduce a new abstract theory which allows to get error estimates.

[1]  B. Auld,et al.  Accoustic Fields And Waves In Solids Vol-2 , 1973 .

[2]  Claes Johnson,et al.  Some equilibrium finite element methods for two-dimensional elasticity problems , 1978 .

[3]  Eliane Bécache,et al.  Étude d'un nouvel élément fini mixte permettant la condensation de masse , 1997 .

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

[5]  F. Collino.  Perfectly Matched Absorbing Layers for the Paraxial Equations , 1997 .

[6]  J. Bérenger.  Three-Dimensional Perfectly Matched Layer for the Absorption of Electromagnetic Waves , 1996 .

[7]  R. Glowinski,et al.  A fictitious domain method for Dirichlet problem and applications , 1994 .

[8]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[9]  An Optimal Absorbing Boundary Condition For Elastic Wave Modeling , 1993 .

[10]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[11]  T. Dupont.  $L^2 $-Estimates for Galerkin Methods for Second Order Hyperbolic Equations , 1973 .

[12]  Vassilios A. Dougalis,et al.  The Effect of Quadrature Errors on Finite Element Approximations for Second Order Hyperbolic Equations , 1976 .

[13]  R. Stenberg.  A family of mixed finite elements for the elasticity problem , 1988 .

[14]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[15]  R. Glowinski,et al.  Error analysis of a fictitious domain method applied to a Dirichlet problem , 1995 .

[16]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[17]  Nathalie Tordjman,et al.  Éléments finis d'ordre élevé avec condensation de masse pour l'équation des ondes , 1994 .

[18]  C. Tsogka,et al.  Some New Mixed Finite Elements in View of the Numerical Solution of Time Dependent Wave Propagation Problems , 1998 .

[19]  Chrysoula Tsogka,et al.  Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic hete , 1998 .

[20]  M. E. Morley.  A family of mixed finite elements for linear elasticity , 1989 .

[21]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[22]  Rolf Stenberg,et al.  A technique for analysing finite element methods for viscous incompressible flow , 1990 .

[23]  F. Collino,et al.  Fictitious Domain Method for Unsteady Problems: Application to Electromagnetic Scattering , 1996 .

[24]  J. Douglas,et al.  PEERS: A new mixed finite element for plane elasticity , 1984 .