Finite element methods for a composite model in elastodynamics. (English) Zbl 0684.73034

Santos, Juan Enrique; Douglas, Jim jun.; Calderon, Alberto Pedro

SIAM J. Numer. Anal. 25, No. 3, 513-532 (1988).

The propagation of elastic waves through an isotropic composite system consisting of an elastic solid with an embedded porous medium saturated by a compressible viscous fluid is studied. Boit’s low frequency dynamic equations are chosen to describe the propagation of waves within the porous medium while standard elastic wave equation is used for the rest of the medium. The weak form of the problem is described and the existence and uniqueness of the solutions are established. The continuous-time and discrete-time Galerkin procedures are defined and the corresponding error analyses are performed. The displacement vector components for the solid part of the medium are approximated by standard finite element subspaces, whereas the mixed finite element subspaces are employed to approximate the vector displacements for the porous medium. Certain smoothness assumptions for the solution of different problems are made to derive the convergence of optimal order for the methods.

Reviewer: V.K.Arya

MSC:

74S05	Finite element methods applied to problems in solid mechanics
65C20	Probabilistic models, generic numerical methods in probability and statistics
74J10	Bulk waves in solid mechanics
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
86-08	Computational methods for problems pertaining to geophysics
76S05	Flows in porous media; filtration; seepage
35B65	Smoothness and regularity of solutions to PDEs
74S30	Other numerical methods in solid mechanics (MSC2010)
74G30	Uniqueness of solutions of equilibrium problems in solid mechanics
74H25	Uniqueness of solutions of dynamical problems in solid mechanics
74L10	Soil and rock mechanics

Keywords:

effect of dissipation; relative motion of Poiseuille type; energy- preserving boundary conditions; contact surface; elastic waves; isotropic composite system; elastic solid; embedded porous medium saturated by a compressible viscous fluid; Boit’s low frequency dynamic equations; standard elastic wave equation; weak form; existence; continuous-time; discrete-time Galerkin procedures; error analyses; displacement vector components; solid part; finite element subspaces; mixed finite element subspaces; convergence of optimal order