Accurate pressure post-process of a finite element method for elastoacoustics

Abstract

This paper deals with a post-process to obtain a more accurate approximation of the fluid pressure from a finite element computation of the vibration modes of a fluid-structure coupled system. The underlying finite element method, based on a displacement formulation for both media, consists of using Raviart-Thomas elements for the fluid combined with standard continuous elements for the solid.An easy to compute post-process of the pressure is derived. The relation between this post-process and an alternative finite element approximation of the problem based on discretizing the fluid pressure by enriched CrouzeixRaviart elements is studied. Higher order estimates for the L 2 norm of the post-processed pressure are proved by exploiting this relation. As a by-product, higher order L 2 estimates for the solid displacements obtained with the original method are also proved. Member of CIC, Provincia de Buenos Aires, Argentina Member of CONICET, Argentina. Partially supported by FONDECYT 7.990.075 and FONDAP in Applied Mathematics, Chile Partially supported by FONDECYT 1.990.346 and FONDAP in Applied Mathematics, Chile 390 A. Alonso et al.

Mixed approximation of eigenvalue problems: A superconvergence result

“…This alternative finite element formulation has been also used in the context of elastoacoustics vibration problem to obtain an accurate pressure post-process (see [3]). …”

Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods

“…Also, there is a closed relationship between mixed methods and nonconforming finite element methods for second order elliptical problems (see [1]). This relationship can be further exploited for deriving efficient solvers for the mixed formulations (see [9,3]) . We mention also that the present theory allows the analysis of a large class of discontinuous finite element methods when they are used for the approximation of spectral problems. This justifies the generality of our abstract approach.…”

A Posteriori Error Estimates for Maxwell’s Eigenvalue Problem

“…It also uses a superconvergent approximation of the primal variable, which is constructed by exploiting the equivalency between the lowest-order Raviart-Thomas mixed discretization and a nonconforming method for the primal problem based on the Crouzeix-Raviart space enriched by bubble functions. This approach has been extended to fluid-structure vibration problems in Alonso et al (2001Alonso et al ( , 2004. However, in spite of many existing analogies, a direct extension of these ideas to Maxwell's eigenvalue problem does not seem feasible, because no element that could play the role of Crouzeix-Raviart's in Durán et al (1999) is known.…”