On the adaptive coupling of FEM and BEM in 2-d-elasticity

Abstract

This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier-Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi-optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh-refinement we state adaptive algorithms for the h-version of the FEM/BEM-coupling. Illustrating numerical results are included. Subject Classification (1991): 65N35, 65R20, 65D07, 45L10 Mathematics

Efficiency of a posteriori BEM–error estimates for first-kind integral equations on quasi–uniform meshes

“…The efficiency quotients γ h are computed for various numerical examples reported in [5,6,7,8]. From this, the efficiency results of this paper are confirmed; one observes efficiency in practice.…”

“…The questions of how and where to perform the refinement and whether this is "efficient" (a concept to be defined) is subject of many papers, and we refer, e.g., to [1,14,15,16,17,18,23,24,28,29,30] and the references quoted therein. The framework of adaptive methods, introduced by Eriksson and Johnson [14,15] for finite elements, is studied in [3,4,5,6,7,8] for boundary element methods (BEM) and covers weakly singular and hypersingular integral equations, integral equations for transmission problems, and the coupling of finite elements and boundary elements. However, the questions of efficiency of the adaptive algorithms and the sharpness of the a posteriori error estimates have been studied by numerical experiments only.…”

Adaptive FE–BE coupling for an electromagnetic problem in ℝ3—A residual error estimator

“…using (16). For the boundary error term e, we remark that e = curl for some ∈ H 1/2 ( ) / C, since e ∈ H −1/2 (div 0, ) (cf.…”

FE/BE coupling for an acoustic fluid–structure interaction problem. Residual a posteriori error estimates

“…The residual error estimate is formulated in the L 2 -norm using standard techniques for FE methods, see [11] and techniques for FE/BE coupling methods e.g. [20,12,13,14]. The derived error indicators are used later for the implementation of adaptive algorithms.…”