Mixed methods and the marriage between “mixed” finite elements and boundary elements

Abstract

When fields that describe a physical situation extend over the whole space, but with complex behavior (nonlinearity, coupling, etc.) only in a bounded region and simple behavior in the rest of space, it may be worthwhile to treat the inner field by a finite elements procedure and the outer field by boundary elements. We address this "marriage" problem in the case of mixed elements. The model problem adopted for this discussion is mugnetostutics. A new approach to the question of mixed elements is used, which emphasizes their parenthood with a classical concept of differential geometry, Whifney forms.

Comparison of several discretization methods of the Steklov–Poincaré operator

“…The Steklov-Poincare´operator is a non-local linear operator which leads, after discretization, to a full matrix whose size is governed by the number of unknowns on @O: The aim of this paper is to compare from the numerical point of view, three ways of discretizing P in three dimensions, giving for each the advantages and drawbacks. These methods are based on the ballooning technique (first introduced by Sylvester et al [8]), integral equations (see References [4,5] Calderon equations which leads to a priori good discretization properties of P: To our knowledge, this last one is a new method of discretization for the Steklov-Poincare´operator. The outline of the paper is as follows: the theory is recalled in Section 2, Section 3 is devoted to the explanation of the three methods, and Section 4 gives numerical examples and the comparison of the methods.…”

“…and it is possible to solve this equation from the boundary values of j: The reader is referred to References [2][3][4][5][6][7]. More precisely, the variational formulation of the problem involves integrals on O c such as Z where n is the inward normal with respect to O: Considering P the exterior Steklov-Poincare´operator, we may rewrite (6) as…”

Mixed FEM and BEM coupling for the three‐dimensional magnetostatic problem

“…We denote by nai the number of internal edges, nsg the number of vertices on ⌫ and nag the number edges of ⌫. We have h ϭ On the other hand, the discretization of v h in tetrahedron is [4]:…”

“…Another procedure is obtained by coupling the boundary integral method and the finite element method. Such coupled procedures have been proposed in [4] and in [5]. In these methods, we split the whole domain into two parts: one bounded part where we use the finite element method to describe the nonlinearity and another unbounded part in which the integral method is applied.…”

Mixed and hybrid formulations for the three‐dimensional magnetostatic problem

“…The method detailed in this article is a variant of a method described by Bossavit in [2]. The approach to study the physical problem is the same, but in [2], the methods used in the proofs are different.…”