Nonsymmetric coupling of BEM and mixed FEM on polyhedral interfaces

Abstract

Abstract. In this paper we propose and analyze some new methods for coupling mixed finite element and boundary element methods for the model problem of the Laplace equation in free space or in the exterior of a bounded domain. As opposed to the existing methods, which use the complete matrix of operators of the Calderón projector to obtain a symmetric coupled system, we propose methods with only one integral equation. The system can be considered as a further generalization of the Johnson-Nédélec coupling of BEM-FEM to the case of mixed formulations in the bounded domain. Using some recent analytical tools we are able to prove stability and convergence of Galerkin methods with very general conditions on the discrete spaces and no restriction relating the finite and boundary element spaces. This can be done for general Lipschitz interfaces and in particular, the coupling boundary can be taken to be a Lipschitz polyhedron. Both the indirect and the direct approaches for the boundary integral formulation are explored.

When topological derivatives met regularized Gauss-Newton iterations in holographic 3D imaging

“…Here, B R = B(0, R) is a sphere with radius R large enough to contain the objects and detectors and L ke stands for the Dirichlet to Neumann operator for Maxwell equations [49,Chapter 10]. To be able to use the transmission boundary conditions forĖ we combine (47)- (48) to get:…”

“…[6] succeeds in producing first guesses of objects and of their optical properties from the holograms they generate by combining topological derivative and gradient based optimization procedures, provided the size of the objects is of the same order or smaller than the employed light wavelength. Helmholtz equations for the forward problems are solved in [6] by coupled boundary element (BEM)/finite element (FEM) methods [48,51], whereas shapes are constructed by means of blobby molecule coverings and signed distance functions as in [5], without imposing any specific parametrization.…”

The Validity of Johnson--Nédélec's BEM--FEM Coupling on Polygonal Interfaces

“…With the same cut-off and compactness argument that is used for Neumann boundary conditions, it is easy to show that this incoming flood of energy can be controlled by discretization; i.e., once the finite and boundary element spaces are refined enough to take care of the area that separates the interface Γ from the region with different diffusivity, stability is restored. This paper motivated some additional work on nonsymmetric coupling of mixed FEM with BEM [13] and discontinuous Galerkin methods (of the interior penalty family) with BEM [8]. The group of Dirk Praetorius in Vienna has given the corresponding a posteriori error estimates of nonsymmetric BEM-FEM formulations and has worked on the associated adaptive algorithms [16,15].…”

Relaxing the hypotheses of Bielak–MacCamy’s BEM–FEM coupling

“…where · ∞, int denotes the (L ∞ ( int )) d norm, then [19,Lemma 19] proves that there exists 0 < C < 2 (depending on ∇w ∞, int /c) such that…”

“…The technique employed in [22] is based on rewriting the discrete equations as a non-standard transmission problem in free space, an idea that had already been used in [16] to deal with BEM and BEM-FEM discretizations in the resolvent set of the Laplace operator. It was also the origin of the method in [19], which gives a non-symmetric coupling of BEM with mixed FEM on any Lipschitz interface.…”