Martin Costabel

Boundary integral operators on Lipschitz domains: elementary results

The simple and double layer potentials for second order linear strongly elliptic differential operators on Lipschitz domains are studied and it is shown that in a certain range of Sobolev spaces, r...

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Girding’s inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only. 0 1985

On traces for H(curl, Ω) in Lipschitz domains

Abstract We study tangential vector fields on the boundary of a bounded Lipschitz domain Ω in R 3 . Our attention is focused on the definition of suitable Hilbert spaces corresponding to fractional Sobolev regularities and also on the construction of tangential differential operators on the non-smooth manifold. The theory is applied to the characterization of tangential traces for the space H ( curl ,Ω) . Hodge decompositions are provided for the corresponding trace spaces, and an integration by parts formula is proved.

Weighted regularization of Maxwell equations in polyhedral domains

Summary. We present a new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the non-density of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves to be numerically efficient.

A coercive bilinear form for Maxwell's equations

Abstract When one wants to treat the time-harmonic Maxwell equations with variational methods, one has to face the problem that the natural bilinear form is not coercive on the whole Sobolev space H1. One can, however, make it coercive by adding a certain bilinear form on the boundary of the domain. This addition causes a change in the natural boundary conditions. The additional bilinear form (see (2.7), (2.21), (3.3)) contains tangential derivatives of the normal and tangential components of the field on the boundary, and it vanishes on the subspaces of H1 that consist of fields with either vanishing tangential components or vanishing normal components on the boundary. Thus the variational formulations of the “electric” or “magnetic” boundary value problems with homogeneous boundary conditions are not changed. A useful change is caused in the method of boundary integral equations for the boundary value problems and for transmission problems where one has to use nonzero boundary data. The idea of this change emerged from the desire to have strongly elliptic boundary integral equations for the “electric” boundary value problem that are suitable for numerical approximation. Subsequently, it was shown how to incorporate the “magnetic” boundary data and to apply the idea to transmission problems. In the present note we present this idea in full generality, also for the anisotropic case, and prove coercivity without using symbols of pseudo-differential operators on the boundary.

Coupling of finite and boundary element methods for an elastoplastic interface problem

A class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain. A typical example is a problem from three-dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material. The nonlinear problem is given in variational form with a strictly convex functional. The linear elliptic problem is described by boundary integral equations on the coupling boundary. The typical saddle point structure of such problems is analyzed. Galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary. The convergence of the Galerkin approximation is based on the saddle-point structure which is shown to hold for the exact as well as the discretized problems.

Boundary integral operators for the heat equation

We study the integral operators on the lateral boundary of a space-time cylinder that are given by the boundary values and the normal derivatives of the single and double layer potentials defined with the fundamental solution of the heat equation. For Lipschitz cylinders we show that the 2×2 matrix of these operators defines a bounded and positive definite bilinear form on certain anisotropic Sobolev spaces. By restriction, this implies the positivity of the single layer heat potential and of the normal derivative of the double layer heat potential. Continuity and bijectivity of these operators in a certain range of Sobolev spaces are also shown. As an application, we derive error estimates for various Galerkin methods. An example is the numerical approximation of an eddy current problem which is an interface problem with the heat equation in one domain and the Laplace equation in a second domain. Results of numerical computations for this problem are presented.

Maxwell and Lamé eigenvalues on polyhedra

In a convex polyhedron, a part of the Lame eigenvalues with hard simple support boundary conditions does not depend on the Lame coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lame coefficients and the associated eigenmodes are the gradients of the Laplace–Dirichlet eigenfunctions. In a non-convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non-H2 singularities of the Laplace–Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non-convex polyhedron, the spectrum cannot be approximated by finite element methods using H1 elements. Similar properties hold in polygons. We give numerical results for two L-shaped domains. Copyright

Boundary element methods for Maxwell's equations on non-smooth domains

Summary. Variational boundary integral equations for Maxwells equations on Lipschitz surfaces in

EXPONENTIAL CONVERGENCE OF hp-FEM FOR MAXWELL EQUATIONS WITH WEIGHTED REGULARIZATION IN POLYGONAL DOMAINS

{\mathbb R}^3