Martin Costabel

University of Rennes, Rennes, France

The Method of Volume Integral Equations


Volume potentials have been applied since a long time for transforming scattering problems with variable coefficients into second kind integral equations on bounded domains. Some instances of these volume integral equations have been widely used for the theoretical analysis of the scattering problems and also for their numerical approximation. In acoustic or electromagnetic scattering by inhomogeneous bodies, the resulting integral operators are strongly singular and their analysis is nontrivial. There has been some recent progress in the analysis of the spectrum of these operators, based on new extension techniques that compare the volume integral equations with a kind of boundary-domain integral equations that are easier to analyze. In particular, for electromagnetic scattering problems the essential spectrum can now be described completely, for smooth domains and for domains that have only a Lipschitz continuous boundary. On the other hand, despite the popularity of the volume integral equation method in some fields of computational physics, where numerical algorithms based on this method are routinely applied in large-scale computations, the analysis of numerical methods for this method is still largely absent and many questions remain open.

Marius Mitrea

University of Missouri-Columbia, United States

The Mighty Double Layer: History and Perspectives


In this talk I will focus on the role of the method of boundary layer potentials as a driving force in the development of present day analysis. This includes recent advances such as a Calderon-Zygmund theory for multi-layer operators associated with higher-order PDE, and the impact of tools from Geometric Measure Theory in potential analysis.

Iain W. Stewart

University of Strathclyde, Glasgow, United Kingdom

Weak Compactness Methods for Integro-Differential Equations in L1


This presentation will begin with an outline of the Dunford-Pettis theorem for weak L1 compactness and related equicontinuity results. The introduction of gauge spaces (which use a semi-metric) that are applicable to function spaces which are not separable allows explicit convergence criteria to be developed for weak convergence. The techniques are quite general and can be applied to a range of linear and nonlinear integral equations. As a particular example, these ideas will be applied to prove the global existence of solutions to a general nonlinear Smoluchowski coagulation-fragmentation integral equation with unbounded kernels; this is achieved by the construction of approximating solutions to integral equations with bounded kernels which converge weakly in L1 to a solution of the fully nonlinear problem with unbounded kernels.

Jon Trevelyan

Durham University, Durham, United Kingdom

Improving Boundary Element Approximations Using Non-polynomial Enrichment


Boundary element approximations are classically formed using piecewise polynomial shape functions as the solution basis. However, this gives rise to difficulties for some problems. For example, in linear elastic fracture mechanics the rapidly varying stress field around the stress singularity at a crack tip requires special care. These cases can be handled in a variety of ways, but one method involves including into the approximation space some enrichment functions based on the asymptotic crack tip displacement fields. This allows excellent results to be obtained on coarse boundary element discretisations. Another example is in wave propagation models and other Helmholtz problems, in which analysis of short wavelengths in the frequency domain imposes stringent demands on the mesh density. Here, use of a partition of unity enrichment based on sets of plane waves can allow us to use multi-wavelength sized elements and achieve a large reduction in the problem size required for a prescribed accuracy. The method can be more closely integrated with CAD systems by introducing an isogeometric formulation.

Wolfgang Wendland

University of Stuttgart, Stuttgart, Germany

Boundary Potentials and Elliptic Boundary Value Problems


Beginning with the classical boundary potentials and boundary integral equations for the Dirichlet and Neumann problems of the Poisson equation, we give a brief overview of the history of these methods in relation to different assumptions for the domain's boundary and function spaces involved. Then the introduction of energy spaces and coerciveness properties lead to strongly elliptic problems with coercive and strongly elliptic boundary integral equations. These concepts can be applied to various problems such as Riesz energy minimization, the Helmholtz equation, the Lam\'e equations of linear elasticity, to stationary Stokes and Brinkman equations. The lecture ends with an outlook to more general regularity properties and to problems on manifolds.