# INVITED SPEAKERS

## Martin Costabel

University of Rennes, Rennes, France

### The Method of Volume Integral Equations

**Abstract**

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

**Abstract**

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

**Abstract**

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

**Abstract**

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

**Abstract**

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.