Eric Michielssen began by discussing some future trends, motivated by the problems that he is asked to solve by industry. Most of these are multi-scale problems, such as the propagation of signals and power through semiconductors, or computing the radar cross-section of ships.

From an algorithmic perspective, he identified some of the important areas of current and future work. These include solvers:

• for computational electromagnetics (CEM) problems that are robust, stable, high order accurate and efficient;

• for high frequency Helmholtz equations (especially applicable to surface problems);

• for coupled boundary element-finite element CEM problems;

• for mixed physics problems, including CEM coupled to simulations of circuits, heat, micro-electro-mechanical systems (MEMS), and fluids;

• that are adaptive and / or "fast", especially for time domain problems.

Oscar Bruno agreed that the development of numerical methods which are fast, accurate and general is a priority. He believes that as mathematicians we are ideally suited to develop the necessary techniques for this. His "wish list" of future solved problems includes:

• the treatment of singular geometries;

• an accurate description of scattering surfaces and volumes;

• preconditioners for inherently ill-conditioned problems (such as volumetric scattering by open surfaces);

• improvements to all solution methods, i.e. those employing integral equations, finite elements and finite differences, possibly through a combination of several techniques;

• hybridisation - methods to solve multi-physics problems;

• accurate and efficient methods for high frequency scattering;

• improved numerical methods for scattering by random structures, including realistic descriptions of complicated random scattering geometries.

The session ended with a wide-ranging debate. There was broad consensus that inverse problems (in both the time and frequency domains) are also important and challenging. One key theme coming from both the presentations and discussion was that some of the most exciting current research concerns the development and analysis of new numerical algorithms that are high order accurate and/or ``fast''. Progress in these areas is allowing realistic engineering problems to be tackled, and in future perhaps accurate solvers will actually enable new technologies.

Another very important area is the development of methods for multi-scale or multi-physics problems, which we agreed is going to require "multi-mathematics".