Polynomial Approximations of Singular Vector Fields

Many problems of mathematical physics require the approximate solution of a partial differential equation in an unbounded domain. Examples are the calculation of scattered acoustic or electromagnetic waves at an obstacle in the three dimensional space.The numerical treatment of such problems poses a particular challenge since the underlying domain has to be discretised by finite meshes and this is not straightforward if the domain is unbounded. Main focus in this project is on the boundary element method. The idea is to discretise, instead of the governing partial differential equation, a related integral equation that lives only on the boundary of the domain. In this way, the problem on an unbounded domain is reduced to a related problem on a bounded surface (if the obstacle is bounded).In many applications, the domain or obstacle under consideration has corners and edges. Solutions to the above mentioned problems are ill-behaved there, so-called singularities appear. These singularities greatly reduce the performance of numerical approximation schemes. In this project we study numerical schemes for the solution of wave problems (in acoustics and electromagnetism) with singularities, e.g. scattering problems at non-smooth obstacles. The additional difficulty with wave problems, apart from appearing singularities, is that numerical approximations suffer from a phase difference with the exact (unknown) solution, the so-called dispersion error. We propose to use high order piecewise polynomials for the approximation to tackle both problems in parallel: reduction of approximation properties by singularities are less severe for high order polynomials and numerical dispersion errors are much more efficiently reduced by high order polynomials than by those of low order.There is almost no mathematical theory available for the use of high order methods to deal with singular wave problems. The aim of this project is to provide this theory and to develop numerical methods that efficiently approximate singular solutions to wave problems. A particular focus is on the analysis and use of boundary integral equations for the solution of scattering problems in unbounded domains.