Past Numerical Analysis Group Internal Seminar

We present a multilevel preconditioner for the mixed finite element discretization of the biharmonic equation of first kind. While for the interior degrees of freedom a standard multigrid methods can be applied, a different approach is required on the boundary. The construction of the preconditioner is based on a BPX type multilevel representation in fractional Sobolev spaces. Numerical examples illustrate the obtained theoretical results.

Linear wave scattering problems (e.g. for acoustic, electromagnetic and elastic waves) are ubiquitous in science and engineering applications. However, conventional numerical methods for such problems (e.g. FEM or BEM with piecewise polynomial basis functions) are prohibitively expensive when the wavelength of scattered wave is small compared to typical lengthscales of the scatterer (the so-called "high frequency" regime). This is because the solution possesses rapid oscillations which are expensive to capture using conventional approximation spaces. In this talk I will outline some of my recent work in the development of "hybrid numerical-asymptotic" methods, which incur significantly reduced computational cost. These methods use approximation spaces containing oscillatory basis functions, carefully chosen to capture the high frequency asymptotic behaviour. In particular I will discuss some of the interesting challenges arising from non convex, penetrable and three-dimensional scatterers.