Mathematikgebäude Raum M 614

Numerical methods defined on computational meshes consisting of polygonal and/or polyhedral (henceforth collectively termed as “polytopic``) elements, with, potentially, many faces, have gained substantial traction in recent years for a number of important reasons. Clearly, a key underlying issue for all classes of FEM is the design of a suitable computational mesh upon which the underlying PDE problem will be discretized. The task of generating the mesh must address two competing issues: on the one hand, the mesh should provide a good representation of the given computational geometry with suffi- cient resolution for the computation of accurate numerical approximations. On the other hand, the mesh should not be so fine that computational turn-around times are too high, or in some cases even intractable, due to the high number of degrees of freedom in the resulting FEM. Traditionally, standard mesh generators generate grids consisting of triangular/quadrilateral elements in 2D and tetrahedral/hexahedral/prismatic/pyramidal elements in 3D. In the presence of essentially lower-dimensional solution features, for example, boundary/internal layers, anisotropic meshing may be exploited. However, in regions of high curvature, the use of such highly-stretched elements may lead to element self-intersection, unless the curvature of the geometry is carefully ‘propagated’ into the interior of the mesh through the use of (computationally expensive) isoparametric element mappings. These issues are particularly pertinent in the context of high-order methods, since in this setting, accuracy is often achieved by exploiting coarse meshes in combination with local high-order polynomial basis functions. I will argue that, by dramatically increasing the flexibility in terms of the set of admissible element shapes present in the computational mesh, the resulting, possibly discontinuous, FEMs can potentially deliver dramatic savings in computational costs. Moreover, I will present some recent theoretical developments in the error analysis of both “conforming” and discontinuous Galerkin finite element methods.

Prof. Georgoulis

Universität Leicester