Anisotropic Adaptivity in Finite Element Modelling

Finite element modelling/method (FEM) is indispensable for the accurate and rigorous numerical solution of continuum models arising in Engineering, Physics, Biology and other disciplines. Despite FEM’s astounding success, large classes of evolution PDE problems remain very expensive or even intractable computationally. The proposed research aims to address this computational challenge with the development of a new paradigm of reduced-complexity FEM, using novel mesh-order (hp-version), space-time, adaptive, discontinuous Galerkin (dG) methods on meshes consisting of, possibly anisotropic, polygonal/polyhedral element shapes.

State of the art. The limited applicability of standard FEM is particularly evident in the numerical approximation of evolution diffusion-type nonlinear problems involving multiple scales posed on complicated, possibly evolving, domains. Indeed, the presence of diffusion typically requires temporally implicit discretisations, while spatio-temporal multiscale features require local numerical resolution. Moreover, classical low-order, global, time-stepping schemes are typically employed to resolve such spatio-temporal multiscale features, thereby increasing unnecessarily the computational cost also in regions of the computational domain where the underlying solution is smoothly varying.