Two-Grid hp-Adaptive Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

Two-Grid hp-Adaptive Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

Thu, 01/03/2012
14:00 		Professor Paul Houston (University of Nottingham) Computational Mathematics and Applications Add to calendar Gibson Grd floor SR
Two-Grid hp-Adaptive Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs
In this talk we present an overview of some recent developments concerning the a posteriori error analysis and adaptive mesh design of $ h $- and $ hp $-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems. In particular, we consider the derivation of computable bounds on the error measured in terms of an appropriate (mesh-dependent) energy norm in the case when a two-grid approximation is employed. In this setting, the fully nonlinear problem is first computed on a coarse finite element space $ V_{H,P} $. The resulting 'coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretization on the finer space $ V_{h,p} $; thereby, only a linear system of equations is solved on the richer space $ V_{h,p} $. Here, an adaptive $ hp $-refinement algorithm is proposed which automatically selects the local mesh size and local polynomial degrees on both the coarse and fine spaces $ V_{H,P} $ and $ V_{h,p} $, respectively. Numerical experiments confirming the reliability and efficiency of the proposed mesh refinement algorithm are presented.