Convergence of an adaptive discontinuous Galerkin method for the Biharmonic problem

Abstract

In this thesis we develop a basic convergence result for an adaptive symmetric interior penalty discontinuous Galerkin discretisation for the Biharmonic problem which provides convergence without rates for arbitrary polynomial degree r≥2, all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. We have to deal with the problem that the spaces consisting of piecewise polynomial functions may possibly contain no proper C^1-conforming subspace. This prevents from a straightforward generalisation of convergence results of adaptive discontinuous Galerkin methods for elliptic PDEs and requires the development of some new key technical tools. The convergence analysis is based on several embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space of the non-conforming discrete spaces, created by the adaptive algorithm. Finally, the convergence result is validated through a number of numerical experiments.