The Residual–Free Bubble Method for Problems with Multiple Scales

The Residual–Free Bubble Method for Problems with Multiple Scales Andrea Cangiani St Hugh’s College Doctor of Philosophy Trinity Term 2004 This thesis is devoted to the numerical analysis and development of the residual–free bubble finite element method. We begin with an overview of known results and properties of the method, showing how techniques used on a range of multiscale problems can be cast into the framework of the residual–free bubble method. Further, we present an a priori error analysis of the method applied to convection–dominated diffusion problems on anisotropic meshes. The result has implications for the problem of parameter–tuning in classical stabilised finite element methods (for instance, the streamline– diffusion finite element method). We show how the local SD–parameter should be chosen on meshes with high aspect–ratios. A new algorithm named RFBe (enhanced residual–free bubble method) is proposed for the resolution of boundary layers on coarse meshes. The residual–free bubble finite element space is augmented locally by ad hoc bubble functions with support on two elements sharing a particular edge. The idea is presented in a general framework to highlight its applicability to a wide range of multiscale problems. Finally, we derive an a posteriori error estimate for the method and describe an associated adaptive algorithm designed to minimise the computational effort required for reducing the error below a prescribed tolerance. Both norm–error and linear–functional–error bounds are considered. We also propose an automatic procedure for switching off bubble stabilisation locally during the mesh refinement process (hb–adaptivity) to further reduce the computational cost.