Lecture SS 25 Advanced Topics in Scientific Computing
Optimality of Adaptive Finite Element Methods
Topics
The aim of the lecture is to give an introduction to a very active field of research in numerical analysis. The lecture may serve as a basis for a master thesis which can also be written in the frame of a current research project at the Institute for Numerical Simulation.
Accurate a posteriori error estimation plays a key role in reliable and efficient scientic computing: First, one may want to check whether the solution of a numerical simulation is accurate enough. The accuracy of numerical approximations to solutions of PDEs, however, usually suffers from singularities and anisotropies of the given data and/or the (unknown) exact solution. Second, if the approximation is thus not sufficiently accurate, one remedy is to use meshes which resolve these singularities appropriately. Such meshes are usually obtained iteratively by adaptive algorithms which are driven by certain a posteriori error estimators.
The ultimate goal of adaptive mesh-refining schemes is to compute a discrete solution with error below a prescribed tolerance at the expense of, up to a multiplicative constant, the minimal computational cost. In recent years, the mathematical understanding of this ultimate goal has matured. The lecture aims to give an overview on the current state of research.
Up-to-date lecture notes written in LaTeX will be made available during the course.
Basic knowledge on FEM (e.g., from Scientific Computing I) is expected.