Adaptive Multiscale Methods for the Numerical Treatment of Systems of PDEs

Abstract

These notes are concerned with numerical analysis issues arising in the solution of certain systems involving stationary and instationary linear variational problems. Standard examples are second order elliptic boundary value problems, where particular emphasis is placed on the treatment of essential boundary conditions, and linear parabolic equations. These operator equations serve as a core ingredient for control problems where in addition to the state, the solution of the PDE, a control is to be determined which together with the state minimizes a certain tracking-type objective functional. Having assured that the variational problems are well-posed, we discuss numerical schemes based on B-splines and B-spline-type wavelets as a particular multiresolution discretization methodology. The guiding principle is to devise fast and efficient solution schemes which are optimal in the number of arithmetic unknowns. We discuss optimal conditioning of the system matrices, numerical stability of discrete formulations, and adaptive approximations.