Analysis of time discretizations for parabolic problems with application to space discretizations

Abstract

This work summarizes some of the theoretical results of the author in last ten years, where the main area of the research was the numerical analysis for the stable higher order time discretization methods applied on parabolic problems. The main discretization scheme is the time discontinuous Galerkin method in combination with the conforming finite element method or the discontinuous Galerkin method in space. The thesis presents a priori error estimates for nonstationary singularly perturbed convection-diffusion problems, stability results for the problems with the domain evolving in time and a posteriori error estimates based on the equilibrated flux reconstructions. The technique presented for a posteriori analysis in time is applied to purely spatial problem and the quality of the recontruction is investigated with respect to the degree of polynomial approximation.