$hp$-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

Abstract

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The stucture of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated.