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

TL;DR

This paper develops an a posteriori error estimator for Galerkin methods applied to nonlinear initial value problems, enabling adaptive h and hp algorithms to accurately approximate blow-up times and analyze convergence behavior.

Contribution

It introduces a novel a posteriori error estimator and adaptive algorithms for nonlinear IVPs, focusing on finite-time blow-up and convergence analysis.

Findings

Adaptive algorithms effectively approximate blow-up times.

Error estimator guides efficient mesh refinement.

Numerical experiments demonstrate convergence rates.

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.