$hp$-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems
For researchers studying blow-up phenomena in nonlinear initial value problems, this work provides adaptive algorithms with error control, though the results are incremental.
This work derives an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems, focusing on finite-time existence and blow-up. The resulting h and hp adaptive algorithms are tested numerically, showing convergence to the blow-up time.
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.