An Adaptive Linear Time Stepping Algorithm for Second-Order Linear Evolution Problems

Abstract

In this paper, we propose and analyze a linear time stepping finite element method for abstract second order linear evolution problems. For such methods, we derive optimal order a posteriori error estimates and sharp a posteriori nodal error estimates using the energy approach and the duality argument. Based on these estimates, we further design an adaptive time stepping strategy for the previous discretization in time. Several numerical experiments are provided to show the reliability and efficiency of the a-posteriori error estimates and to assess the effectiveness of the proposed adaptive time stepping method.