Reduced Element for Adaptive Finite Element Analysis of First-Order IVP with Built-in Error Estimator in Maximum Norm

Abstract

This paper proposes a novel yet simple approach to the adaptive finite element (FE) analysis of the first-order Initial Value Problems (IVPs) in the maximum norm by introducing the reduced element technique. In the present approach, the FE solution uh of the conventional Galerkin element of degree m + 1 is decomposed into two parts: a reduced solution uRh from the reduced element of degree m obtained by ignoring the highest degree term of uh, and a built-in point-wise error estimator εRh directly given by the ignored term. Since the end node solutions of the reduced element are inherited from the full order element, it gains O(h2m+2) accuracy and achieves a nodal/element accuracy ratio as high as two, which greatly enhances its adaptive capability regarding solving IVPs on long time domains. The related error analysis is addressed and a complete adaptivity algorithm is given. Typical numerical examples of both linear and nonlinear IVPs of both single and systems of equations are presented to show the validity and effectiveness of the proposed approach.