A Direct Maximum-Norm Error Estimator for Condensed Elements With Application to Adaptive Analysis of First-Order Initial-Value Problems

This paper proposes a direct maximum-norm error estimator for the condensed element with application to the adaptive finite element (FE) analysis of the first-order initial-value problems (IVPs). Taking advantage of the ultra superconvergent nodal accuracy of the condensed element, the FE errors are analogized with the errors of the corresponding polynomial interpolation, which allows a direct calculation of the points and values of the maximum errors on each element in a simple and straightforward manner. Equipped with this error estimator, a complete adaptivity algorithm based on the condensed element is developed with reliable error control and effective step-size adjustment. Mathematical analysis and representative numerical examples of both linear and nonlinear IVPs of both single and system of equations are presented to show the validity and effectiveness of the proposed approach.