A hybrid adaptive finite difference method powered by a posteriori error estimation technique

Abstract

This study focuses on developing an adaptive finite difference method oriented by the a posteriori error estimator that is originally derived from the finite element analysis, building a novel type of hybrid mesh adaptive technique that has not been reported elsewhere as of today. The time-dependent heat conduction problem is chosen for numerical investigation. Corresponding to this target problem, three major numerical “black boxes” are established, namely, finite difference solver, a posteriori error estimation framework, and re-meshing via simultaneous local mesh refinement and re-coarsening. Selected transient heat conduction problems are solved using the present hybrid finite difference mesh adaptive tools. The accuracy of the obtained adaptive solutions appears fairly good while the computational resources can be greatly saved, clearly demonstrating the usefulness, effectiveness, and robustness of the hybrid mesh adaptive approach developed through this study.