The moving finite element method: Applications to general partial differential equations with multiple large gradients

Abstract

The moving finite element (MFE) method has been reduced to practice in the automatic solution program DYLA for general systems of transient partial differential equations (PDEs) in 1-D. Several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the MFE method. Computational dilemmas of numerical diffusion, Gibbs overshooting and undershooting, zone tangling, and grid remap (or re-connection) aliasing, which occur frequently in conventional PDE methods, are essentially eliminated in the MFE method. Arbitrarily large gradients (or shocks) can be solved with extremely high resolution and accuracy for non-coincident, or even counterdirected, propagating wavefronts. Boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large-scale structures in reactive and non-reactive fluid calculations. The MFE method requires a small fraction of the grid nodes which are used in conventional PDE solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate PDE solutions on entire problem domains. Courant-Friedrichs-Lewy time-step limits are exceeded by wide margins (by factors of two to several thousand). Finally, the extension of the MFE method to 2-D is briefly discussed.