An evaluation of the gradient-weighted moving-finite-element method in one space dimension

Abstract

Moving-grid methods are becoming increasingly popular for solving several kinds of parabolic and hyperbolic partial differential equations involving fine-scale structures such as steep moving fronts and emerging steep layers. An interesting example of such a method is provided by the moving-finite-element (MFE) method. A difficulty with MFE, as with many other existing moving-grid methods, is the threat of grid distortion, which can only be avoided by the use of penalty terms. The involved parameter tuning is known to be very important, not only to provide for a safe automatic grid-point selection, but also for efficiency in the time-stepping process. When compared with MFE, the gradient-weighted MFE (GWMFE) method has some promising properties to reduce the need of tuning. To investigate to what extent GWM FE can be called robust, reliable, and effective for the automatic solution of time-dependent PDEs in one space dimension, we have tested this method extensively on a set of five relevant example problems with various solution characteristics. All tests have been carried out using the BDF time integrator SPGEAR of the existing method-of-lines software package SPRINT.