An implementation of a moving finite element method
This last decade, a lot of papers are devoted to the problem of adaptive gridding procedure for systems of partial differential equations. In particular such procedures are crucial for problems discretized by finite element methods when the solutions contain steep gradients. This paper is concerned with the moving finite element method (MFEM) when applied to one-dimensional spatial problem. This method is analogous to the method of lines (discretization of spatial part while the temporal part is left as a continuous operator) but in this case the spatial nodes move. It is especially suitable for parabolic or evolutionary problems where a steep front sweeps through the spatial domain. For example such methods are very interesting for combustion or oil reservoir problems, where we find phenomena with very different time scales and spatial profiles that are very steep in some regions and essentially flat in other parts of the domain. It has also been used to the solution problem. In this work, the authors i) implement directly the MFEM using a readily available integration package (LSODI); they report very good results with respect to accuracy and simulation cost, specially for Burger's equation \(u_ t=-uu_ x+\mu u_{xx},\) \(\mu =10^{-3}\), \(10^{-4}\). This equation is a simple analogue to the Navier-Stokes equations and serves as a model for many nonlinear wave propagation problems. ii) After an analysis and a review of the current MFEM literature, the authors propose a modification to the algorithm which reduces the computer time necessary to obtain an accurate solution.
scientific article; zbMATH DE number 3965387
scientific article