Moving finite element method: applications to science and engineering problems

Abstract

This paper is concerned with the formulation and development of a numerical moving mesh method to solve time-dependent reaction–diffusion–convection problems. The first part of this contribution gives an overview of the moving finite element method (MFEM) formulated with a piecewise higher degree polynomial basis in space. In the second part, applications are presented in order to give a convincing demonstration that the proposed moving finite element method is a powerful tool to compute numerical solution of a large class of 1D and 2D problems modeled by time-dependent partial differential equations (PDE). Numerical results are described which illustrate some important features of the proposed moving finite element method for solving problems in one and two-dimensional space domains.

Introduction

Many problems in science and engineering are formulated in terms of time-dependent partial differential equations (PDE). It is well known that due to the moving steep fronts present in the solution these problems present serious numerical difficulties. This has been illustrated in the treatment of several chemical engineering problems (Quinta-Ferreira et al., 1992a, Quinta-Ferreira et al., 1992b; Rodrigues & Quinta-Ferreira, 1988, Rodrigues & Quinta-Ferreira, 1990). The usual approach to this kind of problems is based on finite differences schemes for the discretization of the PDE. Finite differences are widely used but oscillations can appear in the solutions. Other techniques such as orthogonal collocations method on finite elements revealed better performance as Quinta-Ferreira et al. (1992b), demonstrated. In order to solve problems with sharp variations of the solution we present an approach where the mesh moves dynamical to capture the sharp front with a small number of space nodes. Moving finite element method (MFEM) is a discretization technique on continuously deforming spatial grids introduced by Miller and Miller (1981) to deal with time-dependent partial differential equations involving fine scale phenomena such as moving fronts, pulses and shocks. In the literature (Baines, 1994) several formulations of the moving finite element method using piecewise linear functions as its finite dimensional approximation are described. In our formulation solutions are calculated using a Galerkin approach with a piecewise higher degree polynomial basis in space (Coimbra, Sereno, & Rodrigues, 2003). Numerical results are described which illustrate some important features of the proposed moving finite element method for solving problems in 1D and 2D dimensional space domains. In order to clarify the effects of the different method parameters we pay special attention to the analysis of nodes movements and its relations with the choice of the initial grid, the degree of the approximations and other parameters such as penalty constants and tolerance errors. In 1D, we choose a problem from mathematical biology, which describes an ionic flow across a semi-infinite nerve membrane and a problem from chemical engineering concerning diffusion-convection and reaction in a catalytic particle. In 2D the moving finite element method will be used in the simulation of a fixed bed heat transfer transient model, which includes axial and radial dispersion.