Modelling Time-Dependent Partial Equations with Moving Bounderies by the Moving Finite Element Method

Abstract

We have developed a numerical algorithm for time dependent partial differential equations (PDE) with moving boundaries based on the moving fmite elements method (MFEM) The use of adaptive grid methods are widely used in many different areas and general classes of time dependent differential equations can be solved efficiently using these methods. In paticular adaptive a i d methods has been found to be suitable for simulating time dependent problems that exhibit s h q transition layers. The MFEM is an adaptive a i d method, especially desim to deal with these problems. In the MFEM, originally developed by Miller and Miller [1], the approximate solution is given by a piecewise linear function depending on the nodal amplitude and on the nodes position. So, the MFEM automatically relocates nodes in order to concentrate them in regions where the solution is steep. In the fmed fmite element method a single set of basis functions are used. To get node movements the MFEM established a second set of basis function to account for the movement of the nodes. In our formulation of MFEM [2] we consider higher order basis functions. The MFEM generates not only the solution but also the adaptive spatial meshes for each dependent variable. To solve efficiently time-dependent problems with moving boundaries a moving boundq technique is developed to treat with the moving interface in the moving finite element mesh. Special domain decomposition is implemented [3] by the addition of a moving node describing the position of the internal moving interface. Numerical tests are investigated to evaluate the method and the performance of the numerical algorithm.