Weighted Moving Finite Elements applied to systems of Partial Differential Equations

Abstract: Solving problems containing complex structures or moving shocks using standard methods can be computationally expensive. Using an adaptive mesh to `track' moving fronts and boundaries has been shown to provide cheaper and more accurate computations, making them a good candidate for solving large scale problems with a wide variety of applications.

In my talk I will introduce an adaptive mesh technique called ``String Gradient Weighted Moving Finite Elements" and present results to several systems of nonlinear Partial Differential Equations (PDEs), including a two dimensional model of a chemical reaction, the porous medium equation, and solutions to the dispersive and non-dispersive nonlinear shallow water equations.

Two deficiencies of the original Moving Finite Element method are (1) possible tangling of the mesh, and (2) absence of a mechanism for global refinement when necessary due to the constant number of degrees of freedom. Recently I have studied the SGWMFE method including uniform remeshing in order to continue computing solutions when the meshes become too distorted. I have also studied the method when implementing global refinement to enable handling of new physical phenomena of a smaller scale which may appear during the solution process. It is shown that the errors in time are kept under control when refinement is necessary. I shall conclude by presenting the results of applying remeshing and refining with SGWMFE to the dispersive shallow water equations with varying degrees of rotation.

The speaker is a UFA candidate in the Department of Mathematics and will also give the SCAIM Seminar, Tuesday, Sept. 25th at 12:30 p.m. in WMAX 216.