String Gradient Weighted Moving Finite Elements in Multiple Dimensions with Applications in Two Dimensions

We formulate the string gradient weighted moving finite element method (SGWMFE) for systems of PDEs in multiple dimensions. Then we illustrate implementation issues for the method using two dimensions. The method is applied successfully to solve the highly nonlinear unsteady porous medium equation in two dimensions and the results are compared to exact solutions. We proceed to present results for the method when applied to the Gray Scott chemical reaction diffusion model in two dimensions. Finally, we apply the method to the shallow water equations in two dimensions and compare the meshes produced with those produced using gradient weighted finite element methods (GWMFE). Our conclusion is that SGWMFE is an easily applied member of the group of moving finite element methods.