1Simon Fraser University, Burnaby, British Columbia, Canada
2University of Kansas, Lawrence, Kansas, U. S. A.
Mesh adaptation is a crucial part for the efficient computation of the solution of PDEs. For finite element methods (FEM), there are three main types of adaptive methods: (1) the h-method, which refines and coarsens elements locally, (2) the p-method, which adjusts the orders of approximation polynomials selectively, and (3) the r-method, which relocates grid points to regions where high resolution is needed. While the h and p methods achieve great success in steady state problems, the r-method is very promising in time dependent problems where the solutions typically change rapidly in a small region moving with the time.
In this talk, we present our recent studies on the r-adaptive FEM based on moving mesh PDEs. Several fundamental and practical issues are addressed: the choice of monitor functions to produce the desired mesh adaptation, the definition of the computational domain and the mesh on it, the movement of the grid points on boundaries. Some practical applications of the adaptive FEM are presented.