Department of Mathematics, UCI

In this talk, I will introduce briefly a moving mesh method based on harmonic mapping. As a rare character, the unique existence of the harmonic mapping is the basic motivation for us to develop this method. The method is implemented in finite element and an iterative procedure is adopted to avoid mesh tangling caused by numerical factors. Our method can move the mesh interior the domain and mesh on the boundary in coupling for both 2D and 3D problems. The moving mesh module can be a black box added on the whole solver to the PDE under consideration that it is very convenient for coding - no modifications to the solver of the PDE are required. The inter-mesh mesh updation is implemented by a linear convection equation instead of generally adopted interpolation methods, thus the method can be easily to applied to problem as incompressible Navier-Stokes equation where the divergence free interpolation can be a big problem, and problem as conservation laws where the conservative interpolation is not trival to be implemented. Numerical results including viscos Burgers equation, reaction-diffusion equation, incompressible Navier-Stokes equation and its coupling with level set method will be shown.