Discontinuous Galerkin Methods with Mesh Adaptivity

摘要

The discontinuous Galerkin (DG) methods have been proved very powerful for solving many practical problems such as the nonlinear conservation laws and Maxwell equations. Since the solutions of these problems may be discontinuous, it is natural for them to be approximated in a discontinuous finite dimensional space. The moving mesh methods involve the solution of the underlying PDE for the physical problem in conjunction with a so-called moving mesh PDE for the mesh itself. The methods keep the total number of grid points unchanged, and can cluster more grid points to areas with singularities or large solution gradients. However, the mesh obtained by using the moving mesh methods may be very irregular. Since there are no continuity requirement among the elements for the DG methods, the geometry of each element can be very flexible, and as a result the DG methods can handle very irregular meshes. It is therefore natural to use the DG methods as the PDE evolution algorithms. In this talk, we will describe how to combine the DG methods with the moving mesh techniques. Several practical issues such as the choice of the monitor functions and the solution interpolation will be carefully investigated. The resulting scheme will be applied to solving nonlinear system of hyperbolic equations.