Mesh Optimization for Discontinuous Galerkin Methods Using a Continuous Mesh Model

A method for anisotropic mesh adaptation and optimization for high-order discontinuous Galerkin schemes is presented. Given the total number of degrees of freedom, a metric-based method is proposed, which aims to globally optimize the mesh with respect to the Lq norm of the error. This is done by minimizing a suitable error model associated with the approximation space. Advantages of using a metric-based method in this context are several. First, it facilitates changing and manipulating the mesh in a general anisotropic way. Second, defining a suitable continuous interpolation operator allows the use of an analytic optimization framework that operates on the metric field, rather than the discrete mesh. The formulation of the method is presented as well as numerical experiments in the context of convection–diffusion systems.