Moving Mesh Methods on Parametric Surfaces

Many phenomena in the applied and natural sciences occur on surfaces. To solve accurately the corresponding partial differential equations (PDEs), it is often necessary to adapt the mesh, based upon the geometry of the surface, or based upon the behaviour of the PDE solution. Moving mesh methods are particularly efficient strategies in many situations. PDEs explicitly involving the mesh speed, called moving mesh PDEs (MMPDEs), offer a robust technique to adapt the mesh. In this work, we implement, with the C++ finite element library deal.II, a mesh adaptation based on Winslow's adaptation functional. We generalize the moving mesh problem to curved surfaces by deriving appropriate mathematical and finite element formulations. Furthermore, a simple method using surface parameterization is developed and implemented using deal.II. The results, for both fixed and dynamically adapting meshes, demonstrate the effectiveness of the method.