High-order unstructured curved mesh generation using the Winslow equations

We propose a method to generate high-order unstructured curved meshes using the classical Winslow equations. We start with an initial straight-sided mesh in a reference domain, and fix the position of the nodes on the boundary on the true curved geometry. In the interior of the domain, we solve the Winslow equations using a new continuous Galerkin finite element discretization. This formulation appears to produce high quality curved elements, which are highly resistant to inversion. In addition, the corresponding nonlinear equations can be solved efficiently using Picard iterations, even for highly stretched boundary layer meshes. Compared to several previously proposed techniques, such as optimization and approaches based on elasticity analogies, this can significantly reduce the computational cost while producing curved elements of similar quality. We show a number of examples in both two and three space dimensions, including complex geometries and stretched boundary layers, and demonstrate the high quality of the generated meshes and the performance of the nonlinear solver.