UC Berkeley

High-order methods are receiving considerable interest from the computational community because they can achieve higher accuracy with reduced computational cost compared to traditional low-order approaches. These methods generally require unstructured meshes of non-inverted curved elements, and the generation of high-order curved meshes in a robust and automatic way is an important and challenging open problem.

We present a method to generate high-order unstructured curved meshes by solving the classical 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.

Another challenge that mesh-based methods face is that the discretization of the domain is usually generated before the solution is known, which can lead to large numerical errors or non-convergent schemes. A tool that can be used to overcome this problem is mesh adaptivity. We use the Winslow variable diffusion equations -- which are a variation of the classical form -- to perform curved and anisotropic unstructured mesh adaptivity.

We use a range of numerical examples to validate our models including complex geometries and stretched boundary layers. We demonstrate the high quality of the generated meshes and the performance of

the nonlinear solver. Finally, we present an example of mesh adaptivity for shock capturing when solving the Euler equations of gas dynamics for supersonic flow.