2-D, 3-D and 4-D anisotropic mesh adaptation for the time-continuous space-time finite element method with applications to the incompressible Navier-Stokes equations

Abstract

A mesh adaptation strategy suitable for unsteady partial differential equations has been developed to control both the spatial and temporal discretization errors in a unified fashion. The aims are to provide a methodology that prevents the accumulation of discretization error associated with time stepping approaches and is also flexible enough to adjust the density of the space-time mesh to varying time scales in the solution domain. The primary focus of this thesis has been the development of anisotropic meshing algorithms that can operate in 2-D, 3-D and 4-D on unstructured simplicial meshes. The mesh modification operators include edge splitting, edge collapsing, simulated edge swapping, and mesh smoothing and are driven by an anisotropic metric field. The mesh adaptation methodology has been coupled with a time-continuous space-time finite element flow solver for the incompressible Navier-Stokes equations. The space and time finite element discretizations have been treated in a fully coupled manner using a Galerkin/Least-Squares formulation on a simplicial mesh that covers the entire space-time solution domain. The anisotropic metric field governing the mesh modification algorithms is constructed from an interpolation based error estimate using a modified Hessian of the magnitude of the velocity in the flow field. It provides a specification of the desired mesh size and orientation for the simplicial elements to refine and coarsen the space-time mesh while stretching the elements in preferred directions to reduce the number of mesh points necessary to achieve a solution of a given accuracy. The anisotropic meshing algorithms have been tested in 2-D, 3-D and 4-D with an analytical metric field and also with a simple heat transfer problem. The resulting element quality was found to be very high for the 2-D cases, comparable to those produced by methods found in the literature for the 3-D cases, but unsatisfactory for the 4-D cases. The ratio for the number of elements to the number of points in the mesh has been found to grow by a factor of about 3 when increasing the space dimension by one. To the best of our knowledge, this is the first time that mesh modifications were shown to operate in a dimension higher than 3 with the ability to modify the boundary mesh. In contrast, previously existing methods that operate on higher dimensional meshes cannot keep track of the boundary of the domain. Verifications for the unified space-time adaptive finite element method have been done using manufactured solutions for a linear heat equation and for the incompressible Navier-Stokes equations. The behaviour of the L2 norm, computed on the entire space-time domain, shows a good agreement between the numerical and the analytical solutions indicating that the unsteady mesh adaptation procedure can control the discretization error in both space and time. Applications to the incompressible Navier-Stokes problems have been shown with unsteady 2-D flows to demonstrate the ability of the method. Numerical solutions are presented for the flow past a circular cylinder at a Reynolds number of 100, the flow over a backward facing step at a Reynolds number of 800 and the flow in a lid-driven cavity at a Reynolds number of 400. For these test cases, the Picard method with the combined mesh adaptation strategy and solution interpolation, introduced to provide a restart solution for the solver after mesh adaptation, exhibit excellent convergence behaviour.