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

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 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.;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.;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.