Approximate L² Error Control by Solution Post-Processing for Finite Element Solutions of PDEs with Higher-Order Adaptive Methods

Abstract

With the substantial computing resources available today, computational fluid dynamics simulations allow scientists and engineers to simulate physical problems very accurately. However, achieving this accuracy requires a sufficiently refined computational mesh, which is a primary driver for the high cost of complex simulations. Mesh adaptation methods provide an automated way to determine the regions where a mesh needs the most refinement and generate a new mesh that efficiently targets these regions. In this thesis, we build on previous work in a posteriori error estimation and mesh adaptation for finite element methods to propose a new mesh adaptation method based on L² error control by solution post-processing. A key feature of our method is its natural extension to higher-order discretizations while providing a problem-independent adaptation methodology. Problem-independent adaptation methods do not depend on specific information about the partial differential equation (PDE) problem being solved, and can therefore be applied to a wide range of problems without modification. We present numerical results applying the approximate L² error control method to a two-dimensional advection-diffusion problem with anisotropic features. These results demonstrate the proposed method’s ability to generate well-adapted anisotropic meshes for solutions with polynomial orders 1, 2, and 3. We also apply the approximate L² error control method to a more complex two-dimensional Reynolds-Averaged Navier-Stokes problem with turbulent flow over a flat plate. We compare the convergence of the drag coefficient and the characteristics of adapted meshes obtained with the proposed method and with an output-based adaptation approach. As expected, the approximate L² error control method is not as effective as the output-based approach in reaching a converged drag coefficient value, but it nevertheless demonstrates the ability to effectively control the approximate L² error in the Mach field.