Pre- and postprocessing techniques for determining goodness of computational meshes

Research in error estimation, mesh conditioning, and solution enhancement for finite element, finite difference, and finite volume methods has been incorporated into AUDITOR, a modern, user-friendly code, which operates on 2D and 3D unstructured neutral files to improve the accuracy and reliability of computational results. Residual error estimation capabilities provide local and global estimates of solution error in the energy norm. Higher order results for derived quantities may be extracted from initial solutions. Within the X-MOTIF graphical user interface, extensive visualization capabilities support critical evaluation of results in linear elasticity, steady state heat transfer, and both compressible and incompressible fluid dynamics

Residual estimates for post-processors in elliptic problems

In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that `tweaks' a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Enhancing filter and Superconvergent Patch Recovery. Extensive numerical tests are conducted that confirm our analytic findings.Comment: 25 pages, 17 figure

Self-Adaptive Methods for PDE

[no abstract available

Robust Methods for Multiscale Coarse Approximations of Diffusion Models in Perforated Domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space which depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Combined with domain decomposition (DD) methods, the coarse space leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains. Numerical experiments showcase the increased precision of the coarse approximation as well as the efficiency and scalability of the coarse space as a component of a DD algorithm.Comment: 32 pages, 14 figures, submitted to Journal of Computational Physic

Cut Finite Element Methods on Overlapping Meshes: Analysis and Applications

This thesis deals with both analysis and applications of cut finite element methods (CutFEMs) on overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a background mesh at the bottom and one or more overlapping meshes that are stacked on top of it. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing geometry. The main content of the thesis is the five appended papers. The thesis consists of an analysis part and an applications part.In the analysis part (Paper I and Paper II), we consider cut finite element methods on overlapping meshes for a time-dependent\ua0parabolic model problem: the heat equation on two overlapping meshes, where one mesh is allowed to move around on top of the other. In Paper I, the overlapping mesh is prescribed a cG(1) movement, meaning that its location as a function of time is continuous and piecewise linear. The cG(1) mesh movement results in a space-time discretization for which existing analysis methodologies either fail or are unsuitable. We therefore propose, to the best of our knowledge, a new energy analysis framework that is general enough to be applicable to the current setting. In Paper II, the overlapping mesh is prescribed a dG(0) movement, meaning that its location as a function of time is\ua0discontinuous\ua0and\ua0piecewise constant. The dG(0) mesh movement results in a space-time discretization for which existing analysis methodologies work with some modifications to handle the shift in the overlapping mesh\u27s location at discrete times.The applications part (Paper III, IV, and V) concerns cut finite element methods on overlapping meshes for\ua0stationary\ua0PDE-problems. We consider two potential applications for CutFEM on overlapping meshes. The first application, presented in Paper III, presents methodology for evaluating configurations of buildings based on wind and view. The wind model is based on a CutFEM on overlapping meshes for Stokes equations. The second application, presented in Paper IV and Paper V, concerns a software application (app). The app lets a user define and solve physical problems governed by PDEs in an immersive and interactive augmented reality environment