Interior a posteriori error estimates for time discrete approximations of parabolic problems

a posteriori error estimates for time discrete approximations o

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media

A conservative flux postprocessing algorithm is presented for both steady-state and dynamic flow models. The postprocessed flux is shown to have the same convergence order as the original flux. An arbitrary flux approximation is projected into a conservative subspace by adding a piecewise constant correction that is minimized in a weighted $L^2$ norm. The application of a weighted norm appears to yield better results for heterogeneous media than the standard $L^2$ norm which has been considered in earlier works. We also study the effect of different flux calculations on the domain boundary. In particular we consider the continuous Galerkin finite element method for solving Darcy flow and couple it with a discontinuous Galerkin finite element method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table

Discontinuous Galerkin methods for convection-diffusion equations and applications in petroleum engineering

This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in Linfinity(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in [J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics (2019)] showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out

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