Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows

In this article we develop both the a priori and a posteriori error analysis of hp–version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain $\Omega \subset R^{d}, d$ = 2,3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm

A posteriori error control for discontinuous Galerkin methods for parabolic problems

We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which requires the derivation of novel a posteriori error bounds. For the analysis of the time-dependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure

Bilinear Immersed Finite Elements for Interface Problems

In this dissertation we discuss bilinear immersed finite elements (IFE) for solving interface problems. The related research works can be categorized into three aspects: (1) the construction of the bilinear immersed finite element spaces; (2) numerical methods based on these IFE spaces for solving interface problems; and (3) the corresponding error analysis. All of these together form a solid foundation for the bilinear IFEs. The research on immersed finite elements is motivated by many real world applications, in which a simulation domain is often formed by several materials separated from each other by curves or surfaces while a mesh independent of interface instead of a body-fitting mesh is preferred. The bilinear IFE spaces are nonconforming finite element spaces and the mesh can be independent of interface. The error estimates for the interpolation of a Sobolev function in a bilinear IFE space indicate that this space has the usual approximation capability expected from bilinear polynomials, which is O(h2) in L2 norm and O(h) in H1 norm. Then the immersed spaces are applied in Galerkin, finite volume element (FVE) and discontinuous Galerkin (DG) methods for solving interface problems. Numerical examples show that these methods based on the bilinear IFE spaces have the same optimal convergence rates as those based on the standard bilinear finite element for solutions with certain smoothness. For the symmetric selective immersed discontinuous Galerkin method based on bilinear IFE, we have established its optimal convergence rate. For the Galerkin method based on bilinear IFE, we have also established its convergence. One of the important advantages of the discontinuous Galerkin method is its flexibility for both p and h mesh refinement. Because IFEs can use a mesh independent of interface, such as a structured mesh, the combination of a DG method and IFEs allows a flexible adaptive mesh independent of interface to be used for solving interface problems. That is, a mesh independent of interface can be refined wherever needed, such as around the interface and the singular source. We also develop an efficient selective immersed discontinuous Galerkin method. It uses the sophisticated discontinuous Galerkin formulation only around the locations needed, but uses the simpler Galerkin formulation everywhere else. This selective formulation leads to an algebraic system with far less unknowns than the immersed DG method without scarifying the accuracy; hence it is far more efficient than the conventional discontinuous Galerkin formulations