Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems

Considering a singularly perturbed convection-diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gau{\ss}-Lobatto type for higher-order streamline diffusion FEM. We show a useful connection between two different types of interpolation, namely a vertex-edge-cell interpolant and a pointwise interpolant. Moreover, different postprocessing operators are analysed and applied to model problems.Comment: 19 page

Greenʼs function estimates for a singularly perturbed convection–diffusion problem

We consider a singularly perturbed convection-diusion problem posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Green's function and its first- and second-order derivatives are derived in the L1 norm. The dependence of these estimates on the small diusion parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem

Error analysis of the Galerkin FEM in L 2 -based norms for problems with layers: On the importance, conception and realization of balancing

In the present thesis it is shown that the most natural choice for a norm for the analysis of the Galerkin FEM, namely the energy norm, fails to capture the boundary layer functions arising in certain reaction-diffusion problems. In view of a formal Definition such reaction-diffusion problems are not singularly perturbed with respect to the energy norm. This observation raises two questions: 1. Does the Galerkin finite element method on standard meshes yield satisfactory approximations for the reaction-diffusion problem with respect to the energy norm? 2. Is it possible to strengthen the energy norm in such a way that the boundary layers are captured and that it can be reconciled with a robust finite element method, i.e.~robust with respect to this strong norm? In Chapter 2 we answer the first question. We show that the Galerkin finite element approximation converges uniformly in the energy norm to the solution of the reaction-diffusion problem on standard shape regular meshes. These results are completely new in two dimensions and are confirmed by numerical experiments. We also study certain convection-diffusion problems with characterisitc layers in which some layers are not well represented in the energy norm. These theoretical findings, validated by numerical experiments, have interesting implications for adaptive methods. Moreover, they lead to a re-evaluation of other results and methods in the literature. In 2011 Lin and Stynes were the first to devise a method for a reaction-diffusion problem posed in the unit square allowing for uniform a priori error estimates in an adequate so-called balanced norm. Thus, the aforementioned second question is answered in the affirmative. Obtaining a non-standard weak formulation by testing also with derivatives of the test function is the key idea which is related to the H^1-Galerkin methods developed in the early 70s. Unfortunately, this direct approach requires excessive smoothness of the finite element space considered. Lin and Stynes circumvent this problem by rewriting their problem into a first order system and applying a mixed method. Now the norm captures the layers. Therefore, they need to be resolved by some layer-adapted mesh. Lin and Stynes obtain optimal error estimates with respect to the balanced norm on Shishkin meshes. However, their method is unable to preserve the symmetry of the problem and they rely on the Raviart-Thomas element for H^div-conformity. In Chapter 4 of the thesis a new continuous interior penalty (CIP) method is present, embracing the approach of Lin and Stynes in the context of a broken Sobolev space. The resulting method induces a balanced norm in which uniform error estimates are proven. In contrast to the mixed method the CIP method uses standard Q_2-elements on the Shishkin meshes. Both methods feature improved stability properties in comparison with the Galerkin FEM. Nevertheless, the latter also yields approximations which can be shown to converge to the true solution in a balanced norm uniformly with respect to diffusion parameter. Again, numerical experiments are conducted that agree with the theoretical findings. In every finite element analysis the approximation error comes into play, eventually. If one seeks to prove any of the results mentioned on an anisotropic family of Shishkin meshes, one will need to take advantage of the different element sizes close to the boundary. While these are ideally suited to reflect the solution behavior, the error analysis is more involved and depends on anisotropic interpolation error estimates. In Chapter 3 the beautiful theory of Apel and Dobrowolski is extended in order to obtain anisotropic interpolation error estimates for macro-element interpolation. This also sheds light on fundamental construction principles for such operators. The thesis introduces a non-standard finite element space that consists of biquadratic C^1-finite elements on macro-elements over tensor product grids, which can be viewed as a rectangular version of the C^1-Powell-Sabin element. As an application of the general theory developed, several interpolation operators mapping into this FE space are analyzed. The insight gained can also be used to prove anisotropic error estimates for the interpolation operator induced by the well-known C^1-Bogner-Fox-Schmidt element. A special modification of Scott-Zhang type and a certain anisotropic interpolation operator are also discussed in detail. The results of this chapter are used to approximate the solution to a recation-diffusion-problem on a Shishkin mesh that features highly anisotropic elements. The obtained approximation features continuous normal derivatives across certain edges of the mesh, enabling the analysis of the aforementioned CIP method.:Notation 1 Introduction 2 Galerkin FEM error estimation in weak norms 2.1 Reaction-diffusion problems 2.2 A convection-diffusion problem with weak characteristic layers and a Neumann outflow condition 2.3 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers 2.3.1 Weakly imposed characteristic boundary conditions 2.4 Numerical experiments 2.4.1 A reaction-diffusion problem with boundary layers 2.4.2 A reaction-diffusion problem with an interior layer 2.4.3 A convection-diffusion problem with characteristic layers and a Neumann outflow condition 2.4.4 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers 3 Macro-interpolation on tensor product meshes 3.1 Introduction 3.2 Univariate C1-P2 macro-element interpolation 3.3 C1-Q2 macro-element interpolation on tensor product meshes 3.4 A theory on anisotropic macro-element interpolation 3.5 C1 macro-interpolation on anisotropic tensor product meshes 3.5.1 A reduced macro-element interpolation operator 3.5.2 The full C1-Q2 interpolation operator 3.5.3 A C1-Q2 macro-element quasi-interpolation operator of Scott-Zhang type on tensor product meshes 3.5.4 Summary: anisotropic C1 (quasi-)interpolation error estimates 3.6 An anisotropic macro-element of tensor product type 3.7 Application of macro-element interpolation on a tensor product Shishkin mesh 4 Balanced norm results for reaction-diffusion 4.1 The balanced finite element method of Lin and Stynes 4.2 A C0 interior penalty method 4.3 Galerkin finite element method 4.3.1 L2-norm error bounds and supercloseness 4.3.2 Maximum-norm error bounds 4.4 Numerical verification 4.5 Further developments and summary Reference

Analytical investigations and numerical experiments for singularly perturbed convection-diffusion problems with layers and singularities using a newly developed FE-software

In the field of singularly perturbed reaction- or convection-diffusion boundary value problems the research area of a priori error analysis for the finite element method, has already been thoroughly investigated. In particular, for mesh adapted methods and/or various stabilization techniques, works have been done that prove optimal rates of convergence or supercloseness uniformly in the perturbation parameter epsilon. Commonly, however, it is assumed that the exact solution behaves nicely in that it obeys certain regularity assumptions although in general, e.g. due to corner singularities, these regularity requirements are not satisfied. So far, insufficient regularity has been met by assuming compatibility conditions on the data. The present thesis originated from the question: What can be shown if these rather unrealistic additional assumptions are dropped? We are interested in epsilon-uniform a priori estimates for convergence and superconvergence that include some regularity parameter that is adjustable to the smoothness of the exact solution. A major difficulty that occurs when seeking the numerical error decay is that the exact solution is not known. Since we strive for reliable rates of convergence we want to avoid the standard approach of the "double-mesh principle". Our choice is to use reference solutions as a substitute for the exact solution. Numerical experiments are intended to confirm the theoretical results and to bring further insights into the interplay between layers and singularities. To computationally realize the thereby arising demanding practical aspects of the finite element method, a new software is developed that turns out to be particularly suited for the needs of the numerical analyst. Its design, features and implementation is described in detail in the second part of the thesis

Mixed formulations for the convection-diffusion equation

This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG.This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG

Singularly perturbed problems modelling reaction - convection - diffusion processes

In this thesis, parameter-uniform numerical methods for certain classes of singularly perturbed differential equations with two small parameters are studied. We initially consider a class of two-parameter ordinary differential equations Parameter explicit bounds on the solution and its derivatives are derived. The solution is decomposed into a sum of regular and singular components and based on this decomposition we construct a numerical algorithm consisting of an upwind finite difference operator and an appropriately chosen piecewise-umform mesh. Parameter-uniform convergence of the numerical approximations is established Some numerical results are given to illustrate this convergence. Two-parameter parabolic and elliptic partial differential equations are considered. We derive parameter explicit bounds on the solutions and their derivatives for both problems, these bounds are analogous to those obtained for the ordinary differential equation. The solutions are decomposed into a sum of regular and singular components but for both problems this decomposition differs from that for the ordinary differential equation. In both cases a numerical algorithm based on an upwind finite difference operator and an appropriate piecewise-umform mesh is constructed. In the case of the parabolic problem, parameter-uniform error bounds for the numerical approximations are established and numerical results illustrating this convergence are given With the elliptic. problem, we show that, given certain assumptions and conjectures our numerical method is parameterunifor