A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

Magister Scientiae - MSc / Many chemical and physical problems are mathematically described by partial differential equations (PDEs). These PDEs are often highly nonlinear and therefore have no closed form solutions. Thus, it is necessary to recourse to numerical approaches to determine suitable approximations to the solution of such equations. For solutions possessing sharp spatial transitions (such as boundary or interior layers), standard numerical methods have shown limitations as they fail to capture large gradients. The method of lines (MOL) is one of the numerical methods used to solve PDEs. It proceeds by the discretization of all but one dimension leading to systems of ordinary di erential equations. In the case of time-dependent PDEs, the MOL consists of discretizing the spatial derivatives only leaving the time variable continuous. The process results in a system to which a numerical method for initial value problems can be applied. In this project we consider various types of singularly perturbed time-dependent PDEs. For each type, using the MOL, the spatial dimensions will be discretized in many different ways following fitted numerical approaches. Each discretisation will be analysed for stability and convergence. Extensive experiments will be conducted to confirm the analyses.

Philosophiae Doctor - PhD / Singularly perturbed problems have been studied extensively over the past few years from different perspectives. The recent research has focussed on the problems whose solutions possess interior layers. These interior layers appear in the interior of the domain, location of which is difficult to determine a-priori and hence making it difficult to investigate these problems analytically. This explains the need for approximation methods to gain some insight into the behaviour of the solution of such problems. Keeping this in mind, in this thesis we would like to explore a special class of numerical methods, namely, fitted finite difference methods to determine reliable solutions. As far as the fitted finite difference methods are concerned, they are grouped into two categories: fitted mesh finite difference methods (FMFDMs) and the fitted operator finite difference methods (FOFDMs). The aim of this thesis is to focus on the former. To this end, we note that FMFDMs have extensively been used for singularly perturbed two-point boundary value problems (TPBVPs) whose solutions possess boundary layers. However, they are not fully explored for problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions possess interior layers and to improve accuracy of these approximation methods via methods like Richardson extrapolation. Then we extend these two ideas to solve such singularly perturbed TPBVPs with variable diffusion coefficients. The overall approach is further extended to parabolic singularly perturbed problems having constant as well as variable diffusion coefficients. / 2023-08-31

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.

Magister Scientiae - MSc / Efficient numerical approaches for parameter dependent problems have been an inter- esting subject to numerical analysts and engineers over the past decades. This is due to the prominent role that these problems play in modeling many real life situations in applied sciences. Often, the choice and the e ciency of the approaches depend on the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These singularly perturbed problems (SPPs) are governed by integro-differential equations in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches zero, the solution undergoes fast transitions across narrow regions of the domain (termed boundary or interior layer) thus affecting the convergence of the standard numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical methods. This work seeks to investigate some "numerical methods that have been used to solve SPVIDEs. It also proposes alternative ones. The various numerical methods are composed of a fitted finite difference scheme used along with suitably chosen interpolating quadrature rules. For each method investigated or designed, we analyse its stability and convergence. Finally, numerical computations are carried out on some test examples to con rm the robustness and competitiveness of the proposed methods.

Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.