Numerical solution of singularly perturbed convection–diffusion problem using parameter uniform B-spline collocation method

AbstractThis paper is concerned with a numerical scheme to solve a singularly perturbed convection–diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ɛ. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects

A collocation method on a Gartland-type mesh for a singularly perturbed reaction-diffusion problem

A singularly perturbed reaction-diffusion one-dimension problemis solved numerically by collocation method with quadratic [C^1]splines. Using appropriately recursively graded mesh the second order of convergence is obtained in the supremum norm uniformly, up to a logarithmic factor, in the singular perturbation parameter. The aim of this paper is to emphasize the importance of using the recursively graded mesh. The former results obtained by mentioned method on the smoothed Shishkin mesh are improved. Numerical experimentssupport the theoretical results. Numerical results for the reaction-diffusion problem in two-dimension show the same order of the convergence as well as in one-dimensional case, but theoretical analysis presents an ongoing work

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

Macro-element interpolation on tensor product meshes

A general theory for obtaining anisotropic interpolation error estimates for macro-element interpolation is developed revealing general construction principles. We apply this theory to interpolation operators on a macro type of biquadratic $C^1$ finite elements on rectangle grids which can be viewed as a rectangular version of the $C^1$ Powell-Sabin element. This theory also shows how interpolation on the Bogner-Fox-Schmidt finite element space (or higher order generalizations) can be analyzed in a unified framework. Moreover we discuss a modification of Scott-Zhang type giving optimal error estimates under the regularity required without imposing quasi uniformity on the family of macro-element meshes used. We introduce and analyze an anisotropic macro-element interpolation operator, which is the tensor product of one-dimensional $C^1-P_2$ macro interpolation and $P_2$ Lagrange interpolation. These results are used to approximate the solution of a singularly perturbed reaction-diffusion problem on a Shishkin mesh that features highly anisotropic elements. Hereby we obtain an approximation whose normal derivative is continuous along certain edges of the mesh, enabling a more sophisticated analysis of a continuous interior penalty method in another paper

A uniformly convergent quadratic B-spline based scheme for singularly perturbed degenerate parabolic problems.

[EN]In this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerate parabolic problems. The degenerative nature of the problem is due to the coefficient of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layer in the neighborhood of the left end side of the domain. The problem is semi-discretized using the Crank–Nicolson scheme, and then the quadratic spline basis functions are used to discretize the semi-discrete problem. A priori bounds for the solution (and its derivatives) of the continuous problem are given, which are necessary to analyze the error. A rigorous error analysis shows that the proposed method is boundary layer resolving and second-order parameter uniformly convergent. Some numerical experiments have been devised to support the theoretical findings and the effectiveness of the proposed scheme