Hans-Görg Roos

Numerical treatment of partial differential equations

Contents Notation 1 Basics 1.1 Classification and Correctness 1.2 Fouriers Method, Integral Transforms 1.3 Maximum Principle, Fundamental Solution 2 Finite Difference Methods 2.1 Basic Concepts 2.2 Illustrative Examples 2.3 Transportation Problems and Conservation Laws 2.4 Elliptic Boundary Value Problems 2.5 Finite Volume Methods as Finite Difference Schemes 2.6 Parabolic Initial-Boundary Value Problems 2.7 Second-Order Hyperbolic Problems 3 Weak Solutions 3.1 Introduction 3.2 Adapted Function Spaces 3.3 VariationalEquationsand conformingApproximation 3.4 WeakeningV-ellipticity 3.5 NonlinearProblems 4 The Finite Element Method 4.1 A First Example 4.2 Finite-Element-Spaces 4.3 Practical Aspects of the Finite Element Method 4.4 Convergence of Conforming Methods 4.5 NonconformingFiniteElementMethods 4.6 Mixed Finite Elements 4.7 Error Estimators and adaptive FEM 4.8 The Discontinuous Galerkin Method 4.9 Further Aspects of the Finite Element Method 5 Finite Element Methods for Unsteady Problems 5.1 Parabolic Problems 5.2 Second-Order Hyperbolic Problems 6 Singularly Perturbed Boundary Value Problems 6.1 Two-Point Boundary Value Problems 6.2 Parabolic Problems, One-dimensional in Space 6.3 Convection-Diffusion Problems in Several Dimensions 7 Variational Inequalities, Optimal Control 7.1 Analytic Properties 7.2 Discretization of Variational Inequalities 7.3 Penalty Methods 7.4 Optimal Control of PDEs 8 Numerical Methods for Discretized Problems 8.1 Some Particular Properties of the Problems 8.2 Direct Methods 8.3 Classical Iterative Methods 8.4 The Conjugate Gradient Method 8.5Multigrid Methods 8.6 Domain Decomposition, Parallel Algorithms Bibliography: Textbooks and Monographs Bibliography: Original Papers Index

Sufficient conditions for uniform convergence on layer-adapted grids

Abstract.We study convergence properties of the simple upwind difference scheme and a Galerkin finite element method on generalized Shishkin grids. We derive conditions on the mesh-characterizing function that are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. These conditions are easy to check and enable one to immediately deduce the rate of convergence. Numerical experiments support these theoretical results and indicate that the estimates are sharp. The analysis is set in one dimension, but can be easily generalized to tensor product meshes in 2D.

The Sdfem for a Convection-diffusion Problem with Two Small Parameters

Abstract A singularly perturbed convection-diffusion problem with two small parameters is considered. The problem is solved using the streamline-diffusion finite element method on a Shishkin mesh. We prove that the method is convergent independently of the perturbation parameters. Numerical experiments support these theoretical results.

A Priori Estimates for the Solution of Convection-Diffusion Problems and Interpolation on Shishkin Meshes

The solution of singularly perturbed convection-diffusion problems can be split into a regular and a singular part containing the boundary layer terms. In dimensions n = 1 and n = 2, sharp estimates of the derivatives of both parts up to order 2 are given. The results are applied to estimate the interpolation error for the solution on Shishkin meshes for piecewise bilinear finite elements on rectangles and piecewise linear elements on triangles. Using the anisotropic interpolation theory it is proved that the interpolation problem on Shishkin meshes is quasi-optimal in L. and in the energy norm.

Defect correction on Shishkin-type meshes

We consider a defect-correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for a model singularly perturbed convection–diffusion problem in one dimension on a class of Shishkin-type meshes. The method is shown to be convergent, uniformly in the diffusion parameter ε, of second order in the discrete maximum norm. As a corollary we derive error bounds for the gradient approximation of the upwind scheme. Numerical experiments support our theoretical results.

A second-order scheme for singularly perturbed differential equations with discontinuous source term

Abstract A Galerkin finite element method that uses piecewise linear functions on Shishkin- and Bakhvalov–Shishkin-type of meshes is applied to a linear reaction-diffusion equation with discontinuous source term. The method is shown to be convergent, uniformly in the perturbation parameter, of order N –2 ln2 N for the Shishkin-type mesh and N –2 for the Bakhvalov–Shishkin-type mesh, where N is the mesh size number. Numerical experiments support our theoretical results.

The streamline-diffusion method for a convection-diffusion problem with a point source

A singularly perturbed convection-diffusion problem with a point source is considered. The problem is solved using the streamline-diffusion finite element method on a class of Shishkin-type meshes. We prove that the method is almost optimal with second order of convergence in the maximum norm, independently of the perturbation parameter. We also prove the existence of superconvergent points for the first derivative. Numerical experiments support these theoretical results.

Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

We present new results in the numerical analysis of singularly perturbed convection-diffusion-reaction problems that have appeared in the last five years. Mainly discussing layer-adapted meshes, we present also a survey on stabilization methods, adaptive methods, and on systems of singularly perturbed equations.

The Capriciousness of Numerical Methods for Singular Perturbations

A collection of typical examples shows the exotic behavior of numerical methods when applied to singular perturbation problems. While standard meshes are used in the first six examples, even on layer-adapted meshes several surprising phenomena are shown to occur.

Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control

We consider an optimal control problem with an 1D singularly perturbed differential state equation. For solving such problems one uses the enhanced system of the state equation and its adjoint form. Thus, we obtain a system of two convection- diffusion equations. Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain. We proof uniform error estimates for this method on meshes of Shishkin type. We present numerical results supporting our analysis.