Martin Stynes

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

Steady-state convection-diffusion problems

In convection-diffusion problems, transport processes dominate while diffusion effects are confined to a relatively small part of the domain. This state of affairs means that one cannot rely on the formal ellipticity of the differential operator to ensure the convergence of standard numerical algorithms. Thus new ideas and approaches are required. The survey begins by examining the asymptotic nature of solutions to stationary convection-diffusion problems. This provides a suitable framework for the understanding of these solutions and the difficulties that numerical techniques will face. Various numerical methods expressly designed for convection-diffusion problems are then presented and extensively discussed. These include finite difference and finite element methods and the use of special meshes.

On the Stability of Residual-Free Bubbles for Convection-Diffusion Problemsand their Approximation by a Two-Level Finite Element Method.

We consider the Galerkin finite element method for partial diffferential equations in two dimensions, where the finite-dimensional space used consists of piecewise (isoparametric) polynomials enriched with bubble functions. Writing

The midpoint upwind scheme

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The SDFEM for a Convection-Diffusion Problem with a Boundary Layer: Optimal Error Analysis and Enhancement of Accuracy

for the differential operator, we show that for elliptic convection-diffusion problems, the component of the bubble enrichment that stabilizes the method is essentially equivalent to a Petrov-Galerkin method with an

A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem

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A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions

-spline (exponentially fitted) trial space and piecewise polynomial test space; the remaining component of the bubble influences the accuracy of the method. A stability inequality recently obtained by Brezzi, Franca and Russo for a limiting case of bubbles applied to convection-diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the SDFEM/SUPG method, thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method. When the trial functions are piecewise linear, we show that residual-free bubbles are as stable as SDFEM/SUPG, and we extend this stability inequality to include positive mesh-Peclet numbers in the convection-dominated regime. Approximate computations of the residual-free bubbles are performed using a two-level finite element method.

Numerical methods on Shishkin meshes for linear convection–diffusion problems

Abstract A modified upwind scheme is considered for a singularly perturbed two-point boundary value problem whose solution has a single boundary layer. The scheme is analyzed on an arbitrary mesh. It is then analyzed on a Shishkin mesh and precise convergence bounds are obtained, which show that the scheme is superior to the standard upwind scheme. A variant of the scheme on the same Shishkin mesh is proved to achieve even better convergence behaviour.

A finite difference analysis of a streamline diffusion method on a Shishkin mesh

The streamline-diffusion finite element method (SDFEM) is applied to a convection-diffusion problem posed on the unit square, using a Shishkin rectangular mesh with piecewise bilinear trial functions. The hypotheses of the problem exclude interior layers but allow exponential boundary layers. An error bound is proved for

A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems

\|u^I-u^N\|_{SD}