Numerical Methods for Elliptic Problems with Singularities

The following sections are included:

• Preface

• Acknowledgments

• Contents

Please refer to full text.

In this part, boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations (see also Li (1986a, 1988), Li and Mathon (1990a, b) and Li et al. (1987)). Piecewise expansions of particular solutions on subdomains are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is provided and error estimates are derived in a natural norm. Stability analysis is also given for choosing better divisions of solution domains. Besides, numerical experiments are reported for the Motz problem, an infinite domain problem, and an interface problem, which yield extremely accurate solutions and very small condition numbers with only modest computational effort.

The following sections are included:

• Introduction

• Notations and Preliminaries

• Approximation Problems

• Error Estimates

• Numerical Experiments

• Concluding Remarks

The following sections are included:

• Introduction

• Particular Solutions on Subdomains
  • Near the crack singularity
  • Near infinity
  • Near the mild singularity

• A Boundary Method

• Numerical Experiments

The following sections are included:

• Introduction

• Complete Eigenfunctions for the Sturm-Liouville System

• Boundary Approximations

• Error Analysis

• Bounds for the Constant k_m,n

• Integral Approximation

• Stability Analysis

• Numerical Experiments

For solving elliptic problems with singularities, a nonconforming combined approach of the Ritz-Galerkin and finite element methods is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where singularities exist; and piecewise linear (or bilinear) functions are chosen to be admissible functions in the rest of that solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element modes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used…

The following sections are included:

• Introduction

• A Nonconforming Approach

• Error Estimates and a New Coupling Strategy

• Stability of Numerical Solutions

• Numerical Experiments for Motz's Problem

• Concluding Remarks

The following sections are included:

• Description of Methods

• Error Bounds and Coupling Strategy

• Proof of Theorem 6.2.1

• Numerical Experiments

The following sections are included:

• Description of Methods

• Error Estimates and Coupling Strategy

• Numerical Experiments

The following sections are included:

• Introduction

• Finite Difference Methods

• Combinations with Finite Difference Methods
  • Combined methods
  • A combination of Ritz-Galerkin and finite difference methods

• Error Estimates for Motz's Problem in the Norm ‖ · ‖_h

• Error Estimates for Motz's Problem in the Norm

• Numerical Experiments

The following sections are included:

• Epilogue

• References

• Glossary of Symbols

• Index