Book contents
- Frontmatter
- Contents
- Preface
- 1 A brief discussion of integral equations
- 2 Degenerate kernel methods
- 3 Projection methods
- 4 The Nyström method
- 5 Solving multivariable integral equations
- 6 Iteration methods
- 7 Boundary integral equations on a smooth planar boundary
- 8 Boundary integral equations on a piecewise smooth planar boundary
- 9 Boundary integral equations in three dimensions
- Appendix: Results from functional analysis
- Bibliography
- Index
8 - Boundary integral equations on a piecewise smooth planar boundary
Published online by Cambridge University Press: 27 March 2010
- Frontmatter
- Contents
- Preface
- 1 A brief discussion of integral equations
- 2 Degenerate kernel methods
- 3 Projection methods
- 4 The Nyström method
- 5 Solving multivariable integral equations
- 6 Iteration methods
- 7 Boundary integral equations on a smooth planar boundary
- 8 Boundary integral equations on a piecewise smooth planar boundary
- 9 Boundary integral equations in three dimensions
- Appendix: Results from functional analysis
- Bibliography
- Index
Summary
We continue with the study of the numerical solution of boundary integral equation reformulations of boundary value problems for Laplace's equation in the plane. The boundary value problems to be studied are the same as were given in §7.1 of Chapter 7, and much of the notation is also the same as that used earlier. In the present chapter we consider such problems on domains D with a boundary S that is only piecewise continuously differentiable. For such a piecewise smooth boundary S, the properties of the BIE of Chapter 7 are changed in significant ways. In this chapter we give ways of studying and numerically solving such BIE. The first major result in understanding such BIE goes back to the work of Radon in [440] in 1919, but most of the important numerical analysis papers on this problem date from 1984 onwards. Some of these papers are cited in the developments of this chapter, and a further discussion of the literature is given at the end of the chapter.
In §8.1 we discuss the properties of the BIE when S is only piecewise smooth. There are significant differences from the smooth boundary case, both in the behavior of the solutions and in the properties of the integral operators. We give some results on the qualitative behavior of the solutions of the BIE in the neighborhood of corner points of the bounday.
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- The Numerical Solution of Integral Equations of the Second Kind , pp. 384 - 426Publisher: Cambridge University PressPrint publication year: 1997