The material of earlier chapters was illustrated with integral equations for functions of a single variable. In this chapter we develop interpolation and numerical integration tools and use them with the projection and Nyström methods, developed in Chapters 3 and 4, to solve multivariable integral equations. Our principal interest will be the solution of integral equations defined on surfaces in R3, with an eye towards solving boundary integral equations on such surfaces. The solution of boundary integral equations on piecewise smooth surfaces is taken up in Chapter 9.

In §5.1 we develop interpolation and numerical integration formulas for multivariable problems, and these are applied to integral equations over planar regions in §5.2. The interpolation and integration results of §5.1 are extended to surface problems in §5.3. Methods for the numerical solution of integral equations on surfaces are given in §§5.4 and 5.5.

Multivariable interpolation and numerical integration

Interpolation for functions of more than one variable is a large topic with applications to many areas. In this section we consider only those aspects of the subject that we need for our work in the numerical solution of integral equations. To simplify the notation and to make more intuitive the development, we consider only functions of two variables. Generalizations to functions in more than two variables should be fairly straightforward for the reader.

Applications of multivariable interpolation are generally based on first breaking up a large planar region R into smaller ones of an especially simple form, and then polynomial interpolation is carried out over these smaller regions.