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On the efficient use of the Galerkin-method to solve Fredholm integral equations. (English) Zbl 0791.65101
Let $$\Gamma$$ be a piecewise smooth two-dimensional manifold in $$\mathbb{R}^ 3$$. The authors consider boundary integral equations of the form $\lambda (x)u(x)+Ku(x)=f(x),\quad x \in \Gamma, \tag{1}$ where $Ku(x):=\int_ \Gamma k(x,y,y-x) u(y)d \Gamma(y),\;k(x,y,y-x)={1 \over | y-x |^{s+t}} \sum_{| \alpha | \geq t} c_ \alpha (x,y)(y-x)^ \alpha,$ with $$s+t$$ odd, $$t \in N_ 0$$ and piecewise smooth functions $$c_ \alpha$$. This covers a wide class of integral equations (including hypersingular kernels) on piecewise smooth surfaces, approximated by splines of arbitrary degree. Using suitable coordinate transformations, the computation of the singular and nearly singular integrals arising in the discretization of (1) via the Galerkin method is discussed.
Furthermore, the panel-clustering method introduced by the first author and Z. P. Nowak [Numer. Math. 54, No. 4, 463-491 (1989; Zbl 0652.65031)] is developed for Galerkin discretizations and the corresponding error analysis is presented, yielding similar asymptotic estimates for the expense as shown in that paper for the collocation method. Numerical test calculations show that the reduction of the panel- clustering method is stronger for the Galerkin procedure than for the collocation method.

##### MSC:
 65R20 Numerical methods for integral equations 65N38 Boundary element methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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