Effective error estimates for the numerical solution of Fredholm integral equations (Q582851)

The paper gives a theoretical basis for an algorithm solving Fredholm integral equations of the second kind with a guaranteed solution accuracy. For this purpose an effective a posteriori error estimation is derived. Let us write the equation in the form \(x-Kx=y\), where \((Kx)(t)=\int^{1}_{0}k(t,s)x(s)ds,\) assuming that the kernel k(t,s) and the right-hand side y(t) of the equation are both differentiable as much as necessary. Let us denote by \(x_ n\) an approximate solution of the equation. Supposing the equation is not singular, so that the operator (I-K) has a bounded inverse, the residual \(\rho_ n=(I-K)x_ n-y\) may be used to error estimation \(\| x-x_ n\| \leq \| (I- K)^{-1}\| \| \rho_ n\|.\) The real problem is to obtain realistic (not too pessimistic) and effectively computable bounds on \(\| (I-K)^{-1}\|.\) Some computable error bounds have been known for some time, but they did not work well for large \(\| K\|\). The authors explore an own approach developed by the second author [Int. J. Comput. Math. 24, 74-81 (1988; Zbl 0659.65140)], using a degenerate kernel approximation based on piecewise linear polynomial on a mesh with equally-spaced knots. They obtain simple lower and upper bound estimations of \(\| (I-K)^{- 1}\|\). In many practical cases these estimates differ relatively little one from other. The method described in the paper was successfully implemented in an algorithm and a FORTRAN program solving the above integral equation; the details are described in a separate report by the second and the third author [A program for solving Fredholm integral equation with guaranteed accuracy. (Technical Report CSE-88-5, Univ. of California, Davis) (1988)]. The program automatically chooses the mesh size to the specified accuracy.