Superconvergence. (English) Zbl 0759.65091

Numerical solution of integral equations, Math. Concepts Methods Sci. Eng. 42, 35-70 (1990).
[For the entire collection see Zbl 0722.00033.]
The abstract of this paper is just one sentence: ‘Superconvergence properties of approximation methods for integral equations of the second kind are reviewed.’
The paper is, hence, a review article on an equation \(y-Ky=f\) where \(K\) is a standard linear integral operator with a kernel \(k(t,s)\). For a definition of superconvergence in the sense of the author we quote from the introduction:
‘Suppose that the equation is solved approximately by the Galerkin or collocation method, with the approximate solution \(y_ h\) being a piecewise polynomial of order \(r\), on a partition with maximum mesh size \(h\). Then the best result we can hope to achieve, in an appropriate \(L_ p\) norm, is that \(\| y_ h-y\|_{L_ p}\) be of order \(O(h^ r)\). Sometimes, however, certain quantities derived from \(y_ h\) show a faster rate of convergence than this. Any quantity computed from \(y_ h\) that has an order of convergence higher than \(O(h^ r)\) is said to be superconvergent.’
The ‘quantities’ which are discussed are \[ y^ 1_ h=f+Ky_ h\tag{1} \] and a functional \[ \int^ b_ ay_ h(t)g(t)dt=(y,g),\quad g \text{ a fixed function}.\tag{2} \] Superconvergence of (1) is treated for both Galerkin method and collocation method. Superconvergence for (2) is considered for Galerkin approximation only. The key idea of the proofs is that both methods can be viewed as a projection method in the form \(P_ h(y_ h-Ky_ h-f)=0\) with a suitable projection \(P_ h\) for both methods. Then the identity \(y^ 1_ h-y=(I-KP_ h)^{-1}(KP_ h-K)y\) is easily obtained. This implies in a Banach space setting an inequality \(\| y^ 1_ h-y\|\leq c\| KP_ h-K\|\cdot\| y_ h- y\|\). The fact that \(\| KP_ h-K\|\to 0\) as \(h\to 0\) produces the superconvergence. In the case of collocation method, the Banach space setting inequality is not helpful. The author rather uses the inequality \(\| y^ 1_ h-y\|\leq c\| KP_ hy-Ky\|.\)
This is the abstract background material which is a start for a series of applications for concrete integral equations described in the text. Some notes on nonlinear equations finish the paper.
Reviewer: E.Bohl (Konstanz)


65R20 Numerical methods for integral equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65J10 Numerical solutions to equations with linear operators
45B05 Fredholm integral equations
47A50 Equations and inequalities involving linear operators, with vector unknowns


Zbl 0722.00033