Quadrature methods for strongly elliptic equations of negative order on smooth closed curves. (English) Zbl 0796.65124

Saranen, J.; Schroderus, L.

SIAM J. Numer. Anal. 30, No. 6, 1769-1795 (1993).

The authors introduce and analyze a special, very simple quadrature method for strongly elliptic pseudo-differential equations of negative order. All classical boundary integral equations belong to this class. The method is based on an approximation of the integral operators by the trapezoidal rule and application of the \(\varepsilon\)-collocation. It is proved to have the stability and the rate \(O(h^{-p})\) of convergence. Here \(p\) is the order of the operator. For some special values of \(\varepsilon\) the rate \(O(h^{-p+1})\) can be achieved. Numerical experiments with an operator of order –3 and a \(2 \times 2\)-system of mixed order show agreement with the theoretical rates of convergence.

Reviewer: A.Pomp (Stuttgart)

MSC:

65N38	Boundary element methods for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65R20	Numerical methods for integral equations
35S15	Boundary value problems for PDEs with pseudodifferential operators

Keywords:

collocation; numerical experiments; quadrature method; strongly elliptic pseudo-differential equations; negative order; boundary integral equations; stability; convergence