Stabilitätskriterien für Näherungsverfahren bei singulären Integralgleichungen in \(L^ p\) (Q1112580)

Collocation and Galerkin methods are considered for the numerical solution of a class of singular integral equations. The operator is the sum of a multiplication operator and the product of a multiplication operator and a singular integral operator defined in terms of a Cauchy principal value defined on closed curves with a Lipschitz continuous tangential direction. The principal issue is the stability, i.e. the uniform invertibility of the matrices resulting from the use of the collocation or Galerkin procedure. In this paper, the authors establish a variety of stability results in \(L^ p\) spaces for \(1<p<\infty\). The principal tool is a representation of the discrete operators as sums of products of diagonal and circulant matrices.