A unified framework is presented for the error concepts stability, consistency and convergence in the context of discretizations of, in particular, boundary integral equations. For well-posed problems the Lax-Richtmyer equivalence theorem relates these concepts. However, discretizations of boundary integral equations do not always give well-posed problems. Ill-posedness typically occurs for boundary integral equations of the first kind. These problems can have a unique solution, but the continuous dependency on the data may be absent, translating into unbounded operators. The definitions of the error concepts and the related equivalence theorems are generalized to include discretizations of ill-posed problems where the ill-posedness is due to unbounded operators. Of course, well-posed problems are included in the theory. The theory is set up in the general framework of linear operator equations on normed vector spaces, so that the theory is not restricted to integral equations.
|Publisher||Faculty of Mathematical Sciences, University of Twente|