Johansson, B. Tomas; Marin, Liviu Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation. (English) Zbl 1231.65237 CMC, Comput. Mater. Continua 13, No. 2, 153-189 (2009). Summary: We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of earlier work applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods. Cited in 3 Documents MSC: 65N38 Boundary element methods for boundary value problems involving PDEs 65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization Keywords:Helmholtz equation; inverse problem; Cauchy problem; alternating iterative algorithms; relaxation procedure; boundary element method (BEM) PDFBibTeX XMLCite \textit{B. T. Johansson} and \textit{L. Marin}, CMC, Comput. Mater. Continua 13, No. 2, 153--189 (2009; Zbl 1231.65237) Full Text: DOI