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Some novel numerical techniques for an inverse Cauchy problem. (English) Zbl 1446.65151
Summary: In this paper, we are interested in solving an elliptic inverse Cauchy problem. As it is well known this problem is one of highly ill posed problem in J. Hadamard’s sense [Lectures on Cauchy’s problem in linear partial differential equations. New York: Dover Publications (1952; Zbl 0049.34805)]. We first establish formally a relationship between the Cauchy problem and an interface problem illustrated in a rectangular structure divided into two domains. This relationship allows us to use classical methods of non-overlapping domain decomposition to develop some regularizing and stable algorithms for solving elliptic inverse Cauchy problem. Taking advantage of this relationship we reformulate this inverse problem into a fixed point one, based on Steklov-Poincaré operator. Thus, using the topological degree of Leray-Schauder we show an existence result. Finally, the efficiency and the accuracy of the developed algorithms are discussed.
MSC:
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47H11 Degree theory for nonlinear operators
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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