An iterative boundary element method for Cauchy inverse problems (Q699842)

The paper is concerned with the Cauchy problem for Laplace's equation: \(\Delta u=0\) in \(\Omega\), \(u=\varphi_d\) on \(\Gamma_d\), \(\partial u/\partial n=\psi_d\) on \(\Gamma_d\) where \(\Omega\) is an open set in \(R^2\) or \(R^3\) with a smooth boundary \(\Gamma \supset \Gamma_d\). The authors study the proximal iterative process \(u^0=0\), \(u^{k+1} \in H(\Gamma): J^k(u^{k+1}) \leq J^k(v) \forall v \in H(\Gamma)\) with \(J^k (v)=\|v-\phi_d \|^2_{\Gamma_d}+ c \|v-u^k \|^2_{\Gamma}, c>0\), \(\phi_d=(\varphi_d, \psi_d) \in H^{1/2}(\Gamma_d) \times H^{-1/2}(\Gamma_d)\). Here \(H(\Gamma)\) is the space of traces \((\nu|_{\Gamma}, \partial \nu/\partial n |_{\Gamma})\) of functions \(\nu\) from \(\{ \nu \in H^1(\Omega): \Delta \nu=0 \}\) and \(\|\cdot \|_{\Gamma}\) stands for the norm of the space \(H^{1/2}(\Gamma) \times H^{-1/2}(\Gamma)\). The sequence \(\{ u^k \}\) is proved to converge to \(u|_{\Gamma}\) weakly in \(H(\Gamma)\). The proposed algorithm is implemented in the framework of boundary elements. The results of numerical tests are presented.