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Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations. (English) Zbl 1047.65097
A classical example of an inverse boundary value problem, i.e. the Cauchy problem, is considered for the two-dimensional Helmholtz and modified Helmholtz equations. This problem is formulated in a variational form where only weak requirements for the Cauchy boundary data are required. The solution on the underspecified boundary is considered as a control in a direct, mixed, well-posed problem while trying to fit the Cauchy data on the overspecified boundary. Hence the solution of the direct problems and the associated adjoint problems are defined in a weak sense, respectively, and a mathematical analysis is undertaken. It is shown that, in order to solve stably the Cauchy problem for Helmholtz-type equations the variational approach proposed needs the gradient of the minimisation functional, which is explicitly obtained via the adjoint problem.
The algorithm proposed consists of solving iteratively three direct, mixed, well-posed problems for Helmholtz-type equations which are reduced to only two direct solutions due to the linearity of the problem. The numerical implementation of the conjugate gradient method (CGM), in conjunction with Nemirovskii’s stopping criterion, is accomplished by using the boundary element method (BEM). Two examples are analysed and the accuracy, convergence and stability of the CGM+BEM are shown numerically.

##### MSC:
 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 65N38 Boundary element methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35R30 Inverse problems for PDEs