An SVD-approach to Jacobi–Davidson solution of nonlinear Helmholtz eigenvalue problems

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Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi–Davidson method. The suggested truncated SVD polynomial approximation seems to be of interest on its own. It can be applied in combination with existing eigensolvers to the problems where the nonlinearity is expensive to evaluate or not explicitly given.