SOLUTION OF THE HELMHOLTZ EQUATION ON THE BASE OF COMBINATION OF NUMERICAL METHODS

Abstract eng:
Nowadays Domain Decomposition (DD) method is one of the common approaches to construct preconditioner for solving 3D Helmholtz equation, especially in geophysical applications. There are numerous papers devoted to construction of optimal transmission conditions to improve convergence of the DD. However, these researches are focused on the differential statements and no perturbation of differential operators is considered. Whereas solution of the 3D Helmholtz equation requires the use of the numerical methods such as finite differences or finite elements, thus a numerical error is introduced in the operators as a result of numerical approximation. Moreover, in some cases it is worth using different numerical methods in adjoint subdomains, which makes the considered perturbations nonsymmetric. In this paper, we consider 2D Helmholtz equation and assume it to be solved using semi-analytical preconditioner (Neklyudov et al. 2010), (Belonosov et al. 2014). To invert the preconditioner pseudospectral methods are used in (Neklyudov et al. 2010), which are applicable for the models with flat sharp interfaces, but the convergences deteriorate if high gradients of the sharp interfaces present. To overcome this we suggest using domain decomposition technique with grid-based methods applied in the upper part of the model (where the sharpest interfaces such as free-surface are) and semi-analytical approach used in the main part of the model. However, use of different techniques leads to nonsymmetrical perturbation of the original operator, thus causes the irreducible error in the solution. The research was done under financial support of the Russian Foundation for Basic Research grants no. 13-05-00076, 13-05-12051, 14-05-00049, 14-05-93090, 15-05-01310, 15-35-20022, 14-01-31340, 14-05-31222 and pertially supported by the fellowship SP-1777.2015.5 of the President of the Russian Federation.

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