Yas'ko M.

The Helmholtz equation arises in many physical applications, in particular in acoustics. The main difficulties are connected with the computations for the high wavenumbers, when we have to use a large boundary element mesh. The memory storage is very large and CPU time is very long for solving of the large linear system with complex-valued coefficients. In this work a new formulation of the boundary element method is suggested for the fast computing of the Helmholtz equation. It is a variant of the direct integral equation formulation based on the real-valued fundamental solutions for the 2D and 3D homogeneous media and the application of the conjugate gradient methods for a resulting linear system. The new approach was used to obtain the resonant frequencies and mode shapes for the boundary-valued problems for the Helmholtz equation. The corresponding boundary-valued problem are solved for the different wavenumbers from some range. For the each wavenumber two integral functions were computed after obtaining numerical solution. These functions have breaks in points conterminous to eigenfrequencies. These breaks reflect influence of eigenmodes on the numerical solution. It is interesting also to note, that the number of iterations neede for convergence of conjugate gradient methods was increased unsignificantly in the neighbourhoods of eigenvalues. The comparison of numerical and analytical data in the application of the method to boundary-valued problems of the first and second kind has shown that the given method is able to find the resonant frequencies in practical problems.