Numerical solution of the Dirichlet problem using a discrete analogue of the double-layer potential.

*(English)*Zbl 0825.65075Summary: This paper proves a method of solving a system of mesh equations approximating the Dirichlet and Neumann problems for Helmholtz’s equation in an arbitrary smooth domain. The solution of mesh equations is sought in a specific form, which makes it possible to reduce the problem to solving equations corresponding only to the points of the strip adjacent to the boundary. It is shown that the matrix of this system has the form \(I - T\), where \(T\) is a symmetrizable matrix, and that the system can be solved by a successive approximations method giving an \(\varepsilon\)- accurate solution after \(O (\ln \varepsilon^{-1 })\) successive approximations. It is also shown that by making use of the generalized stationary Richardson method for solving systems of FEM equations which approximate the first boundary value problem for the elliptic equation, we can arrive at an \(\varepsilon\)-accurate solution at the cost of \(O(h^{- 2} \ln \varepsilon^{- 1} \ln h^{- 1})\) arithmetic operations.

##### MSC:

65N06 | Finite difference methods for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |