A new scheme based on boundary elements method to solve linear Helmholtz and semi-linear Poisson׳s equations

In the current paper the boundary elements method (BEM) has been employed to solve linear Helmholtz and semi-linear Poisson׳s equations. In fact a new idea will be presented to solve linear Poisson׳s and Helmholtz equations with variable coefficient by the use of BEM. And after that two iterative schemes based on the fixed point theorem and Newton׳s method have been studied to solve semi-linear Poisson׳s equation via the new idea. The main concerning problem of this paper is omitting singularity from BEM׳s domain integrals. So the improvement will be done by transforming the singularity to boundary integrals which can be calculated easily. The new scheme is implemented and compared with some well known numerical methods on various computational domains for the two-dimensional problems with Dirichlet and mixed boundary conditions. Numerical examples show that the new scheme is able to solve linear Helmholtz and semi-linear Poisson׳s equations, efficiently.