A finite element method for scalar Helmholtz equation with field singularities

This paper describes a finite element method to obtain an accurate solution of the scalar Helmholtz equation with field singularities. It is known that the spatial derivatives of the eigenfunction of the scalar Helmholtz equation become infinite under certain conditions. These field singularities undermine the accuracy of the numerical solutions obtained by conventional finite element methods based on piecewise polynomials. In this paper, a regularized eigenfunction is introduced by subtracting the field singularities from the original eigenfunction. The finite element method formulated in terms of the regularized eigenfunction is expected to improve the accuracy and convergence of the numerical solutions. The finite element matrices for the present method can be easily evaluated since they do not involve any singular integrands. Moreover, the Dirichlet type boundary conditions are explicitly imposed on the variables using a transform matrix while the Neumann-type boundary conditions are implicitly imposed in the functional. The numerical results for three test problems show that the present method clearly improves the accuracy of the numerical solutions.