Many boundary element integral equation kernels are based on the Green’s functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell’s equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, ...

It is proposed a new approach for solution of the boundary problems for wide class of partial differential equations of mathematical physics, which includes the Laplace, Poisson and Helmholtz equations and parabolic equations. The approach is based on the discovered by author Local Ray Property and leads to new General Ray (GR) Method that uses the explicit analytical formulas with inverse Rado...

The nonlinear effects of the second harmonic generation have been investigated for the propagation of light along the axis of fibers of wagon wheel cross sectional shape. Nodal finite element formulation is utilized to obtain discretized Helmholtz equations under appropriate boundary conditions. The hierarchical p-version nodal elements are used for meshing the cross section of wagon wheel fibe...

Talk Abstract We examine the solution of high-frequency Helmholtz equations using 2nd, 4th and 6th order finite difference schemes. The examples include two problems with known analytic solutions, enabling error evaluation of the different schemes on various grids (9–18 points per wavelength). We use our block-parallel CARP-CG algorithm [Parallel Computing 36, 2010] for solving the equations. T...