Computation of singular solutions to the Helmholtz equation with high order accuracy
Solutions to elliptic PDEs, in particular to the Helmholtz equation, become singular near
the boundary if the boundary data do not possess suﬃcient regularity. In that case, the
convergence of standard numerical approximations may slow down or cease altogether.
We propose a method that maintains a high order of grid convergence even in the
presence of singularities. This is accomplished by an asymptotic expansion that removes
the singularities up to several leading orders, and the remaining regularized part of the
solution can then be computed on the grid with the expected accuracy.
The computation on the grid is rendered by a compact ﬁnite difference scheme combined
with the method of difference potentials. The scheme enables fourth order accuracy on
a narrow 3 Ч 3stencil: it uses only one unknown variable per grid node and requires
only as many boundary conditions as needed for the underlying differential equation
itself. The method of difference potentials enables treatment of non-conforming boundaries
on regular structured grids with no deterioration of accuracy, while the computational
complexity remains comparable to that of a conventional ﬁnite difference scheme on the
same grid. The method of difference potentials can be considered a generalization of the
method of Calderon’s operators in PDE theory.
In the paper, we provide a theoretical analysis of our combined methodology and
demonstrate its numerical performance on a series of tests that involve Dirichlet and
Neumann boundary data with various degrees of “non-regularity”: an actual jump
discontinuity, a discontinuity in the ﬁrst derivative, a discontinuity in the second derivative,
etc. All computations are performed on a Cartesian grid, whereas the boundary of the
domain is a circle, chosen as a simple but non-conforming shape. In all cases, the proposed
methodology restores the design rate of grid convergence, which is fourth order, in spite
of the singularities and regardless of the fact that the boundary is not aligned with the
discretization grid. Moreover, as long as the location of the singularities is known and
remains ﬁxed, a broad spectrum of problems involving different boundary conditions
and/or data on “smooth” segments of the boundary can be solved economically since the discrete counterparts of Calderon’s projections need to be calculated only once and then
can be applied to each individual formulation at very little additional cost.