Quasi-analytical solution of two-dimensional Helmholtz equation

The two-dimensional Helmholtz differential equation governs vibrational problems for a thin membrane and is therefore well studied. Analytical solutions are limited to particular domain shapes, so that in general numerical methods are used when an arbitrary domain is considered. In this paper, a quasi-analytical solution is proposed, suitable to be applied to an arbitrary domain shape. Concretely, the Helmholtz equation is transformed to account for a conformal map between the shape of the physical domain and the unit disk as canonical domain. This way, the transformed Helmholtz equation is solved exploiting well known analytical solutions for a circular domain and the solution in the physical domain is obtained by applying the conformal map. The quasi-analytical approach is compared to analytical solutions for the case of a circular, elliptic and squared domain.     

Some boundary properties of a solution of the Poisson equation

Certain boundary properties of a solution u of the boundary-value problem for the Poisson equation Δ u = f in a disk are studied. In particular, various estimates for integral norms of the solution through the Green capacity of the condenser composed of the support of the function f and the boundary of the disk and also through the growth rate of the function f are given. The proofs are based on the theorem on coverings of supports of Borel measures outside of which the Green potentials of these measures are bounded by unity. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Some special solutions of the Helmholtz equation

Functions are constructed which describe a cylindrical wave diverging from the origin and tangent to a semi-infinite plane wave. The use of such functions is illustrated by the example of the problem of diffraction of a plane wave by a semi-infinite screen.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 197–202, 1975.     

The Neumann problem for the Helmholtz equation in a starlike planar domain

The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson.     

Semiclassical measure for the solution of the dissipative Helmholtz equation

We study the semiclassical measure for the solution of the high-frequency Helmholtz equation in Rn with non-constant absorption index and a source term concentrated on a bounded submanifold of Rn. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption index is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagrangian distributions.     

High-frequency asymptotics for the numerical solution of the Helmholtz equation

It is often noted that the Helmholtz equation is extremely difficult to solve, in particular, for high-frequency solutions for heterogeneous media. Since stability for second-order discretization methods requires one to choose at least 10–12 grid points per wavelength, the discrete problem on the possible coarsest mesh is huge. In a realistic simulation, one is required to choose 20–30 points per wavelength to achieve a reasonable accuracy; this problem is hard to solve. This article is concerned with the high-frequency asymptotic decomposition of the wavefield for an efficient and accurate simulation for the high-frequency numerical solution of the Helmholtz equation. It has been numerically verified that the new method is accurate enough even when one chooses 4–5 grid points per wavelength.     

Dirichlet problem for the Helmholtz equation in an exterior two-dimensional domain with cuts without matching conditions at the cut endpoints

The Dirichlet problem for the Helmholtz equation in a plane exterior domain with cuts is considered for the case in which functions defined on opposite sides of the cuts in the Dirichlet boundary condition do not necessarily satisfy the matching conditions at the cut endpoints and the solution of the problem is not necessarily continuous at the endpoints of the cuts. We give a well-posed statement of the problem, prove existence and uniqueness theorems for a classical solution, derive an integral representation of the solution, and use it to study its properties. We show that the Dirichlet problem in the considered setting does not necessarily have a weak solution, although there exists a classical solution. We derive asymptotic formulas describing the behavior of the gradient of the solution at the endpoints of the cuts.     

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

Summary We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.     

A numerical solution of a two-dimensional transport equation

In this paper we present a variational method for approximating solutions of the Dirichlet problem for the neutron transport equation in the stationary case. Error estimates from numerical examples are used to evaluate an approximation of the solution with respect to the steps of two grids.     

An improved sweeping domain decomposition preconditioner for the Helmholtz equation

In this paper we generalize and improve a recently developed domain decomposition preconditioner for the iterative solution of discretized Helmholtz equations. We introduce an improved method for transmission at the internal boundaries using perfectly matched layers. Simultaneous forward and backward sweeps are introduced, thereby improving the possibilities for parallellization. Finally, the method is combined with an outer two-grid iteration. The method is studied theoretically and with numerical examples. It is shown that the modifications lead to substantial decreases in computation time and memory use, so that computation times become comparable to that of the fastests methods currently in the literature for problems with up to 10⁸ degrees of freedom.     

Stable boundary element domain decomposition methods for the Helmholtz equation

In this paper, we present a stable boundary element domain decomposition method to solve boundary value problems of the Helmholtz equation via a tearing and interconnecting approach. A possible non-uniqueness of the solution of local boundary value problems due to the appearance of local eigensolutions is resolved by using modified interface conditions of Robin type, which results in a Galerkin boundary element discretization which is robust for all local wave numbers. Numerical examples confirm the stability of the proposed approach.     

Potential-based numerical solution of Dirichlet problems for the Helmholtz equation

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.     

Nonlinear nonlocal problems for a parabolic equation in a two-dimensional domain

We establish the convergence of the Rothe method for a parabolic equation with nonlocal boundary conditions and obtain an a priori estimate for the constructed difference scheme in the grid norm on a ball. We prove that the suggested iterative process for the solution of the posed problem converges in the small. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 244–254, February, 1997.