On the solution of the Helmholtz equation on regions with corners

In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

Download Full-text

Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation

Banach Center Publications ◽

10.4064/-15-1-175-251 ◽

1985 ◽

Vol 15 (1) ◽

pp. 175-251 ◽

Cited By ~ 26

Author(s):

Martin Costabel ◽

Ernst Stephan

Keyword(s):

Boundary Value Problems ◽

Integral Equations ◽

Boundary Integral Equations ◽

Mixed Boundary ◽

Boundary Value ◽

Galerkin Approximation ◽

Boundary Integral ◽

Mixed Boundary Value Problems ◽

Polygonal Domains

Download Full-text

Method of Boundary Integral Equations with Hypersingular Integrals in Boundary-Value Problems

Journal of Mathematical Sciences ◽

10.1007/s10958-021-05475-3 ◽

2021 ◽

Author(s):

A. V. Setukha

Keyword(s):

Boundary Value Problems ◽

Integral Equations ◽

Boundary Integral Equations ◽

Boundary Value ◽

Boundary Integral ◽

Hypersingular Integrals

Download Full-text

Sharp High-Frequency Estimates for the Helmholtz Equation and Applications to Boundary Integral Equations

SIAM Journal on Mathematical Analysis ◽

10.1137/15m102530x ◽

2016 ◽

Vol 48 (1) ◽

pp. 229-267 ◽

Cited By ~ 20

Author(s):

Dean Baskin ◽

Euan A. Spence ◽

Jared Wunsch

Keyword(s):

Helmholtz Equation ◽

Integral Equations ◽

High Frequency ◽

Boundary Integral Equations ◽

Boundary Integral

Download Full-text

A fast spectral Galerkin method for hypersingular boundary integral equations in potential theory

Computational Mechanics ◽

10.1007/s00466-009-0373-8 ◽

2009 ◽

Vol 44 (2) ◽

pp. 263-271 ◽

Cited By ~ 8

Author(s):

Sylvain Nintcheu Fata ◽

Leonard J. Gray

Keyword(s):

Galerkin Method ◽

Integral Equations ◽

Potential Theory ◽

Boundary Integral Equations ◽

Boundary Integral ◽

Spectral Galerkin Method

Download Full-text

Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.22.202003.319-332 ◽

2020 ◽

Vol 22 (3) ◽

pp. 319-332

Author(s):

Aleksandr N. Tynda ◽

Konstantin A. Timoshenkov

Keyword(s):

Boundary Value Problems ◽

Integral Equations ◽

Boundary Integral Equations ◽

Boundary Value ◽

Spline Approximation ◽

Boundary Integral ◽

Double Layers ◽

Fredholm Integral Equations ◽

Fredholm Type ◽

Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

Download Full-text

On the unique solvability of a class of modified boundary integral equations

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500022446 ◽

1984 ◽

Vol 27 (3) ◽

pp. 303-311 ◽

Cited By ~ 3

Author(s):

R. E. Kleinman ◽

G. F. Roach

Keyword(s):

Helmholtz Equation ◽

Integral Equations ◽

Unique Solvability ◽

Green's Functions ◽

Green’S Functions ◽

Boundary Integral Equations ◽

Boundary Integral ◽

Transmission Problem

In a recent paper the authors considered the transmission problem for the Helmholtz equation by using a reformulation of the problem in terms of a pair of coupled boundary integral equations with modified Green's functions as kernels. In this note we settle the question of the unique solvability of these modified boundary integral equations.

Download Full-text

Error analysis of boundary integral methods

Acta Numerica ◽

10.1017/s0962492900002294 ◽

1992 ◽

Vol 1 ◽

pp. 287-339 ◽

Cited By ~ 32

Author(s):

Ian H. Sloan

Keyword(s):

Approximate Solution ◽

Partial Differential Equations ◽

Differential Equations ◽

Error Analysis ◽

Boundary Value Problems ◽

Integral Equations ◽

Boundary Integral Equations ◽

Boundary Integral ◽

Boundary Integral Methods ◽

Integral Methods

Many of the boundary value problems traditionally cast as partial differential equations can be reformulated as integral equations over the boundary. After an introduction to boundary integral equations, this review describes some of the methods which have been proposed for their approximate solution. It discusses, as simply as possible, some of the techniques used in their error analysis, and points to areas in which the theory is still unsatisfactory.

Download Full-text