scholarly journals Generalized combined field integral equations for the iterative solution of the three-dimensional Helmholtz equation

2007 ◽  
Vol 41 (1) ◽  
pp. 147-167 ◽  
Author(s):  
Xavier Antoine ◽  
Marion Darbas
Keyword(s):  
Helmholtz Equation ◽  
Integral Equations ◽  
Three Dimensional ◽  
Iterative Solution ◽  
Field Integral

Electromagnetic modeling of 3-D structures by the method of system iteration using integral equations

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1556-1561 ◽  
Author(s):  
Zonghou Xiong
Keyword(s):  
Integral Equations ◽  
Three Dimensional ◽  
Computation Time ◽  
Iterative Solution ◽  
Matrix Inversion ◽  
Electromagnetic Modeling ◽  
Direct Inversion ◽  
Conductivity Structure ◽  
The Matrix ◽  
Earth Conductivity

A new approach for electromagnetic modeling of three‐dimensional (3-D) earth conductivity structures using integral equations is introduced. A conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion. The influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively. This is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method. This method reduces computer memory requirements since only one submatrix at a time needs to be stored. The diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure. The block iterative solution requires much less computation time than direct matrix inversion or conventional point iterative methods as the convergence depends on the number of the submatrices, not on the total number of unknowns in the solution. As the submatrices are independent of each other, this method is suitable for parallel processing.

Iterative solutions of boundary integral equations in acoustics

1988 ◽  
Vol 417 (1852) ◽  
pp. 45-57 ◽  
Keyword(s):  
Helmholtz Equation ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Iterative Solution ◽  
Rates Of Convergence ◽  
Boundary Integral ◽  
Exterior Problems ◽  
Iterative Solutions ◽  
Boundary Integral Operators ◽  
Linear Operator Equations

It is shown that exterior problems for the Helmholtz equation may be solved iteratively for all frequencies. A general class of boundary-value problems for the Helmholtz equation is reduced to boundary integral equations. Using modified Green’s functions these boundary integral equations are known to be uniquely solvable. Even though the boundary integral operators are not self-adjoint they may be transformed into a form appropriate for iterative solution by a method developed for linear operator equations. Rates of convergence of the iteration are given and an example is presented to show the method.

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