A REVIEW OF INFINITE ELEMENT METHODS FOR EXTERIOR  HELMHOLTZ PROBLEMS

This work is devoted to review infinite element discretizations for the Helmholtz equation in exterior domains, which have become popular in recent years, as many research papers on this topic have appeared in the literature. The early contributions were mostly motivated by engineering considerations and the variational formulations have, in general, not been stated in a mathematically precise way. Only recently, theoretical aspects of the infinite element methodology have been analyzed and helped to put the different formulations into a mathematical framework. We build upon this, and present and compare the infinite element formulations within this context.

Download Full-text

The conjugated vs. the unconjugated infinite element method for the Helmholtz equation in exterior domains

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/s0045-7825(97)00186-2 ◽

1998 ◽

Vol 152 (1-2) ◽

pp. 125-145 ◽

Cited By ~ 27

Author(s):

K. Gerdes

Keyword(s):

Helmholtz Equation ◽

Exterior Domains ◽

Infinite Element ◽

Infinite Element Method ◽

Element Method

Download Full-text

Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

SIAM Journal on Numerical Analysis ◽

10.1137/060665737 ◽

2007 ◽

Vol 45 (5) ◽

pp. 1954-1978 ◽

Cited By ~ 37

Author(s):

Jie Shen ◽

Li-Lian Wang

Keyword(s):

Helmholtz Equation ◽

Galerkin Approximation ◽

Exterior Domains

Download Full-text

Difference potentials for the Helmholtz equation in exterior domains

Applied Numerical Mathematics ◽

10.1016/s0168-9274(99)00122-1 ◽

2000 ◽

Vol 33 (1-4) ◽

pp. 533-540 ◽

Cited By ~ 1

Author(s):

Victor S. Ryaben'kii ◽

Ivan L. Sofronov

Keyword(s):

Helmholtz Equation ◽

Exterior Domains ◽

Difference Potentials

Download Full-text

Numerical Methods for the Helmholtz Equation with Non-Reflecting Boundary Conditions in Exterior Domains

The finite element method in the 1990’s ◽

10.1007/978-3-662-10326-5_38 ◽

1991 ◽

pp. 379-388

Author(s):

I. Harari ◽

T. J. R. Hughes

Keyword(s):

Numerical Methods ◽

Boundary Conditions ◽

Helmholtz Equation ◽

Exterior Domains

Download Full-text

Convergence of the infinite element methods for the Helmholtz equation in separable domains

Numerische Mathematik ◽

10.1007/s002110050330 ◽

1998 ◽

Vol 79 (1) ◽

pp. 11-42 ◽

Cited By ~ 25

Author(s):

L. Demkowicz ◽

K. Gerdes

Keyword(s):

Helmholtz Equation ◽

Infinite Element

Download Full-text

A priori estimates for solutions of the Helmholtz equation in exterior domains and their application

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(69)90047-x ◽

1969 ◽

Vol 27 (2) ◽

pp. 253-261 ◽

Cited By ~ 1

Author(s):

David Colton

Keyword(s):

Helmholtz Equation ◽

A Priori Estimates ◽

A Priori ◽

Exterior Domains ◽

Priori Estimates

Download Full-text

THREE-DIMENSIONAL EXTERIOR PROBLEM OF DARWIN MODEL AND ITS NUMERICAL COMPUTATION

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202508003169 ◽

2008 ◽

Vol 18 (10) ◽

pp. 1673-1701 ◽

Cited By ~ 4

Author(s):

NENGSHENG FANG ◽

LUNG-AN YING

Keyword(s):

Boundary Value Problem ◽

Numerical Computation ◽

Existence And Uniqueness ◽

Three Dimensional ◽

Decomposition Theorem ◽

Exterior Problem ◽

Exterior Domains ◽

Numerical Examples ◽

Infinite Element Method ◽

Variational Formulations

Darwin model is a good approximation to Maxwell's equations, and this paper is concerned with the boundary value problem of the Darwin model in three-dimensional exterior domains. Firstly we establish the variational formulations of Darwin model in exterior domains and prove the existence and uniqueness. Then we prove a useful decomposition theorem that any function in unbounded exterior domains belonging to L2(Ω) can be decomposed into the sum of a function's gradient and another function's rotation, such decomposition is crucial in inducing the Darwin model. At last we spend some energy in using the infinite element method to solve the Darwin model in axis-symmetric exterior domains cases. Error estimates are obtained in weighted spaces, and numerical examples verify the convergence once again.

Download Full-text