Geodesic
Partition of unity method for Helmholtz equation: $q$-convergence for plane-wave and wave-band local bases
Applications of Mathematics, Tome 51 (2006) no. 2, pp. 181-204.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

In this paper we study the $q$-version of the Partition of Unity Method for the Helmholtz equation. The method is obtained by employing the standard bilinear finite element basis on a mesh of quadrilaterals discretizing the domain as the Partition of Unity used to paste together local bases of special wave-functions employed at the mesh vertices. The main topic of the paper is the comparison of the performance of the method for two choices of local basis functions, namely a) plane-waves, and b) wave-bands. We establish the $q$-convergence of the method for the class of analytical solutions, with $q$ denoting the number of plane-waves or wave-bands employed at each vertex, for which we get better than exponential convergence for sufficiently small $h$, the mesh-size of the employed mesh. We also discuss the a-posteriori estimation for any solution quantity of interest and the problem of quadrature for all integrals employed. The goal of the paper is to stimulate theoretical development which could explain various numerical features. A main open question is the analysis of the pollution and its disappearance as function of $h$ and $q$.
DOI : 10.1007/s10492-006-0011-0
Classification : 35J05, 65J10, 65N12, 65N30, 65Y20
Keywords: Partition of Unity Method (PUM); Helmholtz equation; exponential convergence; extrapolation; pollution due to wave-number 
@article{10_1007_s10492_006_0011_0,
     author = {Strouboulis, Theofanis and Hidajat, Realino},
     title = {Partition of unity method for {Helmholtz} equation: $q$-convergence for plane-wave and wave-band local bases},
     journal = {Applications of Mathematics},
     pages = {181--204},
     publisher = {mathdoc},
     volume = {51},
     number = {2},
     year = {2006},
     doi = {10.1007/s10492-006-0011-0},
     mrnumber = {2212312},
     zbl = {1164.65505},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-006-0011-0/}
}
TY  - JOUR
AU  - Strouboulis, Theofanis
AU  - Hidajat, Realino
TI  - Partition of unity method for Helmholtz equation: $q$-convergence for plane-wave and wave-band local bases
JO  - Applications of Mathematics
PY  - 2006
SP  - 181
EP  - 204
VL  - 51
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10492-006-0011-0/
DO  - 10.1007/s10492-006-0011-0
LA  - en
ID  - 10_1007_s10492_006_0011_0
ER  - 
%0 Journal Article
%A Strouboulis, Theofanis
%A Hidajat, Realino
%T Partition of unity method for Helmholtz equation: $q$-convergence for plane-wave and wave-band local bases
%J Applications of Mathematics
%D 2006
%P 181-204
%V 51
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1007/s10492-006-0011-0/
%R 10.1007/s10492-006-0011-0
%G en
%F 10_1007_s10492_006_0011_0
Strouboulis, Theofanis; Hidajat, Realino. Partition of unity method for Helmholtz equation: $q$-convergence for plane-wave and wave-band local bases. Applications of Mathematics, Tome 51 (2006) no. 2, pp. 181-204. doi : 10.1007/s10492-006-0011-0. http://geodesic.mathdoc.fr/articles/10.1007/s10492-006-0011-0/

Cité par Sources :