Partition of unity method for Helmholtz equation: q -convergence for plane-wave and wave-band local bases

Theofanis Strouboulis; Realino Hidajat

Applications of Mathematics (2006)

  • Volume: 51, Issue: 2, page 181-204
  • ISSN: 0862-7940

Abstract

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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 .

How to cite

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Strouboulis, Theofanis, and Hidajat, Realino. "Partition of unity method for Helmholtz equation: $q$-convergence for plane-wave and wave-band local bases." Applications of Mathematics 51.2 (2006): 181-204. <http://eudml.org/doc/33250>.

@article{Strouboulis2006,
abstract = {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 $q$.},
author = {Strouboulis, Theofanis, Hidajat, Realino},
journal = {Applications of Mathematics},
keywords = {Partition of Unity Method (PUM); Helmholtz equation; exponential convergence; extrapolation; pollution due to wave-number; Helmholtz equation; exponential convergence; extrapolation; pollution due to wave-number; partition of unity method; finite element; performance; basis functions; plane-waves; wave-bands},
language = {eng},
number = {2},
pages = {181-204},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Partition of unity method for Helmholtz equation: $q$-convergence for plane-wave and wave-band local bases},
url = {http://eudml.org/doc/33250},
volume = {51},
year = {2006},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 2
SP - 181
EP - 204
AB - 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 $q$.
LA - eng
KW - Partition of Unity Method (PUM); Helmholtz equation; exponential convergence; extrapolation; pollution due to wave-number; Helmholtz equation; exponential convergence; extrapolation; pollution due to wave-number; partition of unity method; finite element; performance; basis functions; plane-waves; wave-bands
UR - http://eudml.org/doc/33250
ER -

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