### Abstract

The boundary value problem of Helmholtz's equation on a n + 1 dimensional thin slab is approximated by appropriate systems of the n-dimensional boundary value problem. The very detailed estimates for modeling error in the H^{1}-norm demonstrate convergence when the thickness of the slab approaches 0 as well as when the size of the systems approaches infinity. Shape functions through the thickness are first selected by finitely many eigenfunctions, and the tail is then selected to consist of polynomials. The presence of two types of functions gives rise to a certain choice in the selection of a particular set of shape functions. Numerical results provide a good illustration of the effect of different choices for specific problems.

Original language | English |
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Pages (from-to) | 169-190 |

Number of pages | 22 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1999 Mar |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

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*Numerical Methods for Partial Differential Equations*, vol. 15, no. 2, pp. 169-190. https://doi.org/10.1002/(SICI)1098-2426(199903)15:2<169::AID-NUM3>3.0.CO;2-X

**Selections of shape functions for dimensional reduction to Helmholtz's equation.** / Liu, Kang Man; Babuška, Ivo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Selections of shape functions for dimensional reduction to Helmholtz's equation

AU - Liu, Kang Man

AU - Babuška, Ivo

PY - 1999/3

Y1 - 1999/3

N2 - The boundary value problem of Helmholtz's equation on a n + 1 dimensional thin slab is approximated by appropriate systems of the n-dimensional boundary value problem. The very detailed estimates for modeling error in the H1-norm demonstrate convergence when the thickness of the slab approaches 0 as well as when the size of the systems approaches infinity. Shape functions through the thickness are first selected by finitely many eigenfunctions, and the tail is then selected to consist of polynomials. The presence of two types of functions gives rise to a certain choice in the selection of a particular set of shape functions. Numerical results provide a good illustration of the effect of different choices for specific problems.

AB - The boundary value problem of Helmholtz's equation on a n + 1 dimensional thin slab is approximated by appropriate systems of the n-dimensional boundary value problem. The very detailed estimates for modeling error in the H1-norm demonstrate convergence when the thickness of the slab approaches 0 as well as when the size of the systems approaches infinity. Shape functions through the thickness are first selected by finitely many eigenfunctions, and the tail is then selected to consist of polynomials. The presence of two types of functions gives rise to a certain choice in the selection of a particular set of shape functions. Numerical results provide a good illustration of the effect of different choices for specific problems.

UR - http://www.scopus.com/inward/record.url?scp=0009726479&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0009726479&partnerID=8YFLogxK

U2 - 10.1002/(SICI)1098-2426(199903)15:2<169::AID-NUM3>3.0.CO;2-X

DO - 10.1002/(SICI)1098-2426(199903)15:2<169::AID-NUM3>3.0.CO;2-X

M3 - Article

AN - SCOPUS:0009726479

VL - 15

SP - 169

EP - 190

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 2

ER -