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Selections of shape functions for dimensional reduction to Helmholtz's equation
Author(s) -
Liu Kang–Man,
Babus̆ka Ivo
Publication year - 1999
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199903)15:2<169::aid-num3>3.0.co;2-x
Subject(s) - mathematics , helmholtz equation , boundary value problem , eigenfunction , norm (philosophy) , mathematical analysis , partial differential equation , reduction (mathematics) , uniform norm , convergence (economics) , geometry , eigenvalues and eigenvectors , physics , quantum mechanics , political science , law , economics , economic growth
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. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 169–190, 1999