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Least‐squares Trefftz‐type elements for the Helmholtz equation
Author(s) -
Stojek Małgorzata
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19980315)41:5<831::aid-nme311>3.0.co;2-v
Subject(s) - helmholtz equation , mathematics , finite element method , hermitian matrix , mathematical analysis , robustness (evolution) , a priori and a posteriori , convergence (economics) , boundary value problem , least squares function approximation , helmholtz free energy , physics , pure mathematics , economics , biochemistry , chemistry , philosophy , statistics , epistemology , quantum mechanics , estimator , gene , thermodynamics , economic growth
Trefftz‐type elements, or T‐elements, are finite elements the internal field of which fulfills the governing differential equations of the problem a priori whereas the prescribed boundary conditions and the interelement continuity must be enforced by some suitable method. In this paper, the relevant matching is achieved by means of a least‐squares procedure. The so‐called ‘frameless’ or least‐squares T‐elements for Helmholtz's equation (related to the scattering of waves by offshore structures) in 2‐D are developed and studied. The required accuracy of the solution can be obtained by increasing the number of either the subdomains or T‐functions, which can be regarded as the h ‐ or p ‐type approach, respectively. Convergence studies are performed with much attention to the use of special purpose elements for a doubly connected domain with a circular hole and for an angular corner subdomain. The most attractive features of the presented formulation are its simplicity and robustness. The matrix of the resulting linear system is always Hermitian and positive definite. © 1998 John Wiley & Sons, Ltd.