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Least-squares Trefftz-type elements for the Helmholtz equation. (English) Zbl 0909.76052

Summary: 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. We develop the so-called ‘frameless’ or least-squares \(T\)-elements for Helmholtz’s equation (related to the scattering of waves by offshore structures) in two dimensions. 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 matrix of the resulting linear system is always Hermitian and positive definite.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Boundary Methods. An Algebraic Theory, Pitman Publishing Ltd, London, 1984.
[2] ’Ein Gegenstück zum Ritz’schen Verfahren’, in Proc. 2nd Int. Congr. Appl. Mech., Zurich, 1926, pp. 131-137.
[3] ’Die Kombination des modifizierten Trefftzschen Verfahrens mit der Methode der finiten Elemente’, in and (eds.), Finite Elemente Congr. in der Static., W. Ernst & Sohn, Berlin, 1973, pp. 172-185.
[4] ’Die praktische Berechnung der Kopplungsmatrizen bei der Kombination der Trefftzschen Methode und der Methode der finiten Elemente bei flachen Schalen’, in and (eds.), Finite Elemente Congress in der Static., W. Ernst & Sohn, Berlin, 1973, pp. 242-259.
[5] Pian, JAIAA 2 pp 1332– (1964)
[6] Pian, Int. J. Numer. Meth. Engng. 1 pp 3– (1969)
[7] Tong, Int. J. Numer. Engng. 7 pp 297– (1973)
[8] Lin, Int. J. Numer. Meth. Engng. 15 pp 1343– (1980)
[9] Tolley, C. R. Acad. Sc. Paris 287 pp 875– (1978)
[10] Hendry, Comput. Meth. Appl. Mech. Engng. 21 pp 1– (1980)
[11] Jirousek, Comput. Meth. Appl. Mech. Engng. 12 pp 77– (1977)
[12] Jirousek, Comput. Meth. Appl. Mech. Engng. 14 pp 65– (1978)
[13] Zienkiewicz, Int. J. Numer. Meth. Engng. 11 pp 355– (1977)
[14] and , ’Marriage à la mode–the best of both worlds (finite elements and boundary integrals)’, in and (eds.), Energy Methods in Finite Element Analysis, Wiley, Chichester, 1979, pp. 81-107.
[15] Jirousek, Arch. Comp. Meth. Engng. 3 pp 323– (1996)
[16] and , ’Approximation of the Poisson problem and of the eigenvalue problem for the Laplace operator by the method of the large singular finite elements’, Research Report 81-01, Zürich, 1981.
[17] Descloux, Comput. Meth. Appl. Mech. Engng. 39 pp 37– (1983)
[18] Sánchez-Sesma, Bull. Seism. Soc. Am. 72 pp 473– (1982)
[19] Zieliński, Int. J. Numer. Meth. Engng. 21 pp 509– (1985)
[20] ’Equivalent FE and BE forms of a substructure oriented boundary solution approach’, Publicatión CIMNE No. 20, Mayo 1992, International Center for Numerical Methods in Engineering, Barcelona, 1992.
[21] Jirousek, Comm. Numer. Meth. Engng. 10 pp 21– (1994)
[22] Jirousek, Comput. Struct. 57 pp 367– (1995)
[23] Herrera, Proc. Nat. Acad. Sci. USA 75 pp 2059– (1978)
[24] ’Finite T-elements for the Poisson and Helmholtz equation’, Thèse no 1491. Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1996.
[25] and , Handbook of Mathematical Functions, Dover Publications, New York, 1965.
[26] ’Manuel d’utilisateur du programme SAFE’, LSC, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1991.
[27] and , ’Wave forces on a pile: a diffraction theory’, Tech. Memo. No., 69, U.S. Army Board, U.S. Army Corp. of Eng., 1954.
[28] , The Applied Dynamics of Ocean Surface Waves, World Scientific, Singapore, 1992.
[29] Infinite Elements, Penshaw Press, Sunderland, 1992.
[30] Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.
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