Trefftz method

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One of the variational methods for solving boundary value problems. Suppose one has to solve the boundary value problem


where is the boundary of a domain . The solution of the problem (*) minimizes the functional

over all functions satisfying the boundary condition . Trefftz' method consists in the following. Suppose one is given a sequence of harmonic functions in that are square summable in together with their first derivatives. An approximate solution is sought in the form

the coefficients being determined from the condition that is minimal, where is the exact solution of (*). This leads to the following system of equations for :

where is the outward normal to .

Trefftz' method can be generalized to various boundary value problems (see [2][4]).

The method was proposed by E. Trefftz (see [1]).


[1] E. Trefftz, "Ein Gegenstück zum Ritzschen Verfahren" , Verhandl. 2er Internat. Kongress. Techn. Mechanik Zürich, 1926, 12–17 Sept. , O. Füssli (1927) pp. 131–137
[2] S.G. [S.G. Mikhlin] Michlin, "Variationsmethoden der mathematischen Physik" , Akademie Verlag (1962) (Translated from Russian)
[3] V.I. Krylov, V.V. Bobkov, P.I. Monastyrnyi, "Computing methods of higher mathematics" , 2 , Minsk (1975) (In Russian)
[4] M.Sh. Birman, "Variational methods for solving boundary value problems analogous to Trefftz' method" Vestnik Leningrad. Gos. Univ. Ser. mat. Mekh. i Astr. , 11 : 13 (1956) pp. 69–89 (In Russian)



[a1] K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 1056–1058
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Trefftz method. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by G.M. Vainikko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article