# Trefftz method

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

$$\tag{* } \Delta u = 0,\ \ u \mid _ {S} = \phi ,$$

where $S$ is the boundary of a domain $\Omega \subset \mathbf R ^ {m}$. The solution of the problem (*) minimizes the functional

$$J ( u) = \int\limits _ \Omega ( \mathop{\rm grad} u ( x)) ^ {2} dx$$

over all functions satisfying the boundary condition $u \mid _ {S} = \phi$. Trefftz' method consists in the following. Suppose one is given a sequence of harmonic functions $w _ {1} , w _ {2} \dots$ in $\Omega$ that are square summable in $\Omega$ together with their first derivatives. An approximate solution is sought in the form

$$u _ {n} = \ \sum _ {j = 1 } ^ { n } c _ {j} w _ {j} ,$$

the coefficients $c _ {j}$ being determined from the condition that $J ( u _ {n} - u)$ is minimal, where $u$ is the exact solution of (*). This leads to the following system of equations for $c _ {1} \dots c _ {n}$:

$$\sum _ {j = 1 } ^ { n } c _ {j} \int\limits _ { S } w _ {j} \frac{\partial w _ {i} }{\partial \nu } \ dS = \ \int\limits _ { S } \phi \frac{\partial w _ {i} }{\partial \nu } dS,\ \ i = 1 \dots n,$$

where $\nu$ is the outward normal to $S$.

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

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

#### References

 [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)