# Difference between revisions of "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 ).

The method was proposed by E. Trefftz (see ).

How to Cite This Entry:
Trefftz method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trefftz_method&oldid=12688
This article was adapted from an original article by G.M. Vainikko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article