Implementation of Trefftz method for the solution of some elliptic boundary value problems

摘要

The purpose of this paper is to implement the well-known Trefftz method for the approximate solution of some elliptic boundary value problems for the two-dimensional Laplace's and the biharmonic operators through the use of polar harmonics as trial functions. Four applications treating four different situations are presented: the Dirichlet problem for the steady-state temperature distribution in an infinite elliptic cylinder (giving rise to Poisson's or Laplace's equations) with discontinuous boundary function for the complete and the truncated ellipses (cases 1, 2) and the first and the second fundamental problems of the plane strain theory of elastostatics for an infinite cylinder with rectangular normal cross-section (cases 3, 4). For two of the examples (cases 1, 3), the results could be checked against known exact solutions to show, in particular, that the method can sometimes give the exact solution, while for the other two (cases 2, 4) the method has provided solutions for new problems. It is shown that the use of polar harmonics, possibly together with other harmonics belonging to different coordinate systems, can handle many situations in an efficient way, i.e., by using few trial functions.