SUMMARYThe paper presents various ways of fitting the boundary conditions in the T-complete functions method. The authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit. The convergence of the Collatz error measures and the conditioning of the solution matrices are investigated in detail. INTRODIJCTIONThere are two main approaches for the formulation of boundary methods; one is based on boundary integral equations (boundary element methods, as well as the boundary series method'-3) and the other is based on the use of complete systems of solutions (Trefftz m e t h~d~-~) . This article is concerned with the latter approach.Complete systems of shape functions can be constructed in many alternative ways. Several aspects of such questions, such as completeness and convergence of integral least squares fitting of boundary conditions have been extensively studied by one of the author^.^^^^^-" Co nstruction of finite elements with this kind of shape function has also been investigated."To fit the boundary conditions one can use a direct or, alternatively, an indirect a p p r~a c h .~ To be more specific, consider the Dirichlet problem for the Laplace equation on a region 0 with boundary r. In this case the boundary condition is u = U on r, where u is a prescribed function.Let K = &/an be the unknown complementary boundary values. To solve such a problem by Trefftz method, one has available a T-complete system of functions { U l , U 2 , . . .}. Then, in the direct approach one constructs a linear combination li = xF= ai U i which approximates the prescribed boundary values U, whereas in the indirect approach Q is required to be such that d Q p n approximates the unknown normal derivative K, on the boundary. The indirect procedure has been used in previous works and has been called boundary fitting using opposite weightst2 However, it had not been realized that such a procedure is tantamount to approximating the unknown boundary derivatives. To see that this is indeed so, recall the well-known reciprocity relation which holds for harmonic functions in SZ. If we impose the condition

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SUMMARYThe use of a complete and nonsingular set of Trefftz functions in the solution of quasi-harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary-type approximation. Procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated.

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