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A precise definition of Trefftz method is proposed and, starting with it, a general theory is briefly explained. This leads to formulating numerical methods from a domain decomposition perspective. An important feature of this approach is the systematic use of “fully discontinuous functions” and the treatment of a general boundary value problem with prescribed jumps. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces in which the functions together with their derivatives may have jump discontinuities and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz—Jirousek) and indirect (Trefftz—Herrera) methods. In turn, each one of them can be divided into overlapping and nonoverlapping. The generality of the resulting theory is remarkable, because it is applicable to any partial (or ordinary) differential equation or system of such equations, which is linear. The article is dedicated to Professor Jiroslav Jirousek, who has been a very important driving force in the modern development of Trefftz method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 561–580, 2000

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

This is the second of a sequence of papers devoted to applying the localized adjoint method (LAM), in space-time, to problems of advective-diffusive transport. We refer to the resulting methodology as the Eulerian-Lagrangian localized adjoint method (ELLAM). The ELLAM approach yields a general formulation that subsumes many specific methods based on combined Lagrangian and Eulerian approaches, so-called characteristic methods (CM). In the first paper of this series the emphasis was placed in the numerical implementation and a careful treatment of implementation of boundary conditions was presented for one-dimensional problems. The final ELLAM approximation was shown to possess the conservation of mass property, unlike typical characteristic methods. The emphasis of the present paper is on the theoretical aspects of the method. The theory, based on Herrera's algebraic theory of boundary value problems, is presented for advection-diffusion equations in both onc-dimensional and multidimensional systems. This provides a generalized ELLAM formulation. The generality of the method is also demonstrated by a treatment of systems of equations as well as a dcrivation of mixed methods. 0 1993 John Wiley & Sons, Inc.

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