The general boundary element method and its further generalizations

In this paper, the basic ideas of the general boundary element method (BEM) proposed by Liao [in Boundary Elements XVII, Computational Mechanics Publi cations, Southampton, MA, 1995, pp. 67-74; Int. J. Numer. Methods Fluids, 2 3, 739-751 (1996), 24, 863-873 (1997); Comput. Mech., 20, 397-406 (1997)] a nd Liao and Chwang [Int. J. Numer. Methods Fluids, 23, 467-483 (1996)] are further generalized by introducing a non-zero parameter fi. Some related ma thematical theorems are proposed. This general BEM contains the traditional BEM in logic, but is valid for non-linear problems, including those whose governing equations and boundary conditions have no linear terms. Furthermo re, the general BEM can solve non-linear differential equations by means of no iterations. This disturbs the absolutely governing place of iterative m ethodology of the BEM for non-linear problems. The general BEM can greatly enlarge application areas of the BEM as a kind of numerical technique. Two non-linear problems are used to illustrate the validity and potential of th e further generalized BEM. Copyright (C) 1999 John Wiley & Sons, Ltd.