| Abstract: | It is well known th a t the m ethod of subsolutions and supersolutions for semilinear elliptic boundary value problems leads to a monotone iteration scheme, which can serve as a numerical algorithm. In this dissertation, the monotone iteration scheme implemented numerically through an indirect boundary element m ethod is analyzed. It is found th at the sequence of simple layer densities in this boundary element - m onotone iteration is of antimonotone property. Meanwhile, a working assum ption [H] is formulated to obtain a rate of convergence and allow an arbitrary small margin 8 for the numerical im plementation of boundary elements within the range of monotone convergence. Based on the Aubin-Nitsche lemma and the triangle inequalities, optim al order error estim ates for the Galerkin boundary element iterates with respect to the norms of the Sobolev spaces # r (0 ), 0 < r < 2, are derived by relating several approxim ate solutions. Furtherm ore, as a peculiarity, it is shown th at for the nonlinearities th a t are of separable type, "higher than optimal order" error estim ates can be obtained with respect to the mesh param eter h. F inally, several examples of semilinear elliptic partial differential equations featuring different situations of existence, uniqueness, m ultiplicity and stability are discussed, com puted, and the graphics of their numerical solutions are also illustrated. |
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