TY - JOUR
T1 - Boundary element monotone iteration scheme for semilinear elliptic partial differential equations
AU - Deng, Yuanhua
AU - Chen, Goong
AU - Ni, Wei Ming
AU - Zhou, Jianxin
PY - 1996/7
Y1 - 1996/7
N2 - The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin δ for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the Hr(Ω), 0 ≤ r ≤ 2, Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, "higher than optimal order" error estimates can be obtained with respect to the mesh parameter h. Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.
AB - The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin δ for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the Hr(Ω), 0 ≤ r ≤ 2, Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, "higher than optimal order" error estimates can be obtained with respect to the mesh parameter h. Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.
KW - Boundary elements
KW - Elliptic type
KW - Nonlinear PDE
KW - Numerical PDE
KW - Potential theory
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U2 - 10.1090/S0025-5718-96-00743-0
DO - 10.1090/S0025-5718-96-00743-0
M3 - Article
AN - SCOPUS:0030353044
SN - 0025-5718
VL - 65
SP - 943
EP - 982
JO - Mathematics of Computation
JF - Mathematics of Computation
IS - 215
ER -