Block monotone iterative methods
for numerical solutions of nonlinear
elliptic equations |
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Authors: | CV Pao |
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Institution: | (1) Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205 , US |
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Abstract: | Summary.
Two block monotone iterative schemes for a nonlinear
algebraic system, which is a finite difference approximation of a
nonlinear elliptic boundary-value problem, are presented and are
shown to converge monotonically either from above or from below to
a solution of the system. This monotone convergence result yields
a computational algorithm for numerical solutions as well as an
existence-comparison theorem of the system, including a sufficient
condition for the uniqueness of the solution. An advantage of the
block iterative schemes is that the Thomas algorithm can be used to
compute numerical solutions of the sequence of iterations in the
same fashion as for one-dimensional problems. The block iterative
schemes are compared with the point monotone iterative schemes of
Picard, Jacobi and Gauss-Seidel, and various theoretical comparison
results among these monotone iterative schemes are given. These
comparison results demonstrate that the sequence of iterations from
the block iterative schemes converges faster than the corresponding
sequence given by the point iterative schemes. Application of the
iterative schemes is given to a logistic model problem in ecology
and numerical ressults for a test problem with known analytical
solution are given.
Received
August 1, 1993 / Revised version received November 7, 1994 |
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Keywords: | Mathematics Subject Classification (1991):65N22 65N06 |
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