Abstract
In this paper, we consider the numerical solution for the discretization of semilinear elliptic complementarity problems. A monotone algorithm is established based on the upper and lower solutions of the problem. It is proved that iterates, generated by the algorithm, are a pair of upper and lower solution iterates and converge monotonically from above and below, respectively, to the solution of the problem. Moreover, we investigate the convergence rate for the monotone algorithm and prove quadratic convergence of the algorithm. The monotone and quadratic convergence results are also extended to the discrete problems of the two-sided obstacle problems with a semilinear elliptic operator. We also present some simple numerical experiments.
Highlights
In this paper, we consider the following semilinear elliptic complementarity problem of finding U ∈ K = {V ∈ H ( ) : V ≥ φ, a.e. in } such that a(U, V – U ) + F (U, ·), V – U ≥, ∀V ∈ K, ( . )where ∈ R is a bounded convex polygonal with boundary ∂, F (V, x) is continuously differentiable in variable V with ∂F ∂V ≥ C∗ ≥ on K× ̄, φ ∈ H (
In Sections and, we propose a monotone iterative algorithm and deal with the quadratic convergence of the monotone iterates, respectively
3 Monotone iterative algorithm for complementarity problem we propose an algorithm for solving the nonlinear complementarity problem ( . ) and discuss its monotone convergence
Summary
MSC: complementarity problem; obstacle problem; upper and lower solution; monotone iteration; quadratic convergence Any generated iterate is an upper (or lower) solution sequence which converges to the solution monotonically. By using a pair of upper and lower solutions as two initial iterates, one can construct two monotone sequences which converge monotonically from above and below, respectively, to the solutions of the problems. The initial iterative solutions in the monotone iterative algorithms can be obtained directly by solving two discrete linear complementarity problems without any knowledge of the exact solution.
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