Some nonlinear complementarity systems algorithms and applications to unilateral boundary-value problems

Summary We consider an open bounded set and the unilateral Dirichlet problem with one or two obstacles, involving a nonlinear differential operator A. Using the finite affine triangular element method, we discretize the corresponding variational inequalities and obtain the complementarity systems with one or two constraints related to a nonlinear finite-dimensional operator F. In 21], 26], we have constructed monotone algorithms for solving such complementarity systems in the linear case, under the hypothesis: F is a P-matrix with nonpositive off-diagonal elements. In the present, we extend the applicability of the mentioned algorithms to the nonlinear case, under the hypothesis: F is continuous, coercive, off-diagonal antitone P-function in R^N. Moreover when F is a convex differentiable operator we apply the Newton method and a global linearization technique to overcome the numerical difficulties due to the nonlinearity of F. Finally we give some applications involving the pseudo-laplace operator 12] and the glaciology operator 24]. In the case of pseudo-laplace operator and a square configuration of , using an optimal regularity theorem due to El Kolli 7] we obtain also an estimate of the discretization error.