Keywords: domain decomposition, nonlinear elliptic partial differential equations.

In this paper we prove the existence of weak solutions for some quasi-linear parabolic and periodic problems and present a method to compute a numerical solution. More precisely we consider the following quasi-linear parabolic and periodic equations, referred to as Problem 1. We are particularly interested in situations where the growth of J, the nonlinearity with respect to the gradient, is critical (quadratic).

The main result of the paper shows that if a supersolution of Problem 1 exists, then a weak solution u of Problem 1 exists such that 0 <= u <= w.

In the last section of the paper we present a numerical iterative method to solve Problem 1. Formally the iterative method constructs a sequence of numerical solutions of the Yosida approximation of Problem 1, with a first estimate which is a supersolution of the considered problem [1].

The algorithm to compute the supersolution is based on the Schwarz overlapping domain decomposition method, combined with the finite element method. The domain partition should be determined by the behavior of the non-linearity F(t,x,u). In a second step a weak solution of the non-linear problem is computed using a Newton method.

1

N. Alaa, J.R. Roche, "Theoretical and Numerical Analysis of a Class of Nonlinear Elliptic Equations", Mediterr. J. Math, 2, 327-344, 2005. doi:10.1007/s00009-005-0048-4

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