A domain decomposition method for second order nonlinear equations
Abstract
In this communication we give a result of existence and present a numerical analysis of weak solutions for the following quasi-linear elliptic problem in one and two dimensions: −Au(x) + G(x,Du(x)) = F(x, u(x)) + f(x) in Ω, u(x) = 0 on ðΩ where A is a second order derivatives operator in one dimension and the Laplace operator in two dimensions, G, F are Caratheodory non negative functions. The function f is given finite and non negative. The domain Ω ⊂ ℝN, N = 1, 2 is open and bounded. Such problems arise from biological, chemical and physical systems and various methods have been proposed to study the existence, uniqueness, qualitative properties and numerical simulation of solutions. In the one dimensional case we consider a situation where f is irregular , more precisely a non negative measure in (0, 1) and where the growth of G with respect to Du = u' and F with respect to u are arbitrary. In the two dimensional case we assume f ∈ L1(Ω), the convexity of s -> G(x, s) and that G(x, s) is sub-quadratic w.r.t. s. The general algorithm for the numerical solution of these equations is one application of the Newton method to the discretized version of problem (30), but at each iteration the resulting system can be indefinite. To overcame this difficulty we introduce a domain decomposition method. Then in the first step of the algorithm we compute a super solution using a domain decomposition method. In the second step we compute a sequence of solutions of an intermediate problem obtained by using the Yosida approximation of G. This sequence converges to the weak solution of the problem (30). We present numerical examples.