Numerical Discrete Algorithm for Some Nonlinear Problems

In this paper we use the eigenfunctions of the Laplacian to approximate the solution of some nonlinear equations which are used to model natural phenomena (as the Navier-Stokes flow equations, for instance). In this respect we propose a numerical algorithm, combining the Uzawa and Arrow-Hurwitz algorithms. The algorithm proposed here shares features from both algorithms, and it has the following advantages upon them: the usage of a single parameter (like in the Uzawa algorithm) and the fact that the approximative equation is linear (like in Arrow-Hurwitz algorithm). We prove the convergence of the approximate solution to the weak solution of the given equation. Next, we apply a Galerkin-type discretization of this algorithm in order to compute the approximate solution. 1 An Arrow-Hurwicz-Uzawa Type Algorithm In this section we develop a numerical algorithm used to approximate the solution of the stationary nonlinear Navier-Stokes system, combining Uzawa and Arrow-Hurwicz algorithms presented in [6], which represent extensions of the classical Uzawa and Arrow-Hurwicz algorithms that appear in nonlinear optimization problems. We describe the algorithm and prove the convergence