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A new nonlinear Galerkin method based on nite element discretization is presented in this paper for a class of second order nonlinear parabolic equations. The new scheme is based on two di erent nite element spaces de ned respectively on one coarse grid with grid size H and one ne grid with grid size h H. Nonlinearity and time dependence are both treated on the coarse space and only a xed stationary equation needs to be solved on the ne space at each time. With linear nite element discretizations, it is proved that the di erence between the new nonlinear Galerkin solution and the standard Galerkin solution in H 1 () norm is of the order of H 3 .

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To cite this version:M. Marion. Qualitative properties of a nonlinear system for laminar flames without ignition temperature. RR-0293, INRIA. 1984.

Summary. With the increase in the computing power and the advent of supercomputers, the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem. In this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements. The algorithm that we propose, called the nonlinear Galerkin method, stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete (finite elements) space. Essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases. Beside a detailed description of the algorithm, the article includes some technical results on finite elements spaces, and a full study of the stability and convergence of the method.

Abstract. In this paper we provide estimates to the rate of convergence of the nonlinear Galerkin approximation method. In particular, and by means of an illustrative example, we show that the nonlinear Galerkin method converges faster than the usual Galerkin method.