Abstract.We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L 2 norm in space and time, and thus has a higher-order than if it were measured in the H 1 norm in space. We present the following results: if h = H 2 = k, then the global error of the two-grid algorithm is of the order of h, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.Mathematics Subject Classification. 35Q30, 74S10, 76D05.
We study a second-order two-grid scheme fully discrete in time and space for solving the Navier-Stokes equations. The two-grid strategy consists in discretizing, in the first step, the fully non-linear problem, in space on a coarse grid with mesh-size H and time step t and, in the second step, in discretizing the linearized problem around the velocity u H computed in the first step, in space on a fine grid with mesh-size h and the same time step. The two-grid method has been applied for an analysis of a first order fully-discrete in time and space algorithm and we extend the method to the second order algorithm. This strategy is motivated by the fact that under suitable assumptions, the contribution of u H to the error in the non-linear term, is measured in the L 2 norm in space and time, and thus has a higher-order than if it were measured in the H 1 norm in space. We present the following results: if h 2 = H 3 = ( t) 2 , then the global error of the two-grid algorithm is of the order of h 2 , the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.
A priori and a posteriori estimates for three-dimensional Stokes equations with non standard boundary conditions. 2010.
A PRIORI AND A POSTERIORI ESTIMATES FOR THREE-DIMENSIONAL STOKES EQUATIONS WITH NON STANDARD BOUNDARY CONDITIONSHYAM ABBOUD † , FIDA EL CHAMI † AND TONI SAYAH ‡ Abstract. In this paper we study the Stokes problem with some non standard boundary conditions. The variational formulation decouples into a system velocity and a Poisson equation for the pressure. The continuous and corresponding discrete system do not need an inf-sup condition. Hence, the velocity is approximated with curl conforming finite elements and the pressure with standard continuous elements. Next, we establish optimal a priori and a posteriori estimates and we finish this paper with numerical tests.
In this work, we propose a bi-grid scheme framework for the Allen-Cahn equation in Finite Element Method. The new methods are based on the use of two FEM spaces, a coarse one and a fine one, and on a decomposition of the solution into mean and fluctuant parts. This separation of the scales, in both space and frequency, allows to build a stabilization on the high modes components: the main computational effort is concentrated on the coarse space on which an implicit scheme is used while the fluctuant components of the fine space are updated with a simple semi-implicit scheme; they are smoothed without damaging the consistency. The numerical examples we give show the good stability and the robustness of the new methods. An important reduction of the computation time is also obtained when comparing our methods with fully implicit ones.
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