Higher order finite element discretizations, although providing higher accuracy, are considered to be computationally expensive and of limited use for large-scale problems. In this paper, we have developed an efficient iterative solver for solving large-scale quadratic finite element problems. The proposed approach shares some common features with geometric multigrid methods but does not need structured grids to create the coarse problem. This leads to a robust method applicable to finite element problems discretized by unstructured meshes such as those from adaptive remeshing strategies. The method is based on specific properties of hierarchical quadratic bases. It can be combined with an algebraic multigrid (AMG) preconditioner or with other algebraic multilevel block factorizations. The algorithm can be accelerated by flexible Krylov subspace methods. We present some numerical results on the convection-diffusion and linear elasticity problems to illustrate the efficiency and the robustness of the presented algorithm. In these experiments, the performance of the proposed method is compared with that of an AMG preconditioner and other iterative solvers. Our approach requires less computing time and less memory storage.In our case, the space V l will refer to the coarse subspace associated with a linear finite elements (P 1 ) discretization, while the space V h will be related to the finer quadratic finite elements (P 2 ) discretization. This is reminiscent of multigrid methods but instead of having a sequence of nested grids, we vary the degree of interpolation on the same grid. In order to simplify the notation, the element of V h will be denoted by u ∈ V h and u l ∈ V l from V l . The restricted discrete problems can now be formulated asThese two discrete systems can be written in a matrix formwhere, from now on, u and u l stand for the nodal vectors. Our goal is to build an iterative method that will solve the global problem (Au = b) efficiently from the solution of the coarse problem A ll u l = b l . Let us decompose the space V h as a direct sum starting from the coarse space V l * l = d l by a direct method or approximatively by an iterative method.

SUMMARYWe present new iterative solvers for large-scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second-order accuracy can be obtained at very small overcost with respect to first-order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2×2 block symmetric indefinite linear system arising from mixed (displacement-pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods.

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SUMMARYWe develop an efficient preconditioning techniques for the solution of large linearized stationary and non-stationary incompressible Navier-Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non-stationary case. The time discretization procedure uses the Gear scheme and the second-order Taylor-Hood element P 2 − P 1 is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r ∇(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P 1 − P 2 hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented.

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