An efficient hierarchical preconditioner for quadratic discretizations of finite element problems

Abstract

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.

Efficient preconditioning techniques for finite‐element quadratic discretization arising from linearized incompressible Navier–Stokes equations

“…Using this structure, El Maliki et al [20,21] have proposed the following hierarchical preconditioner (HP):…”

“…In the literature, many other choices are found to approximate the solution of (10): ILU, SOR, SPAI [22], geometric and algebraic multigrid, and few iterations of Krylov subspace method. We refer to [20,21] for comparison with other methods.…”

“…Hence, a good preconditioner for F will be the one that keeps the iterations count at the same level compared with the exact solver of (10). For quadratic discretization of the velocity field, El Maliki et al [20,21] have proposed an efficient preconditioner especially designed for those systems of the form (10). Let us review briefly the method in the paper [21].…”

“…For quadratic discretization of the velocity field, El Maliki et al [20,21] have proposed an efficient preconditioner especially designed for those systems of the form (10). Let us review briefly the method in the paper [21]. At first, a hierarchical basis is used for the quadratic (P 2 ) finite element discretization of the velocity field.…”

Algebraic Multigrid for Moderate Order Finite Elements

“…Then, partitioning the unknowns according to the degree of the associated finite element basis function, the corresponding matrix expressed in block form possesses some interesting algebraic properties [3]. Combined with the ability to easily solve subproblems associated with linear basis functions (e.g., with an AMG method), these properties allow to easily set up fast and scalable solution algorithms; see [8] for recent results in this direction. Now, using such schemes is of course natural when the discretization is indeed performed in the hierarchical basis.…”