87 research outputs found

    Quasi-optimal Schwarz methods for the conforming spectral element discretization

    Get PDF
    The spectral element method is used to discretize self-adjoint elliptic equations in three-dimensional domains. The domain is decomposed into hexahedral elements, and in each of the elements the discretization space is the set of polynomials of degree N in each variable. A conforming Galerkin formulation is used, the corresponding integrals are computed approximately with Gauss-Lobatto-Legendre (GLL) quadrature rules of order N, and a Lagrange interpolation basis associated with the GLL nodes is used. Fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method. The conforming finite element space on the GLL mesh, consisting of piecewise Q(1) or P-1 functions, produces a stiffness matrix K-h that is known to be spectrally equivalent to the spectral element stiffness matrix K-N. K-h is replaced by a preconditioner Kh which is well adapted to parallel computer architectures. The preconditioned operator is then K-N. Techniques for nonregular meshes are developed, which make it possible to estimate the condition number of K-N, where is a standard finite element preconditioner of K-h, based on the GLL mesh. Two finite element-based preconditioners: the wirebasket method of Smith and the overlapping Schwarz algorithm for the spectral element method are given ay examples of the use of these tools. Numerical experiments performed by Pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions.3462482250

    Domain decomposition methods for domain composition purpose: Chimera, overset, gluing and sliding mesh methods

    Get PDF
    Domain composition methods (DCM) consist in obtaining a solution to a problem, from the formulations of the same problem expressed on various subdomains. These methods have therefore the opposite objective of domain decomposition methods (DDM). Indeed, in contrast to DCM, these last techniques are usually applied to matching meshes as their purpose consists mainly in distributing the work in parallel environments. However, they are sometimes based on the same methodology as after decomposing, DDM have to recompose. As a consequence, in the literature, the term DDM has many times substituted DCM. DCM are powerful techniques that can be used for different purposes: to simplify the meshing of a complex geometry by decomposing it into different meshable pieces; to perform local refinement to adapt to local mesh requirements; to treat subdomains in relative motion (Chimera, sliding mesh); to solve multiphysics or multiscale problems, etc. The term DCM is generic and does not give any clue about how the fragmented solutions on the different subdomains are composed into a global one. In the literature, many methodologies have been proposed: they are mesh-based, equation-based, or algebraic-based. In mesh-based formulations, the coupling is achieved at the mesh level, before the governing equations are assembled into an algebraic system (mesh conforming, Shear-Slip Mesh Update, HERMESH). The equation-based counterpart recomposes the solution from the strong or weak formulation itself, and are implemented during the assembly of the algebraic system on the subdomain meshes. The different coupling techniques can be formulated for the strong formulation at the continuous level, for the weak formulation either at the continuous or at the discrete level (iteration-by-subdomains, mortar element, mesh free interpolation). Although the different methods usually lead to the same solutions at the continuous level, which usually coincide with the solution of the problem on the original domain, they have very different behaviors at the discrete level and can be implemented in many different ways. Eventually, algebraic- based formulations treat the composition of the solutions directly on the matrix and right-hand side of the individual subdomain algebraic systems. The present work introduces mesh-based, equation-based and algebraicbased DCM. It however focusses on algebraic-based domain composition methods, which have many advantages with respect to the others: they are relatively problem independent; their implicit implementation can be hidden in the iterative solver operations, which enables one to avoid intensive code rewriting; they can be implemented in a multi-code environment

    The INTERNODES method for the treatment of non-conforming multipatch geometries in Isogeometric Analysis

    Full text link
    In this paper we apply the INTERNODES method to solve second order elliptic problems discretized by Isogeometric Analysis methods on non-conforming multiple patches in 2D and 3D geometries. INTERNODES is an interpolation-based method that, on each interface of the configuration, exploits two independent interpolation operators to enforce the continuity of the traces and of the normal derivatives. INTERNODES supports non-conformity on NURBS spaces as well as on geometries. We specify how to set up the interpolation matrices on non-conforming interfaces, how to enforce the continuity of the normal derivatives and we give special attention to implementation aspects. The numerical results show that INTERNODES exhibits optimal convergence rate with respect to the mesh size of the NURBS spaces an that it is robust with respect to jumping coefficients.Comment: Accepted for publication in Computer Methods in Applied Mechanics and Engineerin

    Finite element approximation of multi-scale elliptic problems using patches of elements

    Get PDF
    In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presente

    Domain decomposition preconditioners of Neumann-Neumann type for hp‐approximations on boundary layer meshes in three dimensions

    Get PDF
    We develop and analyse Neumann-Neumann methods for hp finite‐element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. These are meshes that are highly anisotropic where the aspect ratio typically grows exponentially with the polynomial degree. The condition number of our preconditioners is shown to be independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. In addition, it only grows polylogarithmically with the polynomial degree, as in the case of p approximations on shape‐regular meshes. This work generalizes our previous one on two‐dimensional problems in Toselli & Vasseur (2003a, submitted to Numerische Mathematik, 2003c to appear in Comput. Methods Appl. Mech. Engng.) and the estimates derived here can be employed to prove condition number bounds for certain types of FETI method

    Substructuring Preconditioners for h-p Mortar FEM

    Get PDF
    International audienceWe build and analyze a substructuring preconditioner for the mortar method in the h-p finite element framework. Particular attention is given to the construction of the coarse component of the preconditioner in this framework, in which continuity at the cross points is not required. Two variants are proposed: the first one is an improved version of a coarse preconditioner already presented in [12]. The second is new and is built by using a Discontinuous Galerkin interior penalty method as coarse problem. A bound of the condition number is proven for both variants and their efficiency and scalability is illustrated by numerical experiments
    corecore