1,721,056 research outputs found
Indefinite overlapping Schwarz methods for time-dependent Stokes problems
No full textA class of indefinite overlapping Schwarz methods is introduced and studied for time-dependent Stokes problems, discretized with either mixed finite elements or mixed spectral elements. The methods proposed are based on the solution of local time-dependent Stokes problems on overlapping subdomains and on the solution of a coarse time-dependent Stokes problem defined on the coarse subdomain mesh. The resulting preconditioner is accelerated by an appropriate Krylov space method, producing a very efficient, scalable and parallel solver. (C) 2000 Elsevier Science S.A. All rights reserved
Preconditioners for spectral discretizations of Helmholtz's equation with Sommerfeld boundary conditions
No full textSome preconditioners for the iterative solution of Helmholtz's equation discretized with spectral Legendre collocation methods are introduced and studied. The preconditioners are based either on a finite element discretization of Helmholtz's equation on the spectral collocation mesh or on replacing the Sommerfeld-like boundary condition on a subset of the boundary with either Neumann or Dirichlet boundary conditions. The convergence rate of the resulting iterative methods is only mildly dependent on the spectral degree N and the wave number k
Overlapping Schwarz Methods for Unstructured Spectral Elements
No full textA parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied. The spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron, pyramid, prism, or cube in three dimensions. The method is based on overlapping Schwarz techniques applied to the Schur complement of the discrete system and is implemented as a preconditioner for a Krylov space method. Numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains. The resulting elliptic solver can be used in Navier–Stokes simulations using the spectral element code NekTar
An explicit second order spectral element method for acoustic waves
No full textThe acoustic wave equation is here discretized by conforming spectral elements in space and by the second order leap-frog method in time. For simplicity, homogeneous boundary conditions are considered. A stability analysis of the resulting method is presented, providing an upper bound for the allowed time step that is proportional to the size of the elements and inversely proportional to the square of their polynomial degree. A convergence analysis is also presented, showing that the convergence error decreases when the time step or the size of the elements decrease or when the polynomial degree increases. Several numerical results illustrating these results are presented
Overlapping Schwarz and spectral element methods for linear elasticity and elastic waves
No full textThe classical overlapping Schwarz algorithm is here extended to the spectral element discretization of linear elastic problems, for both homogeneous and heterogeneous compressible materials. The algorithm solves iteratively the resulting preconditioned system of linear equations by the conjugate gradient or GMRES methods. The overlapping Schwarz preconditioned technique is then applied to the numerical approximation of elastic waves with spectral elements methods in space and implicit Newmark time advancing schemes. The results of several numerical experiments, for both elastostatic and elastodynamic problems, show that the convergence rate of the proposed preconditioning algorithm is independent of the number of spectral elements (scalability), is independent of the spectral degree in case of generous overlap, otherwise it depends inversely on the overlap size. Some results on the convergence properties of the spectral element approximation combined with Newmark schemes for elastic waves are also presented
Domain decomposition methods with small overlap for Qn−Qn−2 spectral elements
No full textAn indefinite overlapping Schwarz method is introduced and studied for Stokes problems discretized with Qn−Qn−2 spectral elements. This results in a parallel and scalable preconditioner for the iterative solution of the discrete system of equations. The preconditioner is based on the solution of local Stokes problems on overlapping subregions and a coarse Stokes problem with Q2−Q0 elements. The efficiency of the method is based on using a nodal basis and quadrature rules associated with Gauss–Lobatto–Legendre nodes. As for h-version finite elements, a small overlap between subregions seems to be the most efficient choice
Balancing Neumann-Neumann methods for incompressible Stokes equations
No full textBalancing Neumann-Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant-pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann-Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf-sup constants of the discrete problem and of the coarse subproblem. Numerical results show that the method is quite fast; they are also fully consistent with the theory
Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems. I: Compressible Linear Elasticity
No full textAn iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed. The pure displacement formulation for compressible materials is discretized with the spectral element method. The resulting stiffness matrix is symmetric and positive definite. The proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse, global solver related to the wire basket of the elements. As in the scalar case, the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree
Parallel multilevel Schwarz and Block preconditioners for the bidomain parabolic-parabolic and parabolic-elliptic formulations
No full textThe aim of this work is to develop parallel multilevel and block preconditioners for the Bidomain model of electrocardiology. The Bidomain model describes the electrical activity of the heart tissue and consists of a system of two parabolic nonlinear partial differential equations (PDEs) of reaction-diffusion type (PP formulation) or alternatively of a system of a parabolic nonlinear PDE and an elliptic linear PDE (PE formulation). In both formulations, the PDEs are coupled with a system of ordinary differential equations, modeling the cellular membrane ionic currents. The first goal of the present study is to construct, analyze, and numerically test a multilevel additive Schwarz preconditioner for the PE formulation of the Bidomain model, extending previous results obtained for the PP formulation. Optimal convergence rate estimates are established and confirmed by 3D numerical test on Linux clusters. The second goal of the present study is to analyze the scalability of multilevel Schwarz block-diagonal and block-factorized preconditioners for both PP and PE formulations of the Bidomain model and to compare them with multilevel Schwarz coupled preconditioners. The 3D parallel numerical tests show that block preconditioners for the PP formulation are not scalable, while they are scalable for the PE formulation, but less efficient than the coupled preconditioners
BDDC and FETI--DP preconditioners for spectral element discretizations of almost incompressible elasticity
No full textWe construct and study a BDDC (Balancing Domain Decomposition by Constraints) algorithm, see [1, 2], for the system of almost incompressible elasticity discretized with Gauss Lobatto Legendre (GLL) spectral elements. Related FETIDP algorithms could be considered as well. We show that sets of primal constraints can be found so that these methods have a condition number that depends only weakly on the polynomial degree, while being independent of the number of subdomains (scalability) and of the Poisson ratio and Youngs modulus of the material considered (robustness)
- …
