An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.

The extended/generalized finite element method: An overview of the method and its applications

The FETI method and its two-level extension (FETI-2) are two numerically scalable domain decomposition methods with Lagrange multipliers for the iterative solution of second-order solid mechanics and fourth-order beam, plate and shell structural problems, respectively.The FETI-2 method distinguishes itself from the basic or one-level FETI method by a second set of Lagrange multipliers that are introduced at the subdomain cross-points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations. In this paper, we present a dual–primal formulation of the FETI-2 concept that eliminates the need for that second set of Lagrange multipliers, and unifies all previously developed one-level and two-level FETI algorithms into a single dual–primal FETI-DP method. We show that this new FETI-DP method is numerically scalable for both second-order and fourth-order problems. We also show that it is more robust and more computationally efficient than existing FETI solvers, particularly when the number of subdomains and/or processors is very large. Copyright © 2001 John Wiley & Sons, Ltd.

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FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method

The first author acknowledges the partial support by a National Science Foundation Graduate Fellowship and the partial support by a National Defense Science and Engineering Graduate Fellowship. The second and third authors acknowledge the partial support by the Motor Sports Division of the Toyota Motor Corporation under Agreement Number 48737, and the partial support by a research grant from the Academic Excellence Alliance program between King Abdullah University of Science and Technology (KAUST) and Stanford University. All authors also acknowledge the constructive comments received during the review process.

Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations

An algebraic multigrid method is presented to solve large systems of linear equations. The coarsen- ing is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the number of variables by a factor slightly less than four. The matching algorithm favors the strongest negative coupling(s), inducing a problem depen- dant coarsening. This aggregation is combined with piecewise constant (unsmoothed) prolongation, ensuring low setup cost and memory requirements. Compared with previous aggregation-based multigrid methods, the scalability is enhanced by using a so-called K-cycle multigrid scheme, providing Krylov subspace acceleration at each level. This paper is the logical continuation of (SIAM J. Sci. Comput., 30 (2008), pp. 1082-1103), where the analysis of a model anisotropic problem shows that aggregation-based two-grid methods may have optimal order convergence, and of (Numer. Lin. Alg. Appl., 15 (2008), pp. 473-487), where it is shown that K-cycle multigrid may provide optimal or near optimal convergence under mild assumptions on the two-grid scheme. Whereas in these papers only model problems with geometric aggregation were considered, here a truly algebraic method is presented and tested on a wide range of discrete second order scalar elliptic PDEs, including nonsymmetric and unstructured problems. Numerical results indicate that the proposed method may be significantly more robust as black box solver than the classical AMG method as implemented in the code AMG1R5 by K. Stuben. The parallel implementation is also discussed. Satisfactory speedups are obtained on a medium size multi-processor cluster that is typical of today com- puter market. A code implemanting the method is freely available for download both as a Fortran program and a Matlab function.

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An aggregation-based algebraic multigrid method

State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite-element methods are described. Stabilized, multiscale, and other wave-based discretization methods developed to reduce this error are reviewed. A review of finite-element methods for acoustic inverse problems and shape optimization is also given.

A review of finite-element methods for time-harmonic acoustics

http://math.ucdenver.edu/~jmandel/papers/dpc.pdf

An algebraic theory for primal and dual substructuring methods by constraints

We analyze the convergence of a substructing iterative method with Lagrange multipliers, propose recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number C(1+log(H/h))^\alpha, \alpha=2 or 3, where h is the characteristic element size, H subdomain size, and C is independent of the number of subdomains.

Convergence of a Substructuring Method with Lagrange Multipliers

In the Dual-Primal FETI method, introduced by Farhat et al. (1999), the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by $C(1+\log^2(H/h))$ for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model.

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On the Convergence of a Dual-Primal Substructuring Method

A dual-primal variant of the FETI-H domain decomposition method is designed for the fast, parallel, iterative solution of large-scale systems of complex equations arising from the discretization of acoustic scattering problems formulated in bounded computational domains. The convergence of this iterative solution method, named here FETI-DPH, is shown to scale with the problem size, the number of subdomains, and the wave number. Its solution time is also shown to scale with the problem size. CPU performance results obtained for the acoustic signature analysis in the mid-frequency regime of mockup submarines reveal that the proposed FETI-DPH solver is significantly faster than the previous generation FETI-H solution algorithm.

Feti-dph: a dual-primal domain decomposition method for acoustic scattering

Recently, a discontinuous Galerkin finite element method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of Helmholtz problems in the mid-frequency regime. In this paper, this method is extended to three dimensions and several new elements are proposed. Computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher-order Galerkin method.

Radek Tezaur

Papers

An algebraic theory for primal and dual substructuring methods by constraints

Convergence of a Substructuring Method with Lagrange Multipliers

On the Convergence of a Dual-Primal Substructuring Method

Feti-dph: a dual-primal domain decomposition method for acoustic scattering

Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems