Marcus Sarkis

A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems

We introduce some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems and show, by numerical examples, that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost when implemented on distributed memory computers. This is especially true for harder problems such as indefinite complex linear systems and systems of convection-diffusion equations from three-dimensional compressible flows. Both sequential and parallel results are reported.

A linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems in all flow regimes

Abstract We present a computational fluid dynamics (CFD)-based linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems. This method is valid in the subsonic, transonic, and supersonic flow regimes, and is insensitive to the frequency or damping level of the sought-after coupled eigenmodes. It is based on the solution by an orthogonal iteration procedure of a complex eigenvalue problem derived from the linearization of a three-field fluid/structure/moving mesh formulation. The key computational features of the proposed method include the reuse of existing unsteady flow solvers, a second-order approximation of the flux Jacobian matrix, and a parallel domain decomposition-based iterative solver for the solution of large-scale systems of discretized fluid/structure equations. While the frequency analysis method proposed here is primarily targeted at the extraction of the eigenpairs of a wet structure, we validate it with the flutter analysis of the AGARD Wing 445.6, for which experimental data is available.

Overlapping Nonmatching Grid Mortar Element Methods for Elliptic Problems

In the first part of the paper, we introduce an overlapping mortar finite element method for solving two-dimensional elliptic problems discretized on overlapping nonmatching grids. We prove an optimal error bound and estimate the condition numbers of certain overlapping Schwarz preconditioned systems for the two-subdomain case. We show that the error bound is independent of the size of the overlap and the ratio of the mesh parameters. In the second part, we introduce three additive Schwarz preconditioned conjugate gradient algorithms based on the trivial and harmonic extensions. We provide estimates for the spectral bounds on the condition numbers of the preconditioned operators. We show that although the error bound is independent of the size of the overlap, the condition number does depend on it. Numerical examples are presented to support our theory.

Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity

We consider a stochastic Darcys pressure equation whose coefficient is generated by a white noise process on a Hilbert space employing the ordinary (rather than the Wick) product. A weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf-sup condition and well-posedness of the problem. We generalize the numerical approximations proposed in Benth and Theting [Stochastic Anal. Appl., 20 (2002), pp. 1191-1223] for Wick stochastic partial differential equations to the ordinary product stochastic pressure equation. We establish discrete inf-sup conditions and provide a priori error estimates for a wide class of norms. The proposed numerical approximation is based on Wiener-Chaos finite element methods and yields a positive definite symmetric linear system. We also improve and generalize the approximation results of Benth and Gjerde [Stochastics Stochastics Rep., 63 (1998), pp. 313-326] and Cao [Stochastics, 78 (2006), pp. 179-187] when a (generalized) process is truncated by a finite Wiener-Chaos expansion. Finally, we present numerical experiments to validate the results.

Balancing Domain Decomposition Methods for Mortar Coupling Stokes-Darcy Systems

We consider Stokes equations in the fluid region Ωf and Darcy equations for the filtration velocity in the porous medium Ωp, and coupled at the interface Γ with adequate transmission conditions. Such problem appears in several applications like well-reservoir coupling in petroleum engineering, transport of substances across groundwater and surface water, and (bio)fluid-organ interactions. There are some works that address numerical analysis issues such as inf-sup and approximation results associated to the continuous and discrete formulations Stokes-Darcy systems [8, 7, 6] and Stokes-Laplacian systems [2, 3], mortar discretizations analysis [12, 6], preconditioning analysis for Stokes-Laplacian systems [4, 1]. Here we are interested on preconditionings for Stokes-Mortar-Darcy with flux boundary conditions, therefore the global system as well as the local systems require flux compatibilities. Here we propose two preconditioners based on balancing domain decomposition methods [9, 11, 5]; in the first one the energy of the preconditioner is controlled by the Stokes system while in the second one it is controlled by the Darcy system. The second is more interesting because it is scalable for the parameters faced in practice. Let Ωf , Ωp ⊂ n be polyhedral subdomains, Ω = int(Ωf ∪ Ωp) and Γ = int(∂Ωf ∪ ∂Ωp), with outward unit normal vectors on ∂Ωj denoted by ηj , j = f, p. The tangent vectors of Γ are denoted by τ 1 (n = 2), or τ l, l = 1, 2 (n = 3). Define Γj := ∂Ωj \ Γ , j = f, p. Fluid velocities are denoted by uj : Ωj → , j = f, p. Pressures are pj : Ωj → , j = f, p. We have:

Restricted Additive Schwarz Preconditioners with Harmonic Overlap for Symmetric Positive Definite Linear Systems

A restricted additive Schwarz (RAS) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems. In this paper, we provide one-level and two-level extensions of RAS for symmetric positive definite problems using the so-called harmonic overlaps (RASHO). Both RAS and RASHO outperform their counterparts of the classical additive Schwarz variants (AS). The design of RASHO is based on a much deeper understanding of the behavior of Schwarz-type methods in overlapping subregions and in the construction of the overlap. In RASHO, the overlap is obtained by extending the nonoverlapping subdomains only in the directions that do not cut the boundaries of other subdomains, and all functions are made harmonic in the overlapping regions. As a result, the subdomain problems in RASHO are smaller than those of AS, and the communication cost is also smaller when implemented on distributed memory computers, since the right-hand sides of discrete harmonic systems are always zero and therefore do not need to be communicated. We also show numerically that RASHO-preconditioned CG takes fewer iterations than the corresponding AS-preconditioned CG. A nearly optimal theory is included for the convergence of RASHO-preconditioned CG for solving elliptic problems discretized with a finite element method.

Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems

In this paper, we describe block matrix algorithms for the iterative solution of a large-scale linear-quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon. We consider an “all at once” discretization of the problem and formulate three iterative algorithms. The first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables; however inner-outer iterations are required. The second algorithm modifies the first algorithm to avoid inner-outer iterations by introducing an auxiliary variable. It yields a symmetric indefinite system with a positive definite block preconditioner. The third algorithm is the central focus of this paper. It modifies the preconditioner in the second algorithm by a parallel-in-time preconditioner based on the parareal algorithm. Theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability. Numerical experiments confirm the theoretical results.

Maximum Norm Analysis of Overlapping Nonmatching Grid Discretizations of Elliptic Equations

In this paper, we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations. We consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from adjacent subdomains. To obtain a global finite difference discretization of the elliptic problem, we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain, by interpolating the discrete solution on adjacent subdomains. If the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large, we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations. Our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains. We describe also how the resulting linear system can be solved iteratively by a parallel Schwarz alternating method or a Schwarz preconditioned Krylov subspace iterative method. Several numerical results are included to support the theory.

A FETI-DP Preconditioner for a Composite Finite Element and Discontinuous Galerkin Method

In this paper a Nitsche-type discretization based on a discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region

Additive Average Schwarz Methods for Discretization of Elliptic Problems with Highly Discontinuous Coefficients

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