Publications

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In this paper, we present and analyze a BDDC algorithm for a class of elliptic problems in the three-dimensional H(curl) space. Compared with existing results, our condition number estimate requires fewer assumptions and also involves two fewer powers of log(H/h), making it consistent with optimal estimates for other elliptic problems. Here, H/h is the maximum of Hi/hi over all subdomains, where Hi and hi are the diameter and the smallest element diameter for the subdomain Ωi. The analysis makes use of two recent developments. The first is a new approach to averaging across the subdomain interfaces, while the second is a new technical tool that allows arguments involving trace classes to be avoided. Numerical examples are presented to confirm the theory and demonstrate the importance of the new averaging approach in certain cases.

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Finite element analysis of transient acoustic phenomena on unbounded exterior domains is very common in engineering analysis. In these problems there is a common need to compute the acoustic pressure at points outside of the acoustic mesh, since meshing to points of interest is impractical in many scenarios. In aeroacoustic calculations, for example, the acoustic pressure may be required at tens or hundreds of meters from the structure. In these cases, a method is needed for post-processing the acoustic results to compute the response at far-field points. In this paper, we compare two methods for computing far-field acoustic pressures, one derived directly from the infinite element solution, and the other from the transient version of the Kirchhoff integral. Here, we show that the infinite element approach alleviates the large storage requirements that are typical of Kirchhoff integral and related procedures, and also does not suffer from loss of accuracy that is an inherent part of computing numerical derivatives in the Kirchhoff integral. In order to further speed up and streamline the process of computing the acoustic response at points outside of the mesh, we also address the nonlinear iterative procedure needed for locating parametric coordinates within the host infinite element of far-field points, the parallelization of the overall process, linear solver requirements, and system stability considerations.

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The first part of this talk provides a basic introduction to the building blocks of domain decomposition solvers. Specific details are given for both the classical overlapping Schwarz (OS) algorithm and a recent iterative substructuring (IS) approach called balancing domain decomposition by constraints (BDDC). A more recent hybrid OS-IS approach is also described. The success of domain decomposition solvers depends critically on the coarse space. Similarities and differences between the coarse spaces for OS and BDDC approaches are discussed, along with how they can be obtained from discrete harmonic extensions. Connections are also made between coarse spaces and multiscale modeling approaches from computational mechanics. As a specific example, details are provided on constructing coarse spaces for incompressible fluid problems. The next part of the talk deals with a variety of implementation details for domain decomposition solvers. These include mesh partitioning options, local and global solver options, reducing the coarse space dimension, dealing with constraint equations, residual weighting to accelerate the convergence of OS methods, and recycling of Krylov spaces to efficiently solve problems with multiple right hand sides. Some potential bottlenecks and remedies for domain decomposition solvers are also discussed. The final part of the talk concerns some recent theoretical advances, new algorithms, and open questions in the analysis of domain decomposition solvers. The focus will be primarily on the work of the speaker and his colleagues on elasticity, fluid mechanics, problems in H(curl), and the analysis of subdomains with irregular boundaries.

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In this talk, we present a collection of domain decomposition algorithms for mixed finite element formulations of elasticity and incompressible fluids. The key component of each of these algorithms is the coarse space. Here, the coarse spaces are obtained in an algebraic manner by harmonically extending coarse boundary data. Various aspects of the coarse spaces are discussed for both continuous and discontinuous interpolation of pressure. Further, both classical overlapping Schwarz and hybrid iterative substructuring preconditioners are described. Numerical results are presented for almost incompressible elasticity and the Navier Stokes equations which demonstrate the utility of the methods for both structured and irregular mesh decompositions. We also discuss a simple residual scaling approach which often leads to significant reductions in iterations for these algorithms.

Advanced computing hardware and software written to exploit massively parallel architectures greatly facilitate the computation of extremely large problems. On the other hand, these tools, though enabling higher fidelity models, have often resulted in much longer run-times and turn-around-times in providing answers to engineering problems. The impediments include smaller elements and consequently smaller time steps, much larger systems of equations to solve, and the inclusion of nonlinearities that had been ignored in days when lower fidelity models were the norm. The research effort reported focuses on the accelerating the analysis process for structural dynamics though combinations of model reduction and mitigation of some factors that lead to over-meshing.

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A simple and effective approach is presented to construct coarse spaces for overlapping Schwarz preconditioners. The approach is based on energy minimizing extensions of coarse trace spaces, and can be viewed as a generalization of earlier work by Dryja, Smith, and Widlund. The use of these coarse spaces in overlapping Schwarz preconditioners leads to condition numbers bounded by C(1 + H/δ)(1 + log(H/h)) for certain problems when coefficient jumps are aligned with subdomain boundaries. For problems without coefficient jumps, it is possible to remove the log(H/h) factor in this bound by a suitable enrichment of the coarse space. Comparisons are made with the coarse spaces of two other substructuring preconditioners. Numerical examples are also presented for a variety of problems.

The Balancing Domain Decomposition by Constraints (BDDC) method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory. © 2008 Springer-Verlag.

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