A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations.

In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code

Block Preconditioners Based on Approximate Commutators

This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible Navier-Stokes equations. We consider the "pressure convection-diffusion preconditioners" proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [16]. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall e#cacy of this methodology. A drawback, however, is that it requires the construction of a convection-diffusion operator (denoted Fp ) projected onto the discrete pressure space. This means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues, something often held only by the developers of the model. As an alternative, we consider automatic ways of computing Fp based on purely algebraic considerations. The new methods are closely related to the "BFBt preconditioner" of Elman [6]. We use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection-diffusion operators, together with methods for computing sparse approximate inverses, to generate the required matrix Fp automatically. We demonstrate that with this strategy, the favorable convergence properties of the preconditioning methodology are retained

Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations

Abstract. This paper introduces two stabilization schemes for the least squares commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth, and Tuminaro [SIAM J. Sci. Comput., 27 (2006), pp. 1651–1668] for the incompressible Navier–Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier–Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to div-stable discretizations of the Navier–Stokes equations. This paper shows how to extend the same methodology to stabilized low-order mixed finite element approximation methods. Key words. preconditioning, Navier–Stokes, iterative algorithm

Developing classification criteria for discoid lupus erythematosus: an update from the World Congress of Dermatology 2015 meeting

Currently, no standardized classification criteria exist for cutaneous lupus erythematosus. With increased interest in studying cutaneous lupus erythematosus, specifically discoid lupus erythematosus, it is our aim to apply previously adopted methods from rheumatology to dermatologic diseases to develop feasible, validated, and standardized classification criteria useful in both academic and community practice. Here we report the progress to date to define discoid lupus erythematosus using clinical, histopathologic, and serologic features by means of a Delphi method—using a series of iterative questionnaires sent to expert stakeholders. We present specific updates from the World Congress of Dermatology 2015 meeting, at which a nominal group of expert stakeholders met to discuss the results of round 1 of the Delphi process to further clarify and harmonize specific classification items for inclusion into round 2

Development of classification criteria for discoid lupus erythematosus: Results of a Delphi exercise

Background: No classification criteria currently exist for discoid lupus erythematosus (DLE), which has led to problematic heterogeneity in both observational and interventional research efforts. Objectives: We sought to develop DLE classification criteria based on consensus of international expert opinion of relevant stakeholders in the field. Methods: Using a Delphi consensus process and nominal group techniques, potential items for classification criteria were generated. Experts ranked items in terms of their appropriateness and ability to discriminate DLE from other diagnoses, and items were subsequently eliminated using consensus exercises. Results: A final list of 12 clinical and histopathologic items was generated for potential inclusion into a set of DLE classification criteria through a formal ongoing validation process. Limitations: The participants are predominantly composed of DLE experts in North America and Europe. Conclusion: This work represents a key step toward the development of formal DLE classification criteria

Developing classification criteria for discoid lupus erythematosus: an update from the World Congress of Dermatology 2015 meeting

Currently, no standardized classification criteria exist for cutaneous lupus erythematosus. With increased interest in studying cutaneous lupus erythematosus, specifically discoid lupus erythematosus, it is our aim to apply previously adopted methods from rheumatology to dermatologic diseases to develop feasible, validated, and standardized classification criteria useful in both academic and community practice. Here we report the progress to date to define discoid lupus erythematosus using clinical, histopathologic, and serologic features by means of a Delphi method—using a series of iterative questionnaires sent to expert stakeholders. We present specific updates from the World Congress of Dermatology 2015 meeting, at which a nominal group of expert stakeholders met to discuss the results of round 1 of the Delphi process to further clarify and harmonize specific classification items for inclusion into round 2. Key words: Discoid Lupus Erythematosus, Cutaneous Lupus Erythematosus, Lupus, Connective Tissue Diseases, Delphi method, classification criteria, World Congress of Dermatolog