A combination of block-Jacobi and deflation preconditioning is used to solve a high-order discontinuous element-based collocation discretization of the Schur complement of the Poisson-Neumann system as arises in the operator splitting of the incompressible Navier-Stokes equations. The ill-posedness of the Poisson-Neumann system manifests as an inconsistency of the Schur complement problem, but it is shown that this can be accounted for with appropriate projections out of the null space of the Schur complement matrix without affecting the accuracy of the solution. The block-Jacobi preconditioner, combined with deflation, is shown to yield GMRES convergence independent of the polynomial order of expansion within an element. Finally, while the number of GMRES iterations does grow as the element size is reduced (e.g. $h$-refinement), the dependence is very mild; the number of GMRES iterations roughly doubles as the element size is divided by a factor of six. In light of these numerical results, the deflated Schur complement approach seems practicable, especially for high-order methods given its convergence independent of polynomial order.
Deflation-accelerated preconditioning of the Poisson-Neumann Schur problem on long domains with a high-order discontinuous element-based collocation method
Cornell University（康奈尔大学）；Applied Research Laboratories（应用研究实验室）
GPU-Accelerated Nested Schur-Complement Solvers for Atmospheric Flow Simulations Using Horizontally-Explicit, Vertically-Implicit Schemes and Discontinuous Galerkin Methods
Johannes Gutenberg University Mainz（约翰内斯·古腾堡美因茨大学）；Universidad Politécnica de Madrid（马德里理工大学）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）；University of Cologne（科隆大学）
Multigrid methods for Hdiv-conforming discontinuous Galerkin methods for the Stokes equations
Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Universität Heidelberg（海德堡大学计算科学跨学科中心）；Department of Mathematics, Texas A&M University（德克萨斯农工大学数学系）
Parameter-free inexact block Schur complement preconditioning for linear poroelasticity under a hybrid Bernardi-Raugel and weak Galerkin finite element discretization
An ADMM-Newton-CNN Numerical Approach to a TV Model for Identifying Discontinuous Diffusion Coefficients in Elliptic Equations: Convex Case with Gradient Observations
A finite volume/discontinuous Galerkin method for the advective Cahn-Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging
Rice University（莱斯大学）；Shell International Exploration and Production Inc.（壳牌国际勘探与生产公司）
A Weak Galerkin Finite Element Method for the Maxwell Equations
Michigan State University（密歇根州立大学）；National Science Foundation（美国国家科学基金会）；University of Arkansas at Little Rock（阿肯色大学小石城分校）；University of Delaware（特拉华大学）
