Chak Shing Lee

A staggered discontinuous Galerkin method for the convection–diffusion equation

This paper is concerned with the staggered discontinuous Galerkin metho d for convection–diffusion equations. Over the past few decades, stagger ed type methods have been applied successfully to many problems, such as wave propagation and fl uid flow problems. A distinctive feature of these methods is that the physical laws arising from th e corresponding partial differential equations are automatically preserved. Nevertheles s, staggered methods for convection–diffusion equations are rarely seen in literature. It is thus the main goal of this paper to develop and analyze a class of staggered numerical schemes for the approximation of convection–diffusion equations. We will prove that our new method pres erv the underlying physical laws in some discrete sense. Moreover, the stability and conver gence of the method are proved. Numerical results are shown to verify the theoretical estimates.

Mixed Generalized Multiscale Finite Element Methods and Applications

In this paper, we present a mixed generalized multiscale finite element method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field. To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface. The snapshot vectors have zero Neumann boundary conditions on the outer boundaries, and we prescribe their values on the common interface. We describe several spectral decompositions in the snapshot space motivated by the analysis. In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM. A main idea of oversampling techniques is to introduce a small dimensional snapshot space. We present numerical results for two-phase flow and...

A Staggered Discontinuous Galerkin Method for the Stokes System

Discontinuous Galerkin (DG) methods are a class of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG...

A BDDC algorithm for a class of staggered discontinuous Galerkin methods

A BDDC (Balancing Domain Decomposition by Constraints) algorithm is developed and analyzed for a staggered discontinuous Galerkin (DG) finite element approximation of second order scalar elliptic problems. On a quite irregular subdomain partition, an optimal condition number bound is proved for two-dimensional problems. In addition, a sub-optimal but scalable condition number bound is obtained for three-dimensional problems. These bounds are shown to be independent of coefficient jumps in the subdomain partition. Numerical results are also included to show the performance of the algorithm.

FETI-DP preconditioners for a staggered discontinuous Galerkin formulation of the two-dimensional Stokes problem

In this paper, a class of FETI-DP preconditioners is developed for a fast solution of the linear system arising from staggered discontinuous Galerkin discretization of the two-dimensional Stokes equations. The discretization has been recently developed and has the distinctive advantages that it is optimally convergent and has a good local conservation property. In order to efficiently solve the linear system, two kinds of FETI-DP preconditioners, namely, lumped and Dirichlet preconditioners, are considered and analyzed. Scalable bounds C ( H / h ) and C ( 1 + log ( H / h ) ) 2 are proved for the lumped and Dirichlet preconditioners, respectively, with the constant C depending on the inf-sup constant of the discrete spaces but independent of any mesh parameters. Here H / h stands for the number of elements across each subdomain. Numerical results are presented to confirm the theoretical estimates.

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

We propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of fine-grid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the inf-sup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coars...

Parallel Solver for H(div) Problems Using Hybridization and AMG

In this paper, a scalable parallel solver is proposed for H(div) problems discretized by arbitrary order finite elements on general unstructured meshes. The solver is based on hybridization and algebraic multigrid (AMG). Unlike some previously studied H(div) solvers, the hybridization solver does not require discrete curl and gradient operators as additional input from the user. Instead, only some element information is needed in the construction of the solver. The hybridization results in a H1-equivalent symmetric positive definite system, which is then rescaled and solved by AMG solvers designed for H1 problems. Weak and strong scaling of the method are examined through several numerical tests. Our numerical results show that the proposed solver provides a promising alternative to ADS, a state-of-the-art solver [12], for H(div) problems. In fact, it outperforms ADS for higher order elements.

Parallel Solver for \boldsymbol{H}(div) Problems Using Hybridization and AMG

A scalable parallel solver for \(\boldsymbol{H}\)(div) problems discretized by arbitrary order finite elements on general unstructured meshes is proposed. The solver is based on hybridization and algebraic multigrid (AMG). The hybridization part of the solver requires the fine-grid element matrix information. Weak and strong scaling are examined through several numerical tests which demonstrate that the proposed solver provides a competitive alternative to ADS (Kolev and Vassilevski, SIAM J Sci Comput 34(6):A3079–A3098, 2012), a state-of-the-art solver for \(\boldsymbol{H}\)(div) problems. In fact, it outperforms ADS for higher order elements.

A mixed generalized multiscale finite element method for planar linear elasticity

Abstract A mixed generalized multiscale finite element method for linear elasticity based on Hellinger Reissner principle with a strong symmetry enforcement for the stress tensor is introduced. The multiscale approximation space for the stress tensor is built on a coarse grid with carefully designed local problems so that the basis functions are also symmetric. Using eigenfunctions of local spectral problems as basis functions allows the approximation error of the multiscale finite element space to have a rapid spectral decay. Together with a properly chosen approximation space for the displacement, the method is shown to be inf–sup stable and robust as the first Lame coefficient λ → ∞ . Numerical experiments are supplemented to demonstrate fast convergence of the method with respect to local enrichment, and robustness of the method with respect to high contrast heterogeneity of the Poisson ratio.

Space-Time Discretizations Using Constrained First-Order System Least Squares (CFOSLS)

Abstract This paper studies finite element discretizations for three types of time-dependent PDEs, namely heat equation, scalar conservation law and wave equation, which we reformulate as first order systems in a least-squares setting, subject to a space-time conservation constraint (coming from the original PDE). Available piecewise polynomial finite element spaces in ( n + 1 ) -dimensions for functional spaces from the ( n + 1 ) -dimensional de Rham sequence for n = 2 , 3 are used for the implementation of the method. Computational results illustrating the error behavior, iteration counts and performance of block-diagonal and monolithic geometric multigrid preconditioners are presented for the discrete CFOSLS system. The results are obtained from a parallel implementation of the methods for which we report reasonable scalability.