On a nonoverlapping additive Schwarz method for h‐p discontinuous Galerkin discretization of elliptic problems

The condition number of a discontinuous Galerkin finite element discretization preconditioned with a nonoverlapping additive Schwarz method is analyzed. We improve the result of Antonietti and Houston (J Sci Comput 46 (2011), 124–149), where a bound has been proved for a two‐level nonoverlapping additive Schwarz method with coarse problem using polynomials of degree on a coarse mesh size . In a more general framework, where the concurrency of the algorithm is increased by applying solvers on subdomains smaller than the coarse grid cells, we prove that the condition number of the preconditioned system is where is the coarse space element degree polynomial and is the size of subdomain where local problems are solved in parallel. Our result also extends to the case of discontinuous coefficient, piecewise constant on the coarse grid, for a composite continuous–discontinuous Galerkin discretization. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1572–1590, 2016     

BDDC methods for discontinuous Galerkin discretization of elliptic problems

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂Ω_i, where the Ω_i are disjoint subregions of the original region Ω, a condition number estimate C(1+max_ilog(H_i/h_i))² is established with C independent of h_i, H_i and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions.     

A selective immersed discontinuous Galerkin method for elliptic interface problems

This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second‐order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems

A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.

     

Error estimates for a discontinuous Galerkin method for elliptic problems

In this paper, we consider an elliptic problem with the homogeneous Dirichlet boundary condition and introduce discontinuous Galerkin approximations of the problem. Optimal error estimates of discontinuous Galerkin approximations are obtained.     

Bubble stabilized discontinuous Galerkin method for parabolic and elliptic problems

In this paper we give an analysis of a bubble stabilized discontinuous Galerkin method for elliptic and parabolic problems. The method consists of stabilizing the numerical scheme by enriching the discontinuous affine finite element space elementwise by quadratic bubbles. This approach leads to optimal convergence in the space and time discretization parameters.     

A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems

E-mail: yangd{at}math.purdue.edu or yangd{at}cs.purdue.edu Present address: Department of Mathematics Wayne State University, Detroit, MI 48202, USA. A parallel iterative nonoverlapping domain decomposition methodis proposed and analyzed for elliptic problems. Each iterationin this method contains two steps. In the first step, at theinterface of two subdomains, one subdomain problem requiresthat Dirichlet data be passed to it from the previous iterationlevel, while the other subdomain problem requires that Neumamdata be passed to it. In the second step, we interchange thetypes of data passing at the interface of the two subdomains.This domain decomposition method is suitable for parallel processingwith coarse granularity. Convergence analysis is demonstratedat the differential level by Hilbert space techniques. Numericalresults are provided to confirm the convergence theory.     

A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary

We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.     

An analysis of discontinuous Galerkin methods for elliptic problems

We study global and local behaviors for three kinds of discontinuous Galerkin schemes for elliptic equations of second order. We particularly investigate several a posteriori error estimations for the discontinuous Galerkin schemes. These theoretical results are applied to develop local/parallel and adaptive finite element methods, based on the discontinuous Galerkin methods. Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem Mathematics subject classifications (2000) 65N12, 65N15, 65N30. Aihui Zhou: Subsidized by the Special Funds for Major State Basic Research Projects, and also partially supported by National Science Foundation of China. Reinhold Schneider: Supported in part by DFG Sonderforschungsbereich SFB 393. Yuesheng Xu: Correspondence author. Supported in part by the US National Science Foundation under grants DMS-9973427 and CCR-0312113, by Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under program “Hundreds Distinguished Young Chinese Scientists”.     

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

Iterative solution of a coupled mixed and standard Galerkin discretization method for elliptic problems

In this paper, we consider approximation of a second‐order elliptic problem defined on a domain in two‐dimensional Euclidean space. Partitioning the domain into two subdomains, we consider a technique proposed by Wieners and Wohlmuth [9] for coupling mixed finite element approximation on one subdomain with a standard finite element approximation on the other. In this paper, we study the iterative solution of the resulting linear system of equations. This system is symmetric and indefinite (of saddle‐point type). The stability estimates for the discretization imply that the algebraic system can be preconditioned by a block diagonal operator involving a preconditioner for H (div) (on the mixed side) and one for the discrete Laplacian (on the finite element side). Alternatively, we provide iterative techniques based on domain decomposition. Utilizing subdomain solvers, the composite problem is reduced to a problem defined only on the interface between the two subdomains. We prove that the interface problem is symmetric, positive definite and well conditioned and hence can be effectively solved by a conjugate gradient iteration. Copyright © 2001 John Wiley & Sons, Ltd.     

On the coupling of finite volume and discontinuous Galerkin method for elliptic problems

The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.     

A discontinuous mixed covolume method for elliptic problems

We develop a discontinuous mixed covolume method for elliptic problems on triangular meshes. An optimal error estimate for the approximation of velocity is obtained in a mesh-dependent norm. First-order L²-error estimates are derived for the approximations of both velocity and pressure.     

A Neumann-Neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients

Summary. Neumann-Neumann algorithm have been well developed for standard finite element discretization of elliptic problems with discontinuous coefficients. In this paper, an algorithm of this kind is designed and analyzed for a mortar finite element discretization of problems in three dimensions. It is established that its rate of convergence is independent of the discretization parameters and jumps of coefficients between subregions. The algorithm is well suited for parallel computations.Mathematics Subject Classification (1991): 65N55, 65N10, 65N30, 65N22.The work was supported in part by the U.S. Department of Energy under contract DE-FG02-92ER25127 and in part by Polish Science Foundation under grant 2P03A00524.AcknowledgmentThe author would like to thank Olof Widlund for many fruitful discussions and valuable remarks and suggestions on how to improve the presentation of our results.     

A robust discontinuous Galerkin method for solving convection-diffusion problems

In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed. Particularly, the 2p ＋ 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.     

对流扩散方程的时空间断有限元方法

研究对流扩散方程的时空间断Galerkin有限元方法,该方法采用时,空两个变量都允许间断的基函数,更适用于移动网格,自适应算法以及并行计算.本文利用拉格朗日欧拉方法,采用F.Brezzi数值流通量,给出对流扩散方程的间断时空有限元离散格式,并证明格式的相容性,强制性,稳定性,解的存在唯一性,以及总体误差估计.     

A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems

We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuous Galerkin methods using polynomials of degree for both the potential as well as the flux, the order of convergence in of both unknowns is . Moreover, both the approximate potential as well as its numerical trace superconverge in -like norms, to suitably chosen projections of the potential, with order . This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order in . The method can be thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.

     

Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions

Summary. Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods. Received April 6, 1994 / Revised version received December 7, 1994