A discontinuous finite volume element method for second‐order elliptic problems

In this article, we propose a new discontinuous finite volume element (DFVE) method for the second‐order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L² ‐norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012     

Neumann‐Neumann methods for a DG discretization on geometrically nonconforming substructures

A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ω_i of size O(H_i). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ω_i, a conforming finite element space associated to a triangulation \begin{align*} {\mathcal{T}}_{h_i}(\Omega_i)\end{align*} is introduced. To handle the nonmatching meshes across ?Ω_i, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ?Ω_i, a condition number estimate \begin{align*} C(1 + \max_i\log \frac{H_i}{h_i})^2\end{align*} is established with C independent of h_i, H_i, h_i/h_j, and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

Error estimates for finite volume element methods for general second‐order elliptic problems

We treat the finite volume element method (FVE) for solving general second order elliptic problems as a perturbation of the linear finite element method (FEM), and obtain the optimal H¹ error estimate, H¹ superconvergence and L^p (1 < p ≤ ∞) error estimates between the solution of the FVE and that of the FEM. In particular, the superconvergence result does not require any extra assumptions on the mesh except quasi‐uniform. Thus the error estimates of the FVE can be derived by the standard error estimates of the FEM. Moreover we consider the effects of numerical integration and prove that the use of barycenter quadrature rule does not decrease the convergence orders of the FVE. The results of this article reveal that the FVE is in close relationship with the FEM. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 693–708, 2003.     

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.     

自适应间断Galerkin 有限元的多水平方法

本文讨论在自适应网格上间断Galerkin 有限元离散系统的局部多水平算法. 对于光滑系数和间断系数情形, 利用Schwarz 理论分析了算法的收敛性. 理论和数值试验均说明算法的收敛率与网格层数以及网格尺寸无关. 对强间断系数情形算法是拟最优的, 即收敛率仅与网格层数有关.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

Discontinuous Galerkin finite volume element methods for second‐order linear elliptic problems

In this article, a one parameter family of discontinuous Galerkin finite volume element methods for approximating the solution of a class of second‐order linear elliptic problems is discussed. Optimal error estimates in L² and broken H¹‐ norms are derived. Numerical results confirm the theoretical order of convergences. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

‐error estimates of discontinuous Galerkin methods with theta time discretization scheme for an evolutionary HJB equations

The main purpose of this paper is to analyze the convergence and regularity of our proposed algorithm of the finite element methods coupled with a theta time discretization scheme for evolutionary Hamilton‐Jacobi‐Bellman equations with linear source terms with respect to the Dirichlet boundary conditions (Appl. Math. Comput., 262 (2015), 42.55 ). Also, an optimal error estimate with an asymptotic behavior in uniform norm is given. Copyright © 2017 John Wiley & Sons, Ltd.     

Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one‐dimensional second‐order wave equation

In this article, we analyze a residual‐based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one‐dimensional second‐order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal L² error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862–901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the L²‐norm under mesh refinement. The order of convergence is proved to be , when p‐degree piecewise polynomials with are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and superconvergent solutions. Our computational results show higher convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the L²‐norm converge to unity at rate while numerically they exhibit and rates, respectively. Numerical experiments are shown to validate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461–1491, 2015     

Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems

In this article, we prove some weighted pointwise estimates for three discontinuous Galerkin methods with lifting operators appearing in their corresponding bilinear forms. We consider a Dirichlet problem with a general second-order elliptic operator.

     

Equilibrated error estimators for discontinuous Galerkin methods

We consider some diffusion problems in domains of ?^d, d = 2 or 3 approximated by a discontinuous Galerkin method with polynomials of any degree. We propose a new a posteriori error estimator based on H(div)‐conforming elements. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. The lower bound is also established with a constant depending on the aspect ratio of the mesh, the dependence with respect to the coefficients being also traced. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Pointwise error estimates of the local discontinuous Galerkin method for a second order elliptic problem

In this paper we derive some pointwise error estimates for the local discontinuous Galerkin (LDG) method for solving second-order elliptic problems in (). Our results show that the pointwise errors of both the vector and scalar approximations of the LDG method are of the same order as those obtained in the norm except for a logarithmic factor when the piecewise linear functions are used in the finite element spaces. Moreover, due to the weighted norms in the bounds, these pointwise error estimates indicate that when at least piecewise quadratic polynomials are used in the finite element spaces, the errors at any point depend very weakly on the true solution and its derivatives in the regions far away from . These localized error estimates are similar to those obtained for the standard conforming finite element method.

     

Improved L2 and H1 error estimates for the Hessian discretization method

The Hessian discretization method (HDM) for fourth-order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit-conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods (ncFEMs), finite volume methods (FVMs) and methods based on gradient recovery operators. A generic error estimate has been established in L², H¹, and H²-like norms in literature. In this paper, we establish improved L² and H¹ error estimates in the framework of HDM and illustrate it on various schemes. Since an improved L² estimate is not expected in general for FVM, a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini ncFEM, in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented.     

Monotone first‐order weighted schemes for scalar conservation laws

In this work, we present a monotone first‐order weighted (FORWE) method for scalar conservation laws using a variational formulation. We prove theoretical properties as consistency, monotonicity, and convergence of the proposed scheme for the one‐dimensional (1D) Cauchy problem. These convergence results are extended to multidimensional scalar conservation laws by a dimensional splitting technique. For the validation of the FORWE method, we consider some standard bench‐mark tests of bidimensional and 1D conservation law equations. Finally, we analyze the accuracy of the method with L¹ and L^∞ error estimates. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Higher‐order discontinuous Galerkin method for pyramidal elements using orthogonal bases

We study finite elements of arbitrarily high‐order defined on pyramids for discontinuous Galerkin methods. We propose a new family of high‐order pyramidal finite elements using orthogonal basis functions which can be used in hybrid meshes including hexahedra, tetrahedra, wedges, and pyramids. We perform a comparison between these orthogonal functions and nodal functions for affine and non‐affine elements. Different strategies for the inversion of the mass matrix are also considered and discussed. Numerical experiments are conducted for the three dimensional Maxwell's equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L^∞(L²) and L^∞(H¹) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L^∞(H¹) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A class of discontinuous Petrov–Galerkin methods. II. Optimal test functions

We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through several theoretical and numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Interior superconvergence error estimates for mixed finite element methods for second order elliptic problem

1.IntroductionLetfibeaplanedomainwithsmoothboundaryonandWm,p(fl)betheusualSobolevspaceonnwithnormWhenp=2,pisusuallyomitted.WeshalldenotetheusualinnerproductinL'(fl)orLa(O)'by','),andinL'(ofl)by't').Weshallusethesamenotationstoindicatethedualltiesbetw...