Weak Galerkin finite element methods combined with Crank-Nicolson scheme for parabolic interface problems

This article is devoted to the a priori error estimates of the fully discrete Crank-Nicolson approximation for the linear parabolic interface problem via weak Galerkin finite element methods (WG-FEM). All the finite element functions are discontinuous for which the usual gradient operator is implemented as distributions in properly defined spaces. Optimal order error estimates in both $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ norms are established for lowest order WG finite element space $({\cal P}_{k}(K),\;{\cal P}_{k-1}(\partial K),\;\big[{\cal P}_{k-1}(K)\big]^2)$. Finally, we give numerical examples to verify the theoretical results.     

On the convergence of finite element solutions to the interface problem for the Stokes system

The Stokes system with a discontinuous coefficient (Stokes interface problem) and its finite element approximations are considered. We firstly show a general error estimate. To derive explicit convergence rates, we introduce some appropriate assumptions on the regularity of exact solutions and on a geometric condition for the triangulation. We mainly deal with the MINI element approximation and then consider P1-iso-P2/P1 element approximation. Results are expected to give an instructive remark in numerical analysis for two-phase flow problems.     

A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation

We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization of the penalty term, we propose reduced and non-reduced integration schemes, and obtain an error estimate for velocity and pressure. The theoretical results are verified by numerical experiments.     

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space $(\mathcal{P}_k(K), \mathcal{P}_{k−1}(∂K), [\mathcal{P}_{k−1}(K)]^2).$ Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in $L^∞(L^2)$ norm. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.     

Two-level Additive Schwarz Preconditioners for Nonconforming Finite Element Methods

Two-level additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar second-order symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergence-free nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap.

     

Shrinkability of Minimal Elements in Sphere Representations of Posets

Let k be a positive integer and let P = (X,≤) be a poset. We call P a k-sphere order provided there is a mapping f which assigns to each element x $ \in$ X a ball f(x) in Rk so that x ≤ y in P if and only if f(x) $ \subseteq$ f(y). We ask: Given that P is a k-sphere order, does there necessarily exist a representation f with the property that every minimal element of P is assigned a ball of radius zero? The answer is “yes” for k = 1, but “no” for all k ≥ 2.     

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.     

全矩阵环的一类基

设P是一个域,Fij(i,j=1,2,…,n)是全矩阵环Mn(P)中n2个n×n矩阵,且满足FijFkl=δjkFil(i,j,k,l=1,2,…,n),其中δij={1,i=j0,i≠j为Kronecker符号.则或者所有Fij(i,j=1,2,…,n)全为零,或者存在可逆矩阵T∈Mn(P),使得Fij=T-1EijT(i,j=1,2,…,n),其中Eij表示(i,j)位置是1,     

Maximum Norm Estimates for Finite Volume Element Method for Non-selfadjoint and Indefinite Elliptic Problems

In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method (FVE) based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems.     

Stokes问题各向异性网格Q2-P1混合元超收敛分析

讨论Stokes问题在各向异性冈格下的Q2-P1混合有限元方法，利用积分恒等式技巧得到了与传统方法相同的超逼近性质，同时基于插值后处理的技巧，构造了速度和压力的一对插值后处理算子，并且前者具有备向异性特征，从而导出了整体超收敛结果．     

非自共轭和不定椭圆问题的有限体积元方法的一致收敛性

In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.     

P1-NONCONFORMING QUADRILATERAL FINITE VOLUME ELEMENT METHOD AND ITS CASCADIC MULTIGRID ALGORITHM FOR ELLIPTIC PROBLEMS

In this paper,we discuss the finite volume element method of P_1-nonconforming quadri-lateral element for elliptic problems and obtain optimal error estimates for general quadri-lateral partition.An optimal eascadie multigrid algorithm is proposed to solve the non-symmetric large-scale system resulting from such discretization.Numerical experimentsare reported to support our theoretical results.     

A posteriori error analysis for locally conservative mixed methods

In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart-Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoiding the saturation assumption made by Braess and Verfürth. Our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries. The main results are applied to the nonconforming finite element method and the interior penalty discontinuous Galerkin method as well as the mixed finite element method.

     

服从幂律的拟牛顿流动稳定化有限元方法

1．引言考虑在R”（N＝2,3）上一个有界单连通域0内的某种流动,该区域具有LIPschitzian边界F．设fEL‘（fi）”,正的函数nEC（0,一,我们讨论下面的边值问题：（P）求（。,P）;使得上述模型反映了定常不可压流体的流动,非线性惯性项已被忽略,f为体力,。为速度,P为压力,P为流体的粘性函数．对于符合幂律（Power－law）的拟牛顿型流动来说,这里幂律指数r＞1为一常数．众所周知,有着较低分子量的物质（如水,空气等）的流动可用Navier－Stokes方程描述,而有较高分子量的物质（如生物流体,润滑剂,涂料,高聚物溶液）的流…     

Comparison of computations of asymptotic flow models in a constricted channel

We aim at comparing computations with asymptotic models issued from incompressible Navier–Stokes at high Reynolds number: the Reduced Navier–Stokes/Prandtl (RNS/P) equations and the Double Deck (DD) equations. We treat the case of the steady two dimensional flow in a constricted pipe. In particular, finite differences and finite element solvers are compared for the RNS/P equations. It results from this study that the two codes compare well. Numerical examples also illustrate the interest of these asymptotic models as well as the flexibility of the finite element solver.     

Multigrid for the mortar element method for P1 nonconforming element

In this paper, a multigrid algorithm is presented for the mortar element method for P1 nonconforming element. Based on the theory developed by Bramble, Pasciak, Xu in [5], we prove that the W-cycle multigrid is optimal, i.e. the convergence rate is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid preconditioner is constructed, which results in a preconditioned system with uniformly bounded condition number. Received May 11, 1999 / Revised version received April 1, 2000 / Published online October 16, 2000     

Superconvergence in H1-norm of a difference finite element method for the heat equation in a 3D spatial domain with almost-uniform mesh

Numerical Algorithms - In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of...     

Rationality and strong reality for Sylow 2-subgroups of Weyl and alternating groups

It is proved that values of all complex characters for a Sylow 2-subgroup P of any Weyl group are rational, and every element of P is a product of two involutions in P. Similar results hold also for Sylow 2-subgroups of alternating groups.Supported by RFBR grant No. 03-01-00905.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 44–53, January–February, 2005.     

Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices

Order - Given a k-tuple P=(x 1,x 2,...,x k ) in a finite lattice X endowed with the lattice metric d, a median of P is an element m of X minimizing the sum ∑ i d(m,x i ). If X is an upper...     

Higher ordermixed finite element approximation of subsurface water flow

This paper focuses on the reliable and efficient numerical approximation of subsurface water flows. The locally mass conservative B rezzi- D ouglas- M arini ( BDM ₁) mixed finite element method is considered and compared to a lowest order R aviart– T homas ( RT ₀) mixed finite element approach and a M ulti P oint F lux A pproximation. Appreciable advantage of the BDM₁ element is that it yields a formally second order accurate flux approximation whereas the RT₀ and MPFA approach are of first order accuracy only. The problem to be analyzed in this work is whether a superiority of the BDM₁ element can also be observed in reservoir simulation where discontinuous and full permeability tensors on non-uniform grids arise and the fluxes lack the regularity that is assumed customarily in optimal order error analyses. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)