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.     

On L p Resolvent Estimates for Elliptic Operators on Compact Manifolds

We prove uniform L^p estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3 Dos Santos Ferreira, D., Kenig, C., and Salo, M., 2014. On L^p resolvent estimates for Laplace-Beltrami operators on compact manifolds, Forum Math. 26 (2014), pp. 815–849.[Crossref], [Web of Science ®] , [Google Scholar]] in the case of Laplace-Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1 Bourgain, J., Shao, P., Sogge, C., and Yao, X., On L^p-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys., to appear.[Web of Science ®] , [Google Scholar]] very closely. We also show that spectral regions in our L^p resolvent estimates are optimal.     

Norm Estimates for Multiplication Operators in Hilbert Algebras

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra with unit over the fields or , the infimum of its norms with respect to all scalar products in this algebra (with ) is either infinite or at most . Sufficient conditions for this bound to be not less than are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras).     

Error Estimates for Mixed Finite Element Methods for Sobolev Equation

1 IntroductionLet fl be a bounded domain in R2 with Lipschitz continuous boundaxy 0fl. For thed0 < T < co, we consider the fo1lowing initial-boun'lar}-ralue problem for thc Sobolevequation:where ut denotes the time derivative of the function (1. Vu denotes the gradient of thefunction u, and divv denotes the divergence of the vect{Jr tulued function v, a1 b1, f, anduo are known functions.The standard finite element method for (1.1) (1.3) llas received considerable attentionand is well studied…     

Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.     

Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems.     

Schauder Estimates for Elliptic and Parabolic Equations

In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.     

Resolvent Estimates for Fleming–Viot Operators with Brownian Drift

This article is a supplement to the paper of D. A. Dawson and P. March (J. Funct. Anal.132(1995), 417–472). We define a two-parameter scale of Banach spaces of functions defined on ₁( ^d), the space of probability measures ond-dimensional euclidean space, using weighted sums of the classical Sobolev norms. We prove that the resolvent of the Fleming–Viot operator with constant diffusion coefficient and Brownian drift acts boundedly between certain members of the scale. These estimates gauge the degree of smoothing performed by the resolvent and separate the contribution due to the diffusion coefficient and that due to the drift coefficient.     

Maximum Principle for Nonlinear Cooperative Elliptic Systems on R^N

We investigate in this work necessary and sufficient conditions for having a Maximum Principle for a cooperative elliptic system on the whole R^N. Moreover, we prove the existence of solutions by an approximation method for the considered system.     

一类四阶奇异非线性椭圆方程的加权模误差估计

利用有限元方法研究了一类四阶奇异非线性椭圆方程,利用有限元的逆性质,给出了考虑数值积分影响时的加权H2模误差估计.     

Lagrange四边形单位分解有限元法的最优误差分析

用构造最优局部逼近空间的方法对Lagrange型四边形单位分解有限元法进行了最优误差分析.单位分解取Lagrange型四边形上的标准双线性基函数,构造了一个特殊的局部多项式逼近空间,给出了具有2阶再生性的Lagrange型四边形单位分解有限元插值格式,从而得到了高于局部逼近阶的最优插值误差.     

三维守恒律有限元方法逼近光滑解的误差估计

我们对一个三维守恒律的显式有限元方法证明了H^1范数的二阶误差估计。     

Discontinuous Hybrid Primal Finite Element Method for Elliptic Equations

A primal hybrid finite element scheme is introduced to produce completely discontinuous solution for diffusion and convection-diffusion problems. Same rate of convergence as classical methods is obtained in suitable norms. Finally an a posteriori error estimator is given.     

高阶椭圆算子Dirichlet本征值的下界

设Ω是R^m(m≥2）中的一个有界区域,其边界足够光滑,考察2p(p≥1）阶椭圆算子（－1）^p ∑│α│＝│β│＝pa^α(Aαβa^β)在Dirichlet边界条件下的本征值问题,给出了其本征值的一个下界,该下界除与维数m有关外仅依赖于区域Ω的体积。     

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We derive superconvergence result for H ¹-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Bab?ska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order 𝒪(h ^k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.     

Recovery Type a Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.     

二次变系数椭圆方程的三次有限体积元方法

In this paper, we present a finite volume framework for second order elliptic equations with variable coefficients based on cubic Hermite element. We prove the optimal H1 norm error estimates. A numerical example is given at the end to show the feasibility of the method.