双线性非协调有限元逼近的梯度恢复后验误差估计

给出了二阶椭圆方程的双线性非协调有限元逼近的梯度恢复后验误差估计.该误差估计是在Q_1非协调元上得到的,并给出了误差的上下界.进一步证明该误差估计在拟一致网格上是渐进精确地.证明依赖于clement插值和Helmholtz分解,数值结果验证了理论的正确性.     

二阶椭圆问题的各向异性混合元简化格式和后验误差估计

对二阶椭圆问题构造了一个非常规各向异性Hermite型矩形单元.并基于泡函数对其构造了一种简化的稳定化混合元格式.同时给出了格式的收敛性分析和后验误差估计.     

特征值问题混合有限元法的一个误差估计

设（λh,σh,uh）是一个混合有限元特征对.Babuska和Osborn建立了（λh,uh）的误差估计.本文导出了σh的抽象误差估计式.并把该估计式应用于二阶椭圆特征值问题Raviart-Thomas混合有限元格式和重调和算子特征值问题Ciarlet-Raviart混合有限元格式,得到了一些新的误差估计.     

各向异性网格下二阶椭圆问题的一个混合元形式

提出了二阶椭圆问题的一个混合变分形式,同时证明了Rariart-Thomas元的各向异性插值性质,并给出了单元的对二阶问题的最优误差估计。     

二阶椭圆问题新混合元模型的超收敛分析及外推

对二阶椭圆问题通过＂增补＂办法导出一个新的混合模型.在各向异性网格下,利用积分恒等式技巧得到了真解与ECHL元近似解的超逼近性质.同时基于插值后处理技术导出了整体超收敛.进一步,通过渐进误差展开和分裂外推,得到了比通常的误差估计更高一阶的收敛速度.     

一个非线性椭圆方程有限元近似的拟范数插值误差估计

运用拟范数方法,获得一个非线性椭圆方程的有限元插值误差估计.该方程源于弹塑性力学中的组合材料问题.为了在更弱正则性条件下得到该方程的优化先验误差界,这个误差估计是非常必要的.一个推广的拟范数被提出,并建立了该范数下的类似结果.     

二阶椭圆问题的混合有限元法的泡函数稳定性及其后验误差估计

本文讨论二阶椭圆问题的混合有限元逼近的一种泡函数稳定性,并给出其基于简化的稳定化格式的先验误差估计和后验误差估计.该方法较通常的格式(例如,Raviaxt-Thomas方法的同阶格式)节省大量的自由度.     

具有迅速振荡系数的非自共轭椭圆问题的二阶双尺度计算方法

为了求解具有迅速振荡系数的非自共轭椭圆问题,考虑了非自共轭椭圆问题二阶双尺度近似解的表示式,推导了二阶双尺度近似解的误差估计式,数值试验结果表明给出的近似解是有效的.     

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.     

非单调型拟线性椭圆问题的非协调元逼近

用非协调有限元来研究非单调型拟线性椭圆问题,使用Aubin-Nitsche对偶技巧,给出了在范数‖.‖h和‖.‖0下的最优误差估计.     

Discontinuous mixed penalty-free Galerkin method for second-order quasilinear elliptic equations

Discrete schemes for finding an approximate solution of the Dirichlet problem for a second-order quasilinear elliptic equation in conservative form are investigated. The schemes are based on the discontinuous Galerkin method (DG schemes) in a mixed formulation and do not involve internal penalty parameters. Error estimates typical of DG schemes with internal penalty are obtained. A new result in the analysis of the schemes is that they are proved to satisfy the Ladyzhenskaya-Babuska-Brezzi condition (inf-sup) condition.     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

Monotone schemes for a class of nonlinear elliptic and parabolic problems

We construct monotone numerical schemes for a class of nonlinear PDE for elliptic and initial value problems for parabolic problems. The elliptic part is closely connected to a linear elliptic operator, which we discretize by monotone schemes, and solve the nonlinear problem by iteration. We assume that the elliptic differential operator is in the divergence form, with measurable coefficients satisfying the strict ellipticity condition, and that the right-hand side is a positive Radon measure. The numerical schemes are not derived from finite difference operators approximating differential operators, but rather from a general principle which ensures the convergence of approximate solutions. The main feature of these schemes is that they possess stencils stretching far from basic grid-rectangles, thus leading to system matrices which are related to M-matrices.     

Nonlinear and adaptive frame approximation schemes for elliptic PDEs: Theory and numerical experiments

This article is concerned with adaptive numerical frame methods for elliptic operator equations. We show how specific noncanonical frame expansions on domains can be constructed. Moreover, we study the approximation order of best n‐term frame approximation, which serves as the benchmark for the performance of adaptive schemes. We also discuss numerical experiments for second order elliptic boundary value problems in polygonal domains where the discretization is based on recent constructions of boundary adapted wavelet bases on the interval. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A priori error estimates for upwind finite volume schemes for two-dimensional linear convection diffusion problems

It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in R² and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h^1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.     

A variant of Boneh-Franklin IBE with a tight reduction in the random oracle model

The first practical identity based encryption (IBE) scheme was published by Boneh and Franklin at Crypto 2001, based on the elliptic curve pairing. Since that time, many other IBE schemes have been published. In this paper, we describe a variant of Boneh-Franklin with a tight reduction in the random oracle model. Our new scheme is quite efficient compared to existing schemes; moreover, upgrading from Boneh-Franklin to our new scheme is straightforward.      

Exponentially fitted discontinuous Galerkin schemes for singularly perturbed problems

New discontinuous Galerkin schemes in mixed form are introduced for symmetric elliptic problems of second order. They exhibit reduced connectivity with respect to the standard ones. The modifications in the choice of the approximation spaces and in the stabilization term do not spoil the error estimates. These methods are then used for designing new exponentially fitted schemes for advection dominated equations. The presented numerical tests show the good performances of the proposed schemes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

On discontinuous Galerkin method and semiregular family of triangulations

Discretization of second order elliptic partial differential equations by discontinuous Galerkin method often results in numerical schemes with penalties. In this paper we analyze these penalized schemes in the context of quite general triangular meshes satisfying only a semiregularity assumption. A new (modified) penalty term is presented and theoretical properties are proven together with illustrative numerical results. This work is a part of the research project MSM 0021620839 financed by MSMT and was partly supported by the project No. 201/04/1503 of the Grant Agency of the Czech Republic.     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

构造高分辨率格式的反扩散混合法

In this paper we study a kind of mixed anti-diffusion method for partial differntial equations. Firstly, we use the method to construct some difference schemes for the conservation laws. The schemes are of second order accuracy and are total variation decreasing (TVD). In particular, there are only three knots involved in the schemes. Secondly, we extend the method to construct a few high accuracy difference schemes for elliptic and parabolic equations. Numerical experiments are carried out to illustrate the efficiency of the method.