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Finite element analysis for a regularized variational inequality of the second kind

Zhang, TieZhang, ShuhuaAzari, Hossein — 2012

Applications of Mathematics 2012

In this paper, we investigate the a priori and the a posteriori error analysis for the finite element approximation to a regularization version of the variational inequality of the second kind. We prove the abstract optimal error estimates in the H 1 - and L 2 -norms, respectively, and also derive the optimal order error estimate in the L -norm under the strongly regular triangulation condition. Moreover, some residual–based a posteriori error estimators are established, which can provide the global upper...

Finite element derivative interpolation recovery technique and superconvergence

Tie Zhu ZhangShu Hua Zhang — 2011

Applications of Mathematics

A new finite element derivative recovery technique is proposed by using the polynomial interpolation method. We show that the recovered derivatives possess superconvergence on the recovery domain and ultraconvergence at the interior mesh points for finite element approximations to elliptic boundary problems. Compared with the well-known Z-Z patch recovery technique, the advantage of our method is that it gives an explicit recovery formula and possesses the ultraconvergence for the odd-order finite...

Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations

Tie Zhu ZhangShu Hua Zhang — 2015

Applications of Mathematics

We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in d space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order k + 1 in the L 2 -norm if the method uses polynomials of order k . Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order k + 1 . Further we consider a residual-based a posteriori...

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu ZhangShu Hua Zhang — 2015

Applications of Mathematics

We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of C -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set S , the gradient approximation possesses the superconvergence: max P S | ( u - ¯ u h ) ( P ) | = O ( h 2 ) | ln h | 3 / 2 , where ¯ denotes the average gradient on elements containing vertex P . Furthermore, by using the interpolation post-processing technique,...

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