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A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems

Lin, QunXie, Hehu — 2012

Applications of Mathematics 2012

Adaptive finite element method based on multilevel correction scheme is proposed to solve Steklov eigenvalue problems. In this method, each adaptive step involves solving associated boundary value problems on the adaptive partitions and small scale eigenvalue problems on the coarsest partitions. Solving eigenvalue problem in the finest partition is not required. Hence the efficiency of solving Steklov eigenvalue problems can be improved to the similar efficiency of the adaptive finite element method...

A parallel method for population balance equations based on the method of characteristics

Li, YuLin, QunXie, Hehu — 2013

Applications of Mathematics 2013

In this paper, we present a parallel scheme to solve the population balance equations based on the method of characteristics and the finite element discretization. The application of the method of characteristics transform the higher dimensional population balance equation into a series of lower dimensional convection-diffusion-reaction equations which can be solved in a parallel way. Some numerical results are presented to show the accuracy and efficiency.

Uniform L 1 error bounds for semi-discrete finite element solutions of evolutionary integral equations

Lin, QunXu, DaZhang, Shuhua — 2012

Applications of Mathematics 2012

In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem u t + 0 t β ( t - s ) A u ( s ) d s = 0 , u ( 0 ) = v , t > 0 , where A is an elliptic partial-differential operator and β ( t ) is positive, nonincreasing and log-convex on ( 0 , ) with 0 β ( ) < β ( 0 + ) . Error estimates are derived in the norm of L t 1 ( 0 , ; L x 2 ) , and some estimates for the first order time derivatives of the errors are also given.

A Superconvergence result for mixed finite element approximations of the eigenvalue problem

Qun LinHehu Xie — 2012

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán [9 (1999) 1165–1178] and Gardini [43 (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic eigenvalue problems by general mixed finite element...

Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method

Wei ChenQun Lin — 2006

Applications of Mathematics

By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally,...

A Superconvergence result for mixed finite element approximations of the eigenvalue problem

Qun LinHehu Xie — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán [ (1999) 1165–1178] and Gardini [ (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic eigenvalue problems...

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

Qin LiQun LinHehu Xie — 2013

Applications of Mathematics

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q 1 rot , E Q 1 rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to...

Multilevel correction adaptive finite element method for semilinear elliptic equation

Qun LinHehu XieFei Xu — 2015

Applications of Mathematics

A type of adaptive finite element method is presented for semilinear elliptic problems based on multilevel correction scheme. The main idea of the method is to transform the semilinear elliptic equation into a sequence of linearized boundary value problems on the adaptive partitions and some semilinear elliptic problems on very low dimensional finite element spaces. Hence, solving the semilinear elliptic problem can reach almost the same efficiency as the adaptive method for the associated boundary...

Superconvergence estimates of finite element methods for American options

Qun LinTang LiuShu Hua Zhang — 2009

Applications of Mathematics

In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation...

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