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International Journal of Modeling, Simulation, and Scientific Computing 2022年第1期13卷 192-213页

作者： Jingyun Lv Key Laboratory of Systems and Control Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 P.R.China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 P.R.China

The aim of this paper is to consider the convergence of the numerical methods for stochastic time-fractional evolution equations driven by fractional Brownian motion.The spatial and temporal regularity of the mild solution is given.The numerical scheme approximates the problem in space by the Galerkin finite element method and in time by the backward Euler convolution quadrature formula,and the noise by the L 2-projection.The strong convergence Error estimates for both semi-discrete and fully discrete schemes are established.A numerical example is presented to verify our theoretical analysis.

关键词： Stochastic time-fractional evolution equations finite element method Error estimates fractional Brownian motion

Optimal Error estimates of the Semi-Discrete Local Discontinuous Galerkin Method and Exponential Time Differencing Schemes for the Thin Film Epitaxy Problem without Slope Selection

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Advances in Applied Mathematics and Mechanics 2023年第3期15卷 545-567页

作者： Danni Zhang Ruihan Guo School of Mathematics and Statistics Zhengzhou UniversityZhengzhouHenan 450001China

In this paper,we prove the optimal Error estimates in L2 norm of the semidiscrete local discontinuous Galerkin(LDG)method for the thin film epitaxy problem without slope selection.To relax the severe time step restriction of explicit time marching methods,we employ a class of exponential time differencing(ETD)schemes for time integration,which is based on a linear convex splitting principle.Numerical experiments of the accuracy and long time simulations are given to show the efficiency and capability of the proposed numerical schemes.

关键词： Local discontinuous Galerkin method thin film epitaxy problem Error estimates exponential time differencing long time simulation

Error estimates FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION

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Journal of Computational Mathematics 2021年第3期39卷 471-492页

作者： Xiaoliang Song Bo Chen Bo Yu Dalian University of Technology Dalian 116024China Department of Mathematics National University of Singapore10 Lower Kent Ridge RoadSingapore School of Mathematical Sciences Dalian University of TechnologyDalian 116024China

Optimization problems with L^(1)-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l^(1)-regularization optimization,the resulting discretized L^(1)norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem.A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L^(1)-norm.In this paper,a new discretized scheme for the L^(1)-norm is presented.Compared to the new discretized scheme for L^(1)-norm with the nodal quadrature formula,the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation.Moreover,finite element Error estimates results for the primal problem with the new discretized scheme for the L^(1)-norm are provided,which confirms that this approximation scheme will not change the order of Error estimates.To solve the new discretized problem,a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD)method is introduced to solve it via its dual.The proposed sGS-mABCD algorithm is illustrated at two numerical examples.Numerical results not only confirm the finite element Error estimates,but also show that our proposed algorithm is efficient.

关键词： Finite element method ABCD method.Approximate discretization Error estimates

Error estimates for the Time Discretization of a Semilinear Integrodifferential Parabolic Problem with Unknown Memory Kernel

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Numerical Mathematics(Theory,Methods and Applications) 2017年第1期10卷 116-144页

作者： Marijke Grimmonprez Karel Van Bockstal Marian Slodicka Department of Mathematical Analysis Ghent UniversityGalglaan 29000 GhentBelgium

This paper is devoted to the study of an inverse problem containing a semilinear integrodifferential parabolic equation with an unknown memory kernel.This equation is accompanied by a Robin boundary condition.The missing kernel can be recovered from an additional global measurement in integral form.In this contribution,an error analysis is performed for a time-discrete numerical scheme based on Backward Euler’s Method.The theoretical results are supported by some numerical experiments.

关键词： Parabolic inverse problem convolution kernel Error estimates

A Numerical Method and its Error estimates for the Decoupled Forward-Backward Stochastic Differential Equations

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Communications in Computational Physics 2014年第3期15卷 618-646页

作者： Weidong Zhao Wei Zhang Lili Ju School of Mathematics Shandong UniversityJinanShandong 250100P.R.China Department of Mathematics University of South CarolinaColumbiaSC 29208USA Beijing Computational Science Research Center Beijing 100084P.R.China

In this paper,a new numerical method for solving the decoupled forwardbackward stochastic differential equations(FBSDEs)is proposed based on some specially derived reference equations.We rigorously analyze errors of the proposed method under general situations.Then we present Error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method;in particular,we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE.Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

关键词： Decoupled forward-backward stochastic differential equations numerical scheme Error estimates

EXPLICIT Error estimates FOR COURANT, CROUZEIX-RAVIART AND RAVIART-THOMAS FINITE ELEMENT METHODS

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Journal of Computational Mathematics 2012年第4期30卷 337-353页

作者： Carsten Carstensen Joscha Gedicke Donsub Rim Institut fur Mathematik Humboldt-Universitiit zu Berlin Unter den Linden 6 10099 Berlin Germany Department of Computational Science and Engineering Yonsei University 120-749 Seoul Korea Institut fur Mathematik Humboldt-Universitat zu Berlin Unter den Linden 6 10099 Berlin Germany Yonsei School of Business and Department of Computational Science and Engineering Yonsei University 120-749 Seoul Korea

The elementary analysis of this paper presents explicit expressions of the constants in the a priori Error estimates for the lowest-order Courant, Crouzeix-Raviart nonconforming and P^viart-Thomas mixed finite element methods in the Poisson model problem. The three constants and their dependences on some maximal angle in the triangulation are indeed all comparable and allow accurate a priori error control.

关键词： Error estimates Conforming Nonconforming Mixed Finite element method.

Error estimates and Superconvergence of RT0Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

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Numerical Mathematics(Theory,Methods and Applications) 2013年第4期6卷 637-656页

作者： Yanping Chen Tianliang Hou School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Research Institute of Hong Kong Baptist University in Shenzhen Shenzhen 518057GuangdongChina

In this paper,we will investigate the Error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control.The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions.We derive some superconvergence properties for the control variable and the state variables.Moreover,we derive L∞-and H−1-Error estimates both for the control variable and the state variables.Finally,a numerical example is given to demonstrate the theoretical results.

关键词： Semilinear elliptic equations optimal control problems superconvergence Error estimates mixed finite element methods

Error estimates for 1D Asymptotic Models in Coaxial Cables with Non-Homogeneous Cross-Section

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Advances in Applied Mathematics and Mechanics 2012年第6期4卷 647-664页

作者： Sébastien Imperiale Patrick Joly Department of Applied Physics and Applied Mathematics Columbia UniversityNY 10027USA INRIA Rocquencourt POemsdomaine de Voluceau78153 Le ChesnayFrance

This paper is the first contribution towards the rigorous justification of asymptotic 1D models for the time-domain simulation of the propagation of electromagnetic waves in coaxial cables.Our general objective is to derive Error estimates between the"exact"solution of the full 3D model and the"approximate"solution of the 1D model known as the Telegraphist’s equation.

关键词： Coaxial cables Telegraphist’s equation Error estimates non-homogeneous crosssection

Error estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity

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Advances in Applied Mathematics and Mechanics 2012年第6期4卷 751-768页

作者： Yanping Chen Tianliang Hou Weishan Zheng School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105HunanChina

In this paper,we investigate the Error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-Thomas mixed fi-nite element spaces and the control variable is approximated by piecewise constant functions.We derive L^(2) and L^(∞)-Error estimates for the control variable.Moreover,using a recovery operator,we also derive some superconvergence results for the control variable.Finally,a numerical example is given to demonstrate the theoretical results.

关键词： Elliptic equations optimal control problems superconvergence Error estimates mixed finite element methods

伪双曲型积分-微分方程的H~1-Galerkin混合元法误差估计

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高等学校计算数学学报 2010年第1期32卷 1-20页

作者：刘洋李宏何斯日古楞内蒙古大学数学科学学院

1引言考虑如下一类具有Lipschitz连续边界(?)Ω的凸有界区域Ω上的伪双曲型积分微分方程其中Ω(?)R~d,(d=1,2,3)J=(0,T],对于固定的T,0

关键词： pseudo-hyperbolic partial integro-differential equation H~1-Galerkin mixed finite element methods existence and uniqueness Error estimates