Article
Mathematics, Applied
Maria Luz Alvarez, Ricardo G. Duran
Summary: This article discusses the application of the Raviart-Thomas mixed finite element method to non-uniform elliptic problems. It introduces an error estimator based on local post-processing and proves its efficiency and reliability, generalizing the theory developed in [24] to degenerate cases. Finally, the authors present numerical computations demonstrating the good performance of an adaptive procedure based on their estimator.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Gilbert Peralta
Summary: This paper analyzes the mixed and hybrid finite element discretizations for distributed optimal control problems governed by an elliptic equation. It also studies a cost functional that keeps track of both the state and its gradient. A priori error estimates and super-convergence properties for the continuous and discrete optimal states, adjoint states, and controls are given, and the approximating finite-dimensional systems are solved by adding penalization terms for the state and the associated Lagrange multipliers.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Sebastian Franz
Summary: The discretisation of H-div-functions on rectangular meshes involves three families of finite elements, namely Raviart-Thomas, Brezzi-Douglas-Marini, and Arnold-Boffi-Falk elements. In order to prove convergence of numerical methods using these elements, sharp interpolation error estimates are important, especially in an anisotropic setting.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Lixiu Wang, Qian Zhang, Zhimin Zhang
Summary: In this paper, we provide a theoretical justification for the previously observed superconvergence phenomena of the curlcurl-conforming finite elements on rectangular domains. We establish a superconvergence theory for these elements on rectangular meshes and show that the convergence rates are one-order higher than the optimal rates. Numerical experiments are conducted to confirm our theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Christian Glusa, Enrique Otarola
Summary: This paper translates the integral definition of the fractional Laplacian and linear-quadratic optimal control problem for the so-called fractional heat equation, including control constraints. By deriving existence, uniqueness results, and first order optimality conditions, a fully discrete scheme is proposed along with a new error estimation method. Finally, the theory is illustrated through one- and two-dimensional numerical experiments.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Ambit Kumar Pany, Morrakot Khebchareon, Amiya K. Pani
Summary: The conforming finite element Galerkin method is used to discretize a class of strongly nonlinear parabolic problems in the spatial direction. Optimal error estimates and superconvergence results are derived through the use of elliptic projection and quasi-projection techniques. Moreover, a priori error estimates in Sobolev spaces of negative index are obtained, along with nodal superconvergence results between the true solution and Galerkin approximation in a single space variable.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Chunjuan Hou, Zuliang Lu, Xuejiao Chen, Fei Huang
Summary: This paper investigates variational discretization for the optimal control problem with nonlinear parabolic equations and control constraints, achieving improved error estimates compared to standard finite element methods with backward Euler. Additionally, the study presents a posteriori error estimates of residual type.
Article
Mathematics
Xiaoling Meng, Huaijun Yang
Summary: In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier-Stokes equations. The superclose error estimates for velocity in H-1-norm and pressure in L-2-norm are derived by carefully dealing with the trilinear term. Global superconvergence results are obtained with the interpolation post-processing technique, and numerical experiments are conducted to support the theoretical findings.
Article
Mathematics, Applied
Stefano Giani, Luka Grubisic, Harri Hakula, Jeffrey S. Ovall
Summary: The study introduces an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems, which is effective in estimating the approximation error in eigenvalue clusters and their corresponding invariant subspaces.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Minqiang Xu, Lufang Zhang, Emran Tohidi
Summary: In this paper, a new method based on the least-squares technique is proposed and analyzed for one-dimensional interface problems in a piecewise polynomial function space of order k (k = 3, 4, 5). By defining a residual functional, a new bilinear form is derived and the minimum residual method is established in an alternative way. The stability of the proposed method is proven. Error estimation and optimal convergence orders are discussed for general non-uniform meshes. The superconvergence behaviors of the method are also discovered. Numerical experiments are conducted to verify the theoretical findings.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Computer Science, Software Engineering
Huilan Zeng, Chen-Song Zhang, Shuo Zhang
Summary: This paper presents a piecewise quadratic finite element method for H-1 problems on rectangular grids. The method is proven to have a convergence rate of O(h(2)) and a lower bound of the L-2-norm error for the source problem. For the eigenvalue problem, the computed eigenvalues by this element are shown to be lower bounds of the exact ones. Numerical results are presented to verify the theoretical findings.
BIT NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Nitin Kumar, Mani Mehra
Summary: This paper presents a new method based on shifted Legendre polynomials for solving a class of two dimension fractional optimal control problem. The necessary optimality conditions are derived as a two-point fractional-order boundary value problem using the Lagrange multiplier method and integration by part formula. The fractional operators of shifted Legendre polynomial are computed and used to convert the necessary optimality conditions into a system of algebraic equations. L-2-error estimates in approximation and convergence analysis of the proposed method are discussed and illustrated with examples.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2022)
Article
Mathematics, Applied
Pratibha Shakya, Rajen Kumar Sinha
Summary: This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Ting Tan, Waixiang Cao, Jing An
Summary: In this paper, an efficient spectral method for the transmission eigenvalue problem is developed and analyzed. By rewriting the original problem into an equivalent fourth order coupled linear eigenvalue problem, a new variational formulation based on a mixed scheme is proposed. Using the spectral theory of compact operators and the approximation property of orthogonal projection operators in non-uniform weighted Soblev spaces, error estimates for both eigenvalues and eigenfunctions approximations are established in the d (d = 2, 3) dimensional setting. Numerical examples demonstrate the exponential convergence, effectiveness, and efficiency of the method for computing the eigenvalues.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Changling Xu, Hongbo Chen
Summary: This paper presents a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The scheme approximates the state, co-state, and control variable, and achieves superclose results and error estimates. The two-grid algorithm reduces the complexity of problem solving and maintains high accuracy.
Article
Mathematics, Applied
Huasheng Wang, Yanping Chen, Yunqing Huang, Wenting Mao
Summary: This paper investigates a boundary value problem of a fractional convection-diffusion equation with general two-sided fractional derivative, studying the well-posedness of the variation formulation under certain assumptions. A Petrov-Galerkin method is developed using Jacobi poly-fractonomials for the trial and test space, allowing for optimal error estimates in properly weighted Sobolev space with a diagonal matrix of the leading term. It is shown that even for smooth data, only algebraic convergence is obtained due to the regularity of the solution, with numerical examples presented to demonstrate the validity of the theoretical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Yanping Chen, Xiuxiu Lin, Yunqing Huang, Qian Lin
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Yonghui Qin, Yanping Chen, Yunqing Huang, Heping Ma
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
Zhaojie Zhou, Jiabin Song, Yanping Chen
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
Wenting Mao, Yanping Chen, Haitao Leng
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2020)
Article
Mathematics, Applied
Huasheng Wang, Yanping Chen, Yunqing Huang, Wenting Mao
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2020)
Article
Mathematics, Applied
Xiulian Shi, Yanping Chen, Yunqing Huang, Fenglin Huang
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2020)
Article
Mathematics, Applied
Jun Liu, Chen Zhu, Yanping Chen, Hongfei Fu
Summary: In this paper, a novel Crank-Nicolson ADI quadratic spline collocation method is developed for the approximation of two-dimensional two-sided Riemann-Liouville space-fractional diffusion equation. The method is unconditionally stable for certain values of alpha and beta, and it is shown to be convergent with second order in time and min{3 - alpha, 3 - beta} order in space. Numerical examples are included to confirm the theoretical results.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Ying Liu, Yanping Chen, Yunqing Huang, Qingfeng Li
Summary: This article presents an efficient two-grid method for solving the mathematical model of semiconductor devices, approximating the electric potential and concentration equations with mixed finite element and standard Galerkin methods. By linearizing the full discrete scheme using Newton iteration, small scaled nonlinear equations are solved on the coarse grid followed by linear equations on the fine grid. Detailed analysis of error estimation for two-grid solutions is provided, demonstrating asymptotically optimal approximations when the mesh size satisfies H = O(h(1/2)). Numerical experiments illustrate the efficiency of the two-grid method.
APPLIED MATHEMATICS AND MECHANICS-ENGLISH EDITION
(2021)
Article
Mathematics, Applied
Wenting Mao, Yanping Chen, Huasheng Wang
Summary: This paper presents an efficient spectral algorithm and method to solve fractional initial value problems, using a special set of general Jacobi functions to form trial and test spaces. Rigorous error analysis is conducted in non-uniformly weighted Sobolev spaces, with optimal error estimates obtained. The postprocessing technique is used and superconvergence estimates are derived, along with asymptotically exact a-posteriori error estimators. Numerical experiments are included for theoretical support.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Li-Bin Liu, Yanping Chen
Summary: This study focuses on a nonlinear fractional differential equation with Caputo fractional derivative and introduces a new monitor function for designing an adaptive grid algorithm to solve this type of equation. Numerical results demonstrate that the presented adaptive method outperforms other monitor functions in solving this kind of nonlinear fractional differential equation.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Qingfeng Li, Yanping Chen, Yunqing Huang, Yang Wang
Summary: This paper presents two efficient two-grid algorithms with L1 scheme for solving two-dimensional nonlinear time fractional diffusion equations, which can reduce computational cost and maintain asymptotically optimal accuracy.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Bo Tang, Yanping Chen, Xiuxiu Lin
Summary: In this paper, a space-time spectral Galerkin method is re-examined for solving multi-term time fractional diffusion equations, with improved a posteriori error estimates proposed and validated through numerical examples to confirm the effectiveness of the theoretical claims.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Yanping Chen, Hanzhang Hu
Summary: This study proposes a combined method of mixed finite element method for the pressure equation and finite element method with characteristics for the concentration equation to solve the coupled system of incompressible two-phase flow in porous media. A two-grid algorithm based on the Newton iteration method is developed and analyzed for the nonlinear coupled system. Theoretical and numerical results demonstrate the efficiency and asymptotically optimal accuracy of the proposed schemes as long as the mesh sizes satisfy a specific condition.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2021)
Article
Mathematics, Applied
Qingfeng Li, Yanping Chen, Yunqing Huang, Yang Wang
APPLIED NUMERICAL MATHEMATICS
(2020)
