Article
Mathematics, Applied
Divay Garg, Kamana Porwal
Summary: The main aim of this article is to analyze the application of the mixed finite element method for the second order Dirichlet boundary control problem. Both a priori and a posteriori error analysis are conducted using the energy space based approach. Optimal order estimates for the a priori errors in the energy norm and L2-norm are obtained with the aid of auxiliary problems. The reliability and efficiency of the proposed a posteriori error estimator are discussed using the Helmholtz decomposition. Numerical experiments are carried out to validate the theoretical findings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Xuqing Zhang, Jiayu Han, Yidu Yang
Summary: The linear elasticity problem is widely used in structural analysis and engineering design. The classical Crouzeix-Raviart finite element method is shown to be unstable for solving the linear elasticity eigenvalue problem with the pure traction boundary condition. In this paper, a new scheme with stabilization terms is proposed, and a priori error estimates for approximate eigenpairs are proved. Numerical experiments using the stabilized scheme demonstrate that it is both locking-free and efficient.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Huoyuan Duan, Can Wang, Zhijie Du
Summary: A new div FOSLS mixed finite element method is proposed and analyzed for the first-order system of the general second-order elliptic problems, offering coercive, symmetric, and versatile features.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Lise-marie Imbert-gerard, Andrea Moiola, Paul Stocker
Summary: This article introduces a quasi-Trefftz discontinuous Galerkin method for the discretization of the acoustic wave equation with piecewise-smooth material parameters. The authors prove that this method has the same excellent approximation properties as the classical Trefftz method and demonstrates stability and high-order convergence of the DG scheme.
MATHEMATICS OF COMPUTATION
(2023)
Article
Mathematics, Applied
Michael Holst, Martin Licht
Summary: This paper presents a new analysis of finite element methods for solving partial differential equations over curved domains. By changing variables, a physical problem over a curved domain is transformed into a parametric problem over a polytopal parametric domain. Despite the low regularity of the coordinate transformation, high-order convergence rates can still be achieved using a broken Bramble-Hilbert lemma. This analysis has practical applications and provides high-order error estimates for isoparametric finite element methods.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Energy & Fuels
Yawen Liang, Shunli Wang, Yongcun Fan, Paul Takyi-Aninakwa, Yanxin Xie, Carlos Fernandez
Summary: This paper proposes a novel ECC-AEKF algorithm for accurate and robust SOC estimation in electric vehicles. The algorithm calculates the maximum likelihood function of the error series to obtain a new priori error covariance, which minimizes estimation error and reduces the effect of noise and inappropriate covariances. Additionally, a PFFRLS method is presented for model parameter identification. The experimental results demonstrate that the proposed method achieves higher accuracy with less computation time compared to other methods.
JOURNAL OF ENERGY STORAGE
(2023)
Article
Mathematics, Applied
Qianqian Ding, Xiaoming He, Xiaonian Long, Shipeng Mao
Summary: In this paper, a finite element projection method for magnetohydrodynamics equations in Lipschitz domain is developed and analyzed. The proposed method achieves accuracy and efficiency, as demonstrated by numerical examples.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Zhihui Zhao, Hong Li
Summary: In this article, the STCG method is used to find the numerical solution for the 2D Burgers' equation. The method achieves high order accuracy in both time and space directions and exhibits good stability. Numerical experiments confirm the efficiency of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Qianqian Ding, Shipeng Mao, Jiaao Sun
Summary: In this paper, a mixed finite element method is proposed to robustly analyze the pressure problem in nonstationary magnetohydrodynamics system. The method features divergence-free velocity approximation and capturing the strongest magnetic singularities. The results demonstrate the scheme's energy stability and well-posedness of the numerical solution, along with a priori error estimates independent of the Reynolds number.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Zhihui Zhao, Hong Li, Jing Wang
Summary: In this article, a space-time continuous Galerkin method is applied to solve the numerical solution of Sobolev equation with space-time variable coefficients. The method achieves high order precision in both spatial and temporal directions and has good stability. Variable spatial grid structures and time steps are allowed, making it suitable for designing adaptive algorithms on unstructured mesh. The article provides a detailed analysis of well-posed numerical solutions and a priori error estimation without restrictions on the space-time grid ratio, along with numerical experiments demonstrating convergence orders and high efficiency.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Zhihui Zhao, Hong Li
Summary: In this paper, the space-time continuous Galerkin (STCG) method is used to analyze the nonlinear Sobolev equation. A detailed analysis is provided including the proof of the existence and uniqueness of the numerical solution and the a priori error estimate. The numerical experiments show that the STCG method is more efficient and stable than the space-time discontinuous Galerkin (STDG) method.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Xu Li, Hongxing Rui
Summary: In this paper, a P-1(c) circle plus RT0 - P0 discretization method is proposed for solving the Stokes equations on general simplicial meshes in two/three dimensions. The method provides an exactly divergence-free and pressure-independent velocity approximation with optimal order. Additionally, the method can be easily transformed into a pressure-robust and stabilized discretization method, which has a much smaller number of degrees of freedom.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Yi Li, Dandan Xue, Yao Rong, Yi Qin
Summary: This paper proposes a new partitioned method for the coupling dual-porosity Stokes equations, which achieves time accuracy by decoupling the problem into three sub-domain equations at each time step. The algorithm also introduces a grad-div stabilization term to improve the accuracy and conservation of mass. Numerical experiments validate the accuracy and advantage of the algorithm and support the theoretical analysis.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Jing Wen, Jian Su, Yinnian He, Hongbin Chen
Summary: In this paper, semi-discrete and fully discrete schemes of the Stokes-Biot model are proposed and analyzed in detail. The existence and uniqueness of the semi-discrete scheme are proved, with a-priori error estimates derived. Numerical tests under matching and non-matching meshes validate the convergence analysis and support the theoretical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Engineering, Multidisciplinary
Karolinne O. Coelho, Philippe R. B. Devloo, Sonia M. Gomes
Summary: The Scaled Boundary Finite Element Method (SBFEM) is a technique for constructing approximation spaces using a semi-analytical approach, with a focus on deriving a priori error estimates for solutions of harmonic test problems. By characterizing SBFEM spaces in the context of Duffy's approximations and investigating similarities with virtual harmonic approximations, optimal convergence rates for smooth solutions have been confirmed through numerical experiments with polytopal meshes. The SBFEM method also shows optimal accuracy rates for approximating a point singular solution and finite element approximations elsewhere.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)