Article
Mathematics, Applied
Lingling Sun, Yidu Yang
Summary: This paper discusses the a posteriori error estimates and adaptive algorithm of non-conforming mixed finite elements for the Stokes eigenvalue problem. The reliability and efficiency of the error estimators are proven. Two adaptive algorithms, direct AFEM and shifted-inverse AFEM, are built based on the error estimators. Numerical experiments and theoretical analysis show that the numerical eigenvalues obtained by these algorithms achieve optimal convergence order and approximate the exact solutions from below.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Felipe Lepe, Gonzalo Rivera, Jesus Vellojin
Summary: The aim of this paper is to analyze a mixed formulation for the two dimensional Stokes eigenvalue problem, where the stress and velocity are the unknowns and the pressure can be recovered through postprocessing. The paper proposes a mixed numerical method using suitable finite elements for stress approximation and piecewise polynomials for velocity approximation. Convergence and spectral correctness of the proposed method are derived using compact operators theory. Additionally, a reliable and efficient a posteriori error estimator is proposed for achieving optimal convergence order in the presence of non-sufficient smooth eigenfunctions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Felipe Lepe, Gonzalo Rivera, Jesus Vellojin
Summary: This paper analyzes a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem and proposes a posteriori estimators for the nearly and perfectly compressible elasticity spectral problems. The reliability and efficiency of the proposed estimators are proven through a post-process argument.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Divay Garg, Kamana Porwal
Summary: This article discusses a posteriori error analysis for HCT and Morley finite element methods applied to the fourth order obstacle problem and the distributed elliptic optimal control problem with pointwise state constraints. The distributed elliptic optimal control problem is transformed into a fourth order obstacle problem by eliminating the control variable. The article examines the reliability and efficiency of the error estimator and presents numerical experiments that demonstrate its effectiveness in guiding adaptive mesh refinement and reducing computational cost.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Jinhua Feng, Shixi Wang, Hai Bi, Yidu Yang
Summary: This paper investigates an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. The a priori error estimates of the approximate eigenfunction are derived in the broken H1 norm and L2 norm, which are optimal in h and suboptimal in p. Convergence rates of the approximate eigenvalues are also discussed, along with a posterior error estimation and adaptive calculation implementation.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Felipe Lepe, Gonzalo Rivera, Jesus Vellojin
Summary: In this paper, we analyze mixed finite element methods for a velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem. The methods approximate the velocity and pressure with piecewise polynomials, and use the Raviart-Thomas and Brezzi-Douglas-Marini elements to approximate the pseudostress. By utilizing the classic spectral theory for compact operators, we prove that our method does not introduce spurious modes and obtain convergence and error estimates. Numerical results are presented to compare the accuracy and robustness of both numerical schemes.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Rohit Khandelwal, Kamana Porwal, Tanvi Wadhawan
Summary: In this paper, the authors present and analyze a posteriori error estimates in the energy norm for a quadratic finite element method used in the frictionless unilateral contact problem. They discuss the reliability and efficiency of the error estimator, highlighting the importance of properly decomposing the discrete spaces V-h(0) and Qh. Numerical results are presented to demonstrate the reliability and efficiency of the proposed error estimator.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Jian Meng, Liquan Mei
Summary: This paper presents the lowest-order virtual element method for the Kirchhoff plate vibration problem, proving spectral approximation and optimal convergence order for eigenvalues using classical spectral approximation theory in functional analysis. Numerical experiments show that the proposed numerical scheme can achieve the optimal convergence order.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Chunguang Xiong, Manting Xie, Fusheng Luo, Hongling Su
Summary: In this paper, an error analysis method and a new procedure to accelerate the convergence of finite element approximation for the Steklov eigenvalue problem are introduced. The error analysis consists of three steps: introducing an optimal residual type a posteriori error estimator, presenting a residual type a priori estimate in terms of derivatives of the eigenfunctions, and proving accurate a priori error estimates by combining the a priori residual estimate and the a posteriori error estimates. The new procedure for accelerating the convergence is achieved through a postprocessing technique involving solving an auxiliary source problem on argument spaces.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Junshan Lin
Summary: An adaptive boundary-integral equation method is proposed for computing the electromagnetic response of wave interactions in hyperbolic metamaterials. By using adaptive mesh refinement and numerical quadrature rules, the method accurately resolves the fast transition of the integral equation solution and its singularity at the propagation cone boundary, reducing the number of degrees of freedom significantly.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
V. Kosin, S. Beuchler, T. Wick
Summary: In this paper, a new mixed method proposed by Rafetseder and Zulehner is investigated for Kirchhoff plates and applied to fourth order eigenvalue problems. This new mixed method uses two auxiliary variables to require only H(1) regularity for the displacement and the auxiliary variables, without the demand of a convex domain. A direct comparison is provided to the C-0-IPG method and Ciarlet-Raviart's mixed method, specifically in view of convergence orders, for vibration problems with clamped and simply supported plates. Numerical experiments are conducted using the open-source finite element library deal.II and incorporating non-trivial boundary conditions with the coupling of finite elements with elements on the boundary.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Felipe Lepe, Jesus Vellojin
Summary: In this paper, we design and analyze a posteriori error estimators for the mixed Stokes eigenvalue problem in two and three dimensions. The unknowns in this mixed formulation are the pseudotress, velocity, and pressure. Using a lowest order mixed finite element scheme and a postprocessing technique, we prove the reliability and efficiency of the proposed estimator. Numerical tests in two and three dimensions are conducted to assess the performance of the estimator.
Article
Mathematics, Applied
Yangshuai Wang, Huajie Chen, Mingjie Liao, Christoph Ortner, Hao Wang, Lei Zhang
Summary: Hybrid quantum/molecular mechanics models, known as QM/MM methods, are commonly used in material and molecular simulations to strike a balance between accuracy and computational cost. Adaptive QM/MM coupling methods, with on-the-fly classification of atoms, allow for real-time updates of the QM and MM subsystems as needed. This study proposes a new adaptive QM/MM method for material defect simulations based on a residual from a posteriori error estimator, showcasing its effectiveness through numerical simulations.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
M. Thamban Nair, Devika Shylaja
Summary: This paper discusses the numerical approximation of the biharmonic inverse source problem with finite-dimensional measurement data, presenting a unified framework that covers both conforming and nonconforming finite element methods (FEMs). The inverse problem is analyzed through the forward problem, with error estimates derived for the forward solution in an abstract setup applicable to both conforming and Morley nonconforming FEMs. Due to the ill-posed nature of the inverse problem, Tikhonov regularization is employed to obtain a stable approximate solution, and error estimates are established for the regularized solution under different regularization schemes. Numerical results that verify the theoretical findings are also provided.
Article
Mathematics, Applied
Yanjun Li, Hai Bi, Yidu Yang
Summary: In this paper, we studied the discontinuous Galerkin finite element method for the Steklov eigenvalue problem in inverse scattering. We presented complete error estimates including both a priori and a posteriori error estimators, and proved the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms. We also analyzed the reliability of estimators for eigenvalues, and conducted numerical experiments in an adaptive fashion to show the optimal convergence order of our method.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
