Article
Physics, Mathematical
Yanjun Li, Yidu Yang, Hai Bi
Summary: This paper discusses the conforming finite element method for a modified interior transmission eigenvalues problem and presents a complete theoretical analysis and numerical experiments. The results indicate that the a posteriori error estimator is effective, the approximations can reach the optimal convergence order, and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2023)
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
X. U. Q. I. N. G. ZHANG, J. I. A. Y. U. HAN, Y. I. D. U. YANG
Summary: This paper discusses the a posteriori error estimates of the H2-conforming finite element method for the elastic transmission eigenvalue problem. The reliability and efficiency of error indicators for primal and dual eigenfunctions are proved, along with the design of an adaptive algorithm. Numerical experiments are conducted to validate the theoretical analysis and demonstrate the robustness of the algorithm.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Fleurianne Bertrand, Gerhard Starke
Summary: The article presents a posteriori error estimates for the Biot problem using a three-field variational formulation, with H(div)-conforming reconstructions of stress and flux for guaranteed error bounds. Emphasis is placed on nearly incompressible materials, with error estimates holding uniformly even in the incompressible limit.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Artificial Intelligence
Nicolas Barnafi, Gabriel N. Gatica, Daniel E. Hurtado, Willian Miranda, Ricardo Ruiz-Baier
Summary: Deformable image registration (DIR) is a popular technique for aligning digital images, especially in medical image analysis. This study proposes adaptive mesh refinement schemes for the finite-element solution of DIR problems, which have shown to significantly reduce the number of degrees of freedom without compromising solution accuracy. The adaptive scheme performs well in numerical convergence on smooth synthetic images and successfully handles volume-constrained registration problems.
SIAM JOURNAL ON IMAGING SCIENCES
(2021)
Article
Engineering, Multidisciplinary
Mary F. Wheeler, Vivette Girault, Hanyu Li
Summary: This paper focuses on the convergence, stability, and reliability and efficiency of error indicators in a coupled Biot poroelastic model and an elastic model in R3. The coupled system is decoupled using a fixed stress splitting algorithm, and the numerical implementation of the residual-based error indicators is simple but suboptimal. The scheme is tested on two benchmark problems through numerical experiments.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Xuqing Zhang, Jiayu Han, Yidu Yang
Summary: In this paper, we discuss the a posteriori error estimates of the H-2-conforming finite element for the fourth-order non-selfadjoint elastic transmission eigenvalue problem with vector-valued eigenfunctions. We introduce error indicators and design an adaptive algorithm, and validate our theoretical analysis and algorithm's robustness through numerical experiments.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
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)
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
Mirjam Walloth, Winnifried Wollner
Summary: This article introduces a residual-type a posteriori error estimator for a time discrete quasi-static phase-field fracture model, focusing on the robustness of the error estimator for the variational inequality governing the phase-field evolution with respect to the phase-field regularization parameter ε. Numerical examples demonstrate the performance of the proposed a posteriori error estimators on three standard test cases.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Mirjam Walloth, Winnifried Wollner
Summary: This article develops a residual-type a posteriori error estimator for a time discrete quasi-static phase-field fracture model, focusing on the robustness of the error estimator with respect to the phase-field regularization parameter epsilon. Numerical examples are provided to demonstrate the performance of the proposed error estimators on three standard test cases.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
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
Huoyuan Duan, Qiuyu Zhang
Summary: This paper proposes and analyzes a residual-based a posteriori error estimator for a new finite element method for solving the time-dependent Ginzburg-Landau equations with the temporal gauge of superconductivity. The reliability of the error estimator is proven using the dual problem of a linearization of the original problem, and an adaptive algorithm with temporal and spatial refining and coarsening steps is proposed. Numerical results illustrate the performance of the error estimator and the adaptive algorithm in convex and nonconvex domains.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
V. Dhanya Varma, Suresh Kumar Nadupuri, Nagaiah Chamakuri
Summary: This work investigates the finite element approximations to the governing equations of heat and mass transfer in fluidized beds, deriving a posteriori error estimates. The time discretization is carried out using the implicit Euler method, and the total residual and error indicators due to spatial discretization, time discretization, and linearization are utilized for deriving the error estimates for all five variables. An adaptive finite element solution is computed for these model equations and its performance is illustrated.
APPLIED NUMERICAL MATHEMATICS
(2021)
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)
