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
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
Engineering, Multidisciplinary
Rodolfo Araya, Cristian Carcamo, Abner H. Poza
Summary: This work introduces and analyzes an adaptive stabilized finite element method for solving a nonlinear Darcy equation with pressure-dependent viscosity and mixed boundary conditions. The well-posedness and optimal error estimates of the discrete problem are established under standard assumptions, along with a residual-based a posteriori error estimator for the stabilized scheme. Numerical examples in two and three dimensions are presented to confirm the theoretical results.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Li-Bin Liu, Ciwen Zhu, Guangqing Long
Summary: An adaptive grid method based on the backward Euler formula is studied for a system of semilinear singularly perturbed initial value problems, showing robustness and a first-order convergence rate. A fully discrete adaptive grid method is constructed with a standard residual-type a posterior error estimation, leading to the design of an adaptive grid algorithm based on an optimal monitor function. The method is extended to nonlinear systems and numerical results demonstrate its effectiveness.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Ying Liu, Yufeng Nie
Summary: This paper derives the a priori and a posteriori error estimates for the weak Galerkin finite element method with CrankNicolson time discretization applied to parabolic equations. The a priori estimates are based on existing results for elliptic projection problems, while the a posteriori estimates use an elliptic reconstruction technique to decompose the true error into elliptic and parabolic components. These estimates are further used to develop a temporal and spatial adaptive algorithm, with numerical results provided to validate the proposed estimators on uniform and adaptive meshes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Gregory Etangsale, Marwan Fahs, Vincent Fontaine, Nalitiana Rajaonison
Summary: In this paper, we improve the a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The theoretical results are supported by numerical evidence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Lothar Banz, Orlando Hernandez, Ernst P. Stephan
Summary: This article analyzes the hp finite element discretization of a second-type Bingham-type variational inequality, proving convergence and presenting reliable a posteriori error estimators applicable to any approximation of the exact solution.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Xuqing Zhang, Jiayu Han
Summary: The modified transmission eigenvalue problem of elastic waves, arising from inverse scattering theory, is analyzed in this study. The well-posedness of the problem is first analyzed, followed by a rigorous error analysis of conforming finite element methods. Two types of multigrid schemes are then established. Numerical research is conducted on the effect of material flaws on the modified elastic transmission eigenvalues, and the multigrid schemes are applied for parallel computing. Numerical experiments validate the theoretical convergence order and the efficiency of the proposed schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
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, 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
Haitao Leng
Summary: In this paper, the hybridized, embedded-hybridized and embedded discontinuous Galerkin methods for the Stokes equations with Dirac measures are analyzed. The velocity, velocity traces, and pressure traces are approximated by polynomials of degree k >= 1 and the pressure is discretized by polynomials of degree k - 1. The discrete velocity field satisfies a property called divergence-free, and the discrete velocity fields obtained by hybridized and embedded-hybridized discontinuous Galerkin methods are H(div)-conforming. A priori and a posteriori error estimates for the velocity in L-2-norm are obtained using duality argument and Oswald interpolation. Additionally, a posteriori error estimates for the velocity in W-1,W-q-seminorm and the pressure in L-q-norm are derived. Numerical examples are provided to validate the theoretical analysis and demonstrate the performance of the obtained a posteriori error estimators.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
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
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
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
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)
