Article
Mathematics, Applied
Pham Quy Muoi, Wee Chin Tan, Viet Ha Hoang
Summary: This study examines a multiscale elliptic eigenvalue problem and utilizes multiscale homogenization to derive a solution containing all possible eigenvalues and eigenfunctions. A sparse tensor product finite element method is developed to solve the problem, achieving the required accuracy in a high dimensional tensorized domain. The method significantly reduces computational costs.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2021)
Article
Operations Research & Management Science
Meilan Zeng
Summary: This paper investigates tensor Z-eigenvalue complementarity problems and proposes a semidefinite relaxation algorithm to solve the complementarity Z-eigenvalues of tensors. The algorithm shows asymptotic and finite convergence for tensors with finitely many complementarity Z-eigenvalues, and numerical experiments demonstrate the efficiency of the proposed method.
COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
(2021)
Article
Mathematics, Applied
S. J. Castillo, J. Karatson
Summary: This paper focuses on the iterative solution of finite element discretizations for second-order elliptic boundary value problems. Mesh independent estimations are provided for the rate of superlinear convergence of preconditioned Krylov methods, exploring the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples are presented to demonstrate the theoretical results.
NUMERICAL ALGORITHMS
(2023)
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
Axel Kroener, Heiko Kroener
Summary: This paper explores error estimates for H2 conforming finite elements for elliptic equations modeling the flow of surfaces by different powers of the mean curvature. The study focuses on the finite element discretization error in higher norms for the regularized equation, providing a detailed analysis for the first time, and also examines the dependencies on the regularization parameter.
APPLIED NUMERICAL MATHEMATICS
(2022)
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
Liming Guo, Chunjia Bi
Summary: This paper studies the adaptive finite element method for second-order nonmonotone quasi-linear elliptic problems with exact solution u ∈ H0^1+α(Ω), α≥1/2. The algorithm is based on residual-based a posteriori error estimators and Dorfler's marking strategy, with convergence and quasi-optimality proven when the initial mesh is sufficiently fine. Numerical experiments validate the findings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yingying Xie, Liuqiang Zhong
Summary: We investigated the AWG finite element method for second order elliptic problems and showed that the error between two consecutive adaptive loops is a contraction. Numerical experiments were conducted to support the theoretical findings.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Fei Xu, Manting Xie, Qiumei Huang, Meiling Yue, Hongkun Ma
Summary: A new type of adaptive multigrid method is proposed for solving multiple eigenvalue problems, achieving the same efficiency as the adaptive multigrid method by solving linear boundary value problems and eigenvalue problems in a low-dimensional space.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Fleurianne Bertrand, Daniele Boffi
Summary: This paper discusses the spectral properties of operators associated with the least-squares finite-element approximation of elliptic partial differential equations. The convergence of discrete eigenvalues and eigenfunctions towards the corresponding continuous eigenmodes is studied and analyzed using appropriate L-2 error estimates. Both a priori and a posteriori estimates are proven.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Interdisciplinary Applications
Timo Sprekeler, Hung Tran
Summary: This study focuses on optimal convergence rates in the periodic homogenization of linear elliptic equations, obtaining the best convergence rate in various norms and providing gradient estimates with correction terms. Numerical experiments using a specific diffusion matrix demonstrate the optimality of the obtained rates, with discussions on extending the results to nonsmooth domains and their utility in numerical homogenization.
MULTISCALE MODELING & SIMULATION
(2021)
Article
Mathematics, Applied
Ingeborg G. Gjerde, Kundan Kumar, Jan M. Nordbotten
Summary: In this work, we analyze the Poisson equation with a line source using a dual-mixed variational formulation. By making assumptions on the problem parameters, we split the solution into higher- and lower-regularity terms, and propose a singularity removal-based mixed finite element method to approximate the higher-regularity terms, which significantly improves the convergence rate compared to approximating the full solution.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Maria E. Cejas, Ricardo G. Duran, Mariana I. Prieto
Summary: The study analyzed the approximation of solutions by mixed finite element methods for equations with degenerate coefficients a. It extended classic error analysis to cases where a belongs to Muckenhoupt class A(2) and applied to mixed finite element spaces with commutative diagram property. Detailed analysis on Raviart-Thomas spaces and optimal error estimates were obtained, along with anisotropic error estimates for problems with boundary layers. Applications to the fractional Laplace equation were also considered.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Mathematics, Applied
Liming Guo, Chunjia Bi
Summary: This paper establishes the convergence and quasi-optimality of an adaptive finite element method for nonmonotone elliptic problems in terms of L2 errors on a sufficiently fine initial mesh. The additional refinements required to maintain the meshes sufficiently mildly graded do not affect the convergence and quasi-optimality of the presented adaptive finite element method. Our theoretical results are supported by numerical examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Xiao-Ping Chen, Wei Wei, Xiao-Ming Pan
Summary: This paper presents the convergence factor of the successive quadratic approximation method for solving nonlinear eigenvalue problems and proposes inexact versions for reducing computational cost. The effectiveness of these modified methods is demonstrated through numerical results and analysis of their convergence properties.
APPLIED NUMERICAL MATHEMATICS
(2021)