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
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
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
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
Engineering, Multidisciplinary
Felipe V. Caro, Vincent Darrigrand, Julen Alvarez-Aramberri, Elisabete Alberdi, David Pardo
Summary: This work extends an automatic energy-norm hp-adaptive strategy to non-elliptic problems and goal-oriented adaptivity. It proposes an error indicator for quasi-optimal hp-unrefinements based on a multi-level hierarchical data structure and alternating h- and p-refinements. The strategy eliminates the basis functions with the lowest contributions to the solution, improving efficiency and accuracy.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(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, Applied
Carsten Carstensen, Rui Ma
Summary: The collective marking strategy combined with alternative refinement indicators has been proven to achieve optimal convergence rates in adaptive mesh refining of LSFEMs. By utilizing explicit identities for the lowest-order Raviart-Thomas and Crouzeix-Raviart finite elements, this study extends the results to arbitrary polynomial degrees and mixed boundary conditions, with novel arguments. The analysis focuses on the Poisson equation in 3D with mixed boundary conditions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Ajinkya Gote, Andreas Fischer, Chuanzeng Zhang, Bernhard Eidel
Summary: The study focuses on reducing computational complexity in numerical homogenization analyses of 3D mesostructures, by systematically adjusting specimen size, resolution, and discretization parameters. By exploring the SRD parameter space, the study aims to find a balance between accuracy and efficiency in analyzing various heterogeneous materials undergoing different deformation regimes.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2022)
Article
Engineering, Petroleum
X. Raynaud, A. Pizzolato, A. Johansson, F. Caresani, A. Ferrari, O. Moyner, H. M. Nilsen, A. Cominelli, K-A Lie
Summary: This study aims to identify discretization errors caused by non-K-orthogonal grids upfront and compare representative, state-of-the-art consistent discretization methods, proposing error indicators and using tracer simulations to assess the impact of these errors. NTPFA and AvgMPFA were found to be the most viable solutions for integration into a commercial simulator, with the linear AvgMPFA method being the least invasive.
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)
Article
Computer Science, Interdisciplinary Applications
Veselin Dobrev, Patrick Knupp, Tzanio Kolev, Ketan Mittal, Vladimir Tomov
Summary: The study introduces an hr-adaptivity framework for optimization of high-order meshes, extending the r-adaptivity method with nonconforming adaptive mesh refinement to better satisfy geometric targets. The methodology is purely algebraic, applicable to various types of meshes and dimensions, and achieves similar accuracy results with significantly fewer mesh nodes.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Rifqi Aziz, Emre Mengi, Matthias Voigt
Summary: This paper first considers the problem of approximating a few eigenvalues of a rational matrix-valued function closest to a prescribed target. It then proposes a subspace framework to solve this problem, which is applied to model reduction and eigenvalue localization. The reliability and efficiency of the framework are verified through numerical experiments.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Gregor Gantner, Dirk Praetorius
Summary: In this study, h-adaptive algorithms are considered in the context of the finite element method and the boundary element method. Under general assumptions on the building blocks SOLVE, ESTIMATE, MARK, and REFINE, it is proven that the adaptive algorithm converges to zero in terms of the underlying a posteriori error estimator. Unlike existing literature, this analysis does not rely on reliability and efficiency estimates but only on the structural properties of the estimator.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Fleurianne Bertrand, Daniele Boffi
Summary: The study focuses on the approximation of the spectrum of least-squares operators in linear elasticity problems. By considering two different formulations and conducting numerical experiments, the theoretical results are confirmed.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Multidisciplinary Sciences
Tatiana Martynova, Galina Muratova, Pavel Oganesyan, Olga Shtein
Summary: The generalized eigenvalue problem for a symmetric definite matrix pencil obtained from finite-element modeling of electroelastic materials is numerically solved using the Lanczos algorithm. In the considered problem, the mass matrix is singular, and hence the semi-inner product defined by this matrix is used in the process. The shift-and-invert Lanczos algorithm is employed to find multiple eigenvalues closest to a given shift and their corresponding eigenvectors. The results of numerical experiments are presented.
Article
Mathematics, Applied
Kevin J. Painter, Thomas Hillen, Jonathan R. Potts
Summary: The use of nonlocal PDE models in describing biological aggregation and movement behavior has gained significant attention. These models capture the self-organizing and spatial sorting characteristics of cell populations and provide insights into how animals perceive and respond to their surroundings. By deriving and analyzing these models, we can better understand biological movement behavior and provide a basis for explaining sociological phenomena.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2024)
Article
Mathematics, Applied
Nicola Bellomo, Massimo Egidi
Summary: This paper focuses on Herbert A. Simon's visionary theory of the Artificial World and proposes a mathematical theory to study the dynamics of organizational learning, highlighting the impact of decomposition and recombination of organizational structures on evolutionary changes.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2024)
Article
Mathematics, Applied
Tayfun E. Tezduyar, Kenji Takizawa, Yuri Bazilevs
Summary: This paper provides an overview of flows with moving boundaries and interfaces (MBI), which include fluid-particle and fluid-structure interactions, multi-fluid flows, and free-surface flows. These problems are frequently encountered in engineering analysis and design, and pose computational challenges that require core computational methods and special methods. The paper focuses on isogeometric analysis, complex geometries, incompressible-flow Space-Time Variational Multiscale (ST-VMS) and Arbitrary Lagrangian-Eulerian VMS (ALE-VMS) methods, and special methods developed in connection with these core methods.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2024)
