Article
Engineering, Multidisciplinary
Gang Wang, Lin Mu, Ying Wang, Yinnian He
Summary: This paper introduces a pressure-robust virtual element method for solving the Stokes problem on convex polygonal meshes. By enhancing the approximation methods for velocity and pressure, pressure-independent velocity approximation is achieved, with numerical experiments validating the theoretical conclusions.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Liming Guo, Wenbin Chen
Summary: In this paper, a decoupled stabilized finite element method is proposed for solving the time-dependent Navier-Stokes/Biot problem. The coupling problem is divided into two subproblems and solved using different numerical methods. The stability analysis and error estimates are provided to validate the effectiveness of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Liming Guo, Wenbin Chen
Summary: The article proposes a decoupled modified characteristic finite element method for solving the time-dependent Navier-Stokes/Biot problem. The method uses implicit backward Euler scheme for time discretization and treats coupling terms explicitly. The stability and error estimates of the fully discrete scheme are established and validated through numerical experiments.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Zhen Guan, Xiaodong Wang, Jie Ouyang
Summary: In this paper, an improved finite difference/finite element method is proposed for the fractional Rayleigh-Stokes problem with a nonlinear source term. The method utilizes a linearized difference scheme along with the second-order backward differentiation formula and weighted Grunwald-Letnikov difference formula for time discretization, achieving higher stability and convergence accuracy than previous works. Numerical examples are also provided to validate the theoretical results.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2021)
Article
Mathematics, Applied
Hui Peng, Qilong Zhai, Ran Zhang, Shangyou Zhang
Summary: This paper proposes a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition, and validates the theoretical analysis through numerical experiments.
SCIENCE CHINA-MATHEMATICS
(2021)
Article
Mathematics, Applied
Mengya Su, Zhiyue Zhang
Summary: In this paper, we study the numerical approximation of an elliptic interface optimal control problem where the control variable acts on the interface. The discontinuity of coefficients across the interface leads to low regularity of the solution over the entire domain. To address this issue, we propose an immersed finite element method based on a uniform mesh to solve the state and adjoint equations, and discretize the control variable using a variational discretization method. Numerical experiments with complex interfaces, constrained control, no exact solution, and variable coefficients demonstrate the effectiveness of this numerical method.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Thirupathi Gudi, Ramesh Ch. Sau
Summary: In this paper, a finite element analysis is presented for a Dirichlet boundary control problem governed by the Stokes equation. The control is considered in a convex closed subset of the energy space H-1(Omega). The authors introduce the Stokes problem with outflow condition and control on the Dirichlet boundary to overcome the limited regularity of previous control formulations. The theoretical results are validated by numerical tests.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Rodolfo Araya, Cristian Carcamo, Abner H. Poza
Summary: In this work, a new stabilized finite element scheme is introduced and analyzed for the Stokes-Temperature coupled problem. The scheme allows for equal-order interpolation to approximate the velocity, pressure, temperature, and stress. An equivalent variational formulation of the coupled problem is analyzed, inspired by ideas proposed in [3]. Existence of the discrete solution is proved, decoupling the proposed stabilized scheme and utilizing continuous dependence results and Brouwer's theorem. Optimal convergence is also proved under classic regularity assumptions of the solution. Numerical examples are presented to demonstrate the quality of the scheme, including a comparison to a standard reference in geosciences described in [38].
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
N. Staili, M. Rhoudaf
Summary: The aim of this paper is to simulate the two-dimensional stationary Stokes problem. The Stokes problem is reduced to a biharmonic one using the vorticity-Stream function formulation. The paper develops an approach to discretize the Laplace operator by the nonconforming P1 finite element. The convergence of the method is shown with the techniques of compactness, and an error estimate is proved for solutions in C-4 ((Ω) over bar). Numerical experiments are performed for the steady-driven cavity problem.
INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS
(2022)
Article
Mathematics, Interdisciplinary Applications
Ying Sheng, Tie Zhang, Zixing Pan
Summary: This paper discusses the superconvergence results of the stable P-1 - P-1 finite element pair solving the Stokes eigenvalue problem, and obtains superconvergence results for pressure and velocity gradient approximations under the strong regular mesh triangulation condition. The proof of the superconvergence rate for the eigenvalue approximation is also provided, along with numerical experiments to validate the theoretical analysis.
CHAOS SOLITONS & FRACTALS
(2021)
Article
Mathematics, Applied
Yuhang Ren, Demin Liu
Summary: The pressure correction finite element method is proposed for the 2D/3D time-dependent thermomicropolar fluid equations in this paper. The first-order and second-order backward difference formulas (BDF) are used to approximate the time derivative term, and the stability analysis and error estimation of the first-order semi-discrete scheme are proven. Finally, numerical examples are provided to demonstrate the effectiveness and reliability of the proposed method, which is capable of simulating problems with high Rayleigh number.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Sergio Caucao, Eligio Colmenares, Gabriel N. Gatica, Cristian Inzunza
Summary: In this paper, a Banach spaces-based approach for numerically solving the stationary chemotaxis-Navier-Stokes problem is introduced and analyzed. The approach involves a fully-mixed finite element method and a coupled system of three saddle point-type problems. The well-posedness and existence of a unique solution are established using fixed-point strategy and Banach theorem. A specific set of finite element subspaces is introduced to ensure stability and approximate local conservation of momentum. Numerical experiments demonstrate the good performance of the method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Bo Zheng, Yueqiang Shang
Summary: This paper introduces two local and parallel finite element algorithms for the 2D/3D steady Stokes equations with a nonlinear damping term, using a two-grid discretization and domain decomposition approach. The algorithms first approximate the low-frequency component on a coarse grid, and then compute the high-frequency component on a locally fine grid through some local and parallel procedures. The proposed algorithms are easy to implement with low communication complexity, and error estimates of the approximate solutions are derived using a technical tool of local a priori estimate for finite element solution.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yuan Li, Rong An
Summary: This paper proposes a linear and decoupled Euler finite element scheme for numerically solving the 3D incompressible Navier-Stokes equations with mass diffusion. The proposed algorithm is unconditionally stable at the full discrete level when the time step size and mesh size are sufficiently small, and optimal temporal-spatial error estimates for velocity and density are provided without any constraint on the time step size and mesh size using error splitting technique.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Astronomy & Astrophysics
Yukitaka Minesaki
Summary: In this article, a second-order integrator that preserves Hill's regions is proposed to accurately simulate the phenomenon of gravitational capture of massless particles.
ASTROPHYSICAL JOURNAL
(2023)
Article
Mathematics, Applied
L. Beirao Da Veiga, D. Mora, G. Rivera
MATHEMATICS OF COMPUTATION
(2019)
Article
Engineering, Multidisciplinary
Heng Chi, Lourenco Beirao da Veiga, Glaucio H. Paulino
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2019)
Article
Mathematics, Applied
L. Beirao da Veiga, A. Russo, G. Vacca
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2019)
Article
Mathematics, Applied
L. Beirao da Veiga, G. Manzini, L. Mascotto
NUMERISCHE MATHEMATIK
(2019)
Article
Mathematics, Applied
L. Beirao da Veiga, D. Mora, G. Vacca
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Engineering, Multidisciplinary
E. Artioli, L. Beirao da Veiga, F. Dassi
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2020)
Article
Mathematics, Applied
L. Beirao da Veiga, F. Dassi, G. Vacca
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2020)
Article
Mathematics, Applied
L. Beirao da Veiga, F. Brezzi, L. D. Marini, A. Russo
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2020)
Review
Mathematics, Applied
E. Artioli, L. Beirao da Veiga, M. Verani
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2020)
Article
Engineering, Multidisciplinary
Fadi Aldakheel, Blaz Hudobivnik, Edoardo Artioli, Lourenco Beirao da Veiga, Peter Wriggers
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2020)
Article
Engineering, Multidisciplinary
L. Beirao da Veiga, A. Pichler, G. Vacca
Summary: This paper presents a virtual element (VE) discretization for a time-dependent coupled system of nonlinear partial differential equations, aiming to investigate the capabilities of virtual element methods (VEM) for complex fluid flow problems. By combining VEM with a time stepping scheme, a theoretical analysis of the method was developed under the assumption of a regular solution. The scheme was then tested on both regular and realistic test cases to validate its effectiveness.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
L. Beirao da Veiga, F. Dassi, G. Manzini, L. Mascotto
Summary: We introduce a low order virtual element discretization for time dependent Maxwell's equations, which allows for the use of general polyhedral meshes. Both the semi- and fully-discrete schemes are considered. We derive optimal a priori estimates and validate them through numerical experiments. As key findings, we discuss novel inequalities associated with de Rahm sequences of nodal, edge, and face virtual element spaces.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
L. Beirao da Veiga, C. Canuto, R. H. Nochetto, G. Vacca
Summary: This study investigates the equilibrium of a hinged rigid leaflet with an attached rotational spring in a stationary incompressible fluid within a rigid channel using theoretical and numerical methods. Sufficient conditions for the existence and uniqueness of equilibrium positions are identified based on properties of the domain functional. The proposed numerical technique utilizes the mesh flexibility of the Virtual Element Method and proves quasi-optimal error estimates through a variety of numerical experiments.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Jian Meng, Lourenco Beirao da Veiga, Lorenzo Mascotto
Summary: In this paper, we establish stability bounds for Stokes-like virtual element spaces in both two and three dimensions. These bounds are crucial for deriving optimal interpolation estimates. In addition, we conduct numerical tests to investigate the behavior of the stability constants from a practical perspective.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Engineering, Multidisciplinary
L. Beirao da Veiga, D. Mora, A. Silgado
Summary: In this paper, a fully-coupled virtual element method is proposed for solving the nonstationary Boussinesq system in 2D. The method utilizes the stream-function and temperature fields and employs C1- and C0-conforming virtual element approaches for spatial discretization. The temporal variable is discretized using a backward Euler scheme. The well-posedness and unconditional stability of the fully-discrete problem are proved, and error estimates in H2- and H1-norms are derived for the stream-function and temperature fields. Benchmark tests are conducted to validate the theoretical error bounds and demonstrate the behavior of the fully-discrete scheme.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
