Article
Mathematics, Applied
Guangzhi Du, Liyun Zuo
Summary: A two-grid method with backtracking is proposed for the mixed Stokes/Darcy system, and theoretical analysis and numerical experiments are conducted to validate the approach. Coarse mesh correction is utilized to improve error bounds for the velocity field and pressure field, with the numerical results supporting the theoretical findings.
JOURNAL OF NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Hongxing Zhao
Summary: This paper investigates the flow of fluid through a thin corrugated domain saturated with porous medium, governed by the Navier-Stokes model. Asymptotic models are derived by comparing the relation between a and the size of the periodic cylinders. The homogenization technique based on the generalized Poincare inequality is used to prove the main results.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Xiaocheng Yang, Yueqiang Shang, Bo Zheng
Summary: A simplified two-level subgrid stabilized method with backtracking technique is proposed for the steady incompressible Navier-Stokes equations at high Reynolds numbers, which combines the best algorithmic characteristics of the standard two-level method with backtracking technique and subgrid stabilized method to achieve an optimal convergence rate.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Eid Wassim, Bo Zheng, Yueqiang Shang
Summary: Based on two-grid discretizations, this paper proposes a parallel finite element method for solving the 2D/3D Navier-Stokes equations with damping. The method solves a fully nonlinear problem on a global coarse grid and then updates the coarse grid solution by solving linearized residual subproblems in overlapping fine grid subdomains using local and parallel procedures. The proposed method's errors are estimated with the help of a local a priori estimate for the finite element solution, and its performance is demonstrated through numerical tests.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics, Applied
Bo Zheng, Yueqiang Shang
Summary: This paper presents and studies three two-grid stabilized quadratic equal-order finite element algorithms based on two local Gauss integrations for the steady Navier-Stokes equations with damping. The algorithms first solve a stabilized nonlinear problem on a coarse grid and then pass the coarse grid solution to a fine grid for solving a stabilized linear problem. The stability of the algorithms is analyzed using nonlinear analysis techniques, and optimal order error estimates of the approximate solutions are derived. Theoretical and numerical results demonstrate that the accuracy of the approximate solutions computed by the two-grid stabilized algorithms is comparable to solving a fully stabilized nonlinear problem on the same fine grid, while saving a significant amount of CPU time compared to the one-grid stabilized algorithm.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
J. K. Djoko, J. Koko, M. Mbehou, Toni Sayah
Summary: In this study, the equations of Stokes and Navier-Stokes under power law slip boundary condition are examined theoretically and numerically. The existence and convergence of solutions are established for both problems, and optimal and sub-optimal a priori error estimates are derived in the finite element approximations. Iterative schemes for solving the nonlinear problems are formulated, and their convergence is studied. The theoretical findings are confirmed by numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov
Summary: This paper studies the lateral flow of a Boussinesq-Scriven fluid on a passively evolving surface embedded in Double-struck capital R-3, and introduces a well-posed weak formulation and a numerical solution method.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Shilin Mi, Guangzhi Du
Summary: In this paper, a two-grid method with backtracking is proposed and analyzed for Magnetohydrodynamic(MHD) equations with low electromagnetic Reynolds number. This algorithm combines the classical two-grid scheme and the backtracking technique, i.e., the coarse grid correction. The optimal error estimates are established both in energy norm and L2 norm.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mustafa E. Danis, Jue Yan
Summary: This study proposes a new formula for the nonlinear viscous numerical flux and extends it to the compressible Navier-Stokes equations using the direct discontinuous Galerkin method with interface correction (DDGIC). The new method simplifies the implementation and enables accurate calculation of physical quantities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Gulnur Hacat, Fikriye Yilmaz, Aytekin Cibik, Songul Kaya
Summary: This paper presents a family of implicit-explicit time stepping scheme for the optimal control problem of the unsteady Navier-Stokes equations. The scheme utilizes discrete curvature to stabilize the optimization problem and provides stability and error analyses for the state, adjoint, and control variables.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Richard M. Hoefer
Summary: This paper studies the solution to the Navier-Stokes equations in perforated media with small particles and no-slip boundary conditions. The behavior of the solution is examined for small values of ε, which depends on the diameter of the particles and the viscosity of the fluid. The results demonstrate that when the local Reynolds number at the particles is negligible, the particles exert an approximately linear friction force on the fluid. The effective macroscopic equations obtained depend on the magnitude of the collective friction.
Article
Mathematics, Applied
Alejandro Allendes, Gabriel R. Barrenechea, Julia Novo
Summary: This work focuses on the finite element discretization of the incompressible Navier-Stokes equations, using a low order stabilized finite element method with piecewise linear continuous discrete velocities and piecewise constant pressures. The modified continuity equation involves a stabilizing bilinear form based on the jumps of the pressure, resulting in a divergence-free velocity field. The stability of the discrete problem is proven without needing to rewrite the convective field in its skew-symmetric way, and error estimates with constants independent of viscosity are established and validated through numerous numerical experiments.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Yueqiang Shang
Summary: A new defect-correction method based on subgrid stabilization is proposed for simulating steady incompressible Navier-Stokes equations with high Reynolds numbers. This method uses a two-grid finite element discretization strategy and involves solving a nonlinear coarse mesh system followed by two linearized fine mesh problems. The effectiveness of the method is demonstrated through numerical results and error bounds estimation.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Zhongwei Shen
Summary: This article examines Darcy's law for an incompressible viscous fluid flowing in a porous medium. The paper establishes the O(root epsilon) convergence rate by constructing two boundary layer correctors to control the boundary layers created by the incompressibility condition and the discrepancy of boundary values.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Medine Demir, Aytekin Cibik, Songul Kaya
Summary: This paper discusses the application of the backward Euler based linear time filtering method in the developed energy-momentum-angular momentum conserving formulation under weakly enforced divergence constraint. The method enhances accuracy and improves approximate solutions by adding time filtering as a post-processing step. The numerical studies confirm the theoretical findings and demonstrate the superiority of the proposed method over the unfiltered case.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
