Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows

2020 ◽  
Vol 145 (1) ◽  
pp. 167-196
Author(s):  
Hongtao Chen ◽  
Jingjing Mao ◽  
Jie Shen
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Auxiliary Variable ◽  
Optimal Error Estimates ◽  
Gradient Flows ◽  
Optimal Error ◽  
Finite Element Schemes

A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains

2018 ◽  
Vol 52 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Yinnian He ◽  
Jun Zou
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Error Estimates ◽  
Mhd Flow ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Discrete Velocity ◽  
Mhd System

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

scholarly journals Numerical analysis for a chemotaxis-Navier-Stokes system

2020 ◽  
Author(s):  
Abelardo Duarte-Rodríguez ◽  
María A. Rodríguez-Bellido ◽  
Diego A. Rueda-Gómez ◽  
Élder J. Villamizar-Roa
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Numerical Scheme ◽  
Stokes System ◽  
Equivalent Model ◽  
Navier Stokes ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Uniform Estimates

In this paper we develop a numerical scheme for approximating a $d$-dimensional chemotaxis-Navier-Stokes system, $d=2,3$, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze optimal error estimates and convergence towards regular solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is  well-posed and mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.

Two-Grid Finite Element Methods for the Steady Navier-Stokes/Darcy Model

2016 ◽  
Vol 6 (1) ◽  
pp. 60-79 ◽  
Author(s):  
Jing Zhao ◽  
Tong Zhang
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Finite Element Methods ◽  
Navier Stokes ◽  
Optimal Error Estimates ◽  
Numerical Computations ◽  
Optimal Error ◽  
Darcy Model ◽  
Efficiency And Effectiveness ◽  
Grid Algorithm

AbstractTwo-grid finite element methods for the steady Navier-Stokes/Darcy model are considered. Stability and optimal error estimates in the H1-norm for velocity and piezometric approximations and the L2-norm for pressure are established under mesh sizes satisfying h = H2. A modified decoupled and linearised two-grid algorithm is developed, together with some associated optimal error estimates. Our method and results extend and improve an earlier investigation, and some numerical computations illustrate the efficiency and effectiveness of the new algorithm.

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