Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations.,

1981 ◽  
Author(s):  
Rolf Rannacher ◽  
Ridgway Scott
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Piecewise Linear ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Finite Element Approximations ◽  
Linear Finite Element

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.

Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry

2019 ◽  
Vol 19 (3) ◽  
pp. 415-430 ◽  
Author(s):  
Fleurianne Bertrand ◽  
Zhiqiang Cai ◽  
Eun Young Park
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Error Estimates ◽  
Stokes Equations ◽  
Displacement Velocity ◽  
Three Dimensions ◽  
Optimal Error Estimates ◽  
Finite Element Approximations ◽  
Solution Methods ◽  
Membrane Problem

AbstractThis paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the{L^{2}}norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of{H(\mathrm{div};\Omega)}for the stress, of{H^{1}(\Omega)}for the displacement/velocity, and of{L^{2}(\Omega)}for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart–Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal{L^{2}}norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook’s membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.

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