Optimal Error Estimate for the Div Least-squares Method with Data $f\inL^2$ and Application to Nonlinear Problems

Authors:
Zhiqiang Cai;JaEun Ku
Affiliations:
zcai@math.purdue.edu;jku@math.okstate.edu
Venue:
SIAM Journal on Numerical Analysis
Year:
2010

Citing 0
Cited 2

Sharp $L_2$-Norm Error Estimates for First-Order div Least-Squares Methods

SIAM Journal on Numerical Analysis
Least-squares solutions as solutions of a perturbation form of the Galerkin methods: Interior pointwise error estimates and pollution effect

Journal of Computational and Applied Mathematics

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Abstract

The div least-squares methods have been studied by many researchers for the second-order elliptic equations, elasticity, and the Stokes equations, and optimal error estimates have been obtained in the $H(\mathrm{div})\times H^1$ norm. However, there is no known convergence rate when the given data $f$ belongs only to $L^2$ space. In this paper, we will establish an optimal error estimate in the $L^2\times H^1$ norm with the given data $f\in L^2$ and, hence, fill a theoretical gap of least-squares methods. As a consequence of this estimate, we will provide a convergence analysis for the linearization process on solving Navier-Stokes equations, which uses the div least-squares method for solving the corresponding Stokes equations.