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An error estimate of the least squares finite element method for the Stokes problem in three dimensions


Author: Ching Lung Chang
Journal: Math. Comp. 63 (1994), 41-50
MSC: Primary 65N15; Secondary 65N30, 76D07, 76M10
DOI: https://doi.org/10.1090/S0025-5718-1994-1234425-7
MathSciNet review: 1234425
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Abstract: In this paper we are concerned with the Stokes problem in three dimensions (see recent works of the author and B. N. Jiang for the two-dimensional case). It is a linear system of four PDEs with velocity $ \underline u $ and pressure p as unknowns. With the additional variable $ \underline \omega = {\operatorname{curl}}\underline u $, the second-order problem is reduced to a first-order system. Considering the compatibility condition $ \operatorname{div} \underline \omega = 0$, we have a system with eight first-order equations and seven unknowns. A least squares method is applied to this extended system, and also to the corresponding boundary conditions. The analysis based on works of Agmon, Douglis, and Nirenberg; Wendland; Zienkiewicz, Owen, and Niles; etc. shows that this method is stable in the h-version. For instance, if we choose continuous piecewise polynomials to approximate $ \underline u ,\underline \omega $, and p, this method achieves optimal rates of convergence in the $ {H^1}$-norms.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1234425-7
Article copyright: © Copyright 1994 American Mathematical Society

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