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

Chang, Ching Lung

doi:10.1090/S0025-5718-1994-1234425-7

An error estimate of the least squares finite element method for the Stokes problem in three dimensions
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Math. Comp. 63 (1994), 41-50 Request permission

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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