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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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An error estimate of the least squares finite element method for the Stokes problem in three dimensions
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by Ching Lung Chang PDF
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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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