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Error estimates for a mixed finite element approximation of the Stokes equations. (English) Zbl 0557.76037
The author considers a mixed finite element method for the Stokes problem in a polygonal domain \(\Omega \subset {\mathbb{R}}^ 2\). He establishes a discrete inf-sup condition, which fits into the abstract error analysis of F. Brezzi [ibid. 8, 129-151 (1974; Zbl 0338.90047)] for mixed problems. Using linear finite elements, \(O(h^{\alpha})\)-error estimates for the energy norm of the velocity and the \(L^ 2\)-norm of the pressure are derived. If the domain \(\Omega\) is convex, the exponent equals 1, otherwise it only depends on the greatest interior angle at a corner of \(\Omega\). When using piecewise quadratic functions for the velocity he gets improved error estimates provided \(f\in H^ 1(\Omega)\) and \(\Omega\) is convex.
Reviewer: J.Groß

MSC:
76D07 Stokes and related (Oseen, etc.) flows
76M99 Basic methods in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:
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