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Asymptotic expansions and extrapolations of eigenvalues for the Stokes problem by mixed finite element methods. (English) Zbl 1149.65090

A higher order convergence for the finite element approximation of the Stokes problem in 2-dimensions are derived by using extrapolation and error expansion. The first results concern Bernardi-Raugel finite elements on rectangles using polynomials \(Q_{21} \times Q_{12}\) for the velocity \(u\) and \(Q_0\) for the pressure \(p\). Using embedded meshes with step-sizes \(h\) and \(h/2\) and extrapolation technique \(O(h^3)\) and \(O(h^4)\) error estimates for a fixed eigenvalue are derived. Similar, but higher order, \(O(h^5)\) error estimates are derived for \(Q_{22} \times Q_{22}\) for the velocity \(u\) and \(P_1\) elements. These rates require, as expected, higher regularity of the corresponding eigenfunctions, namely, \(u \in (H^5(\Omega))^2\) and \(u \in (H^6(\Omega))^2\) for the first and second case, correspondingly.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
35P15 Estimates of eigenvalues in context of PDEs
65N15 Error bounds for boundary value problems involving PDEs
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