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June 2013 Vol. 56 No. 6: 1313–1330 doi: 10.1007/s11425-013-4582-4 Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem YANG YiDu1,∗ & JIANG Wei2 1School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001,China; 2School of Physics and Mechanical & Electrical Engineering, Xiamen University, Xiamen 361005,China Email: [email protected], [email protected] Received November 21, 2011; accepted January 9, 2013; published online January 30, 2013 Abstract This paper discusses conforming mixed finite element approximations for the Stokes eigenvalue problem. Firstly, several mixed finite element identities are proved. Based on these identities, the following new results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P1-P1 element and Q1-Q1 element approximate the exact eigenvalues from above. (2) As for the P1-P1, Q1-Q1 and Q1-P0 element eigenvalues, the asymptotically exact a posteriori error indicators are presented. (3) The reliable and efficient a posteriori error estimator proposed by Verf¨urth is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis. Keywords the Stokes eigenvalue, conforming mixed finite elements, upper spectral bounds, a posteriori error estimates MSC(2010) 65N25, 65N30 Citation: Yang Y D, Jiang W. Upper spectral bounds and a posteriori error analysis of several mixed finite el- ement approximations for the Stokes eigenvalue problem. Sci China Math, 2013, 56: 1313–1330, doi: 10.1007/s11425-013-4582-4 1 Introduction The finite element eigenvalues approximate the exact ones from above or from below is always an active topic in academic community. It is well known that the minimum-maximum principle insures that the conforming finite element eigenvalues approximate the exact eigenvalues from above (see [34]). Unfor- tunately, the classic result is not valid for the nonconforming finite elements. It has been proved that the numerical eigenvalues by many nonconforming element/nonconforming mixed elements approximate exact eigenvalues from below (see [1, 14, 22–24, 27, 40, 42, 43], etc.). Then how about the conforming mixed finite elements? It can be observed from some numerical examples that the numerical eigenvalues obtained by some conforming mixed finite elements have the properties of approximations from above, but there also have converse results. For example, as for the Stokes eigenvalue problem, the numerical results in [44] indicate that the mixed Bernadi-Raugel element eigenvalues approximate from below, while the mixed Q2-P1 element eigenvalues approximate from above. Moreover, the numerical results in [21] demonstrate that, for the Laplace operator eigenvalue problem, the approximate eigenvalues obtained by low order mixed Raviart-Thomas element approximate exact eigenvalues from above. However, there is ∗Corresponding author c Science China Press and Springer-Verlag Berlin Heidelberg 2013 math.scichina.com www.springerlink.com 1314 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 no theoretical proof for these phenomena. The first objective of this paper is to study the approximate properties of numerical eigenvalues obtained by some mixed finite elements for the Stokes eigenvalue problem. The a posteriori error estimates and adaptive computation in finite element methods were first pro- posed by Babuˇska and Rheinboldt in 1978 (see [3]), and have been developed into one of the main fields in finite element method. In the development of this field, a posteriori error estimate of finite element ap- proximation for eigenvalue problem is also an important topic. For elliptic eigenvalue problems, there are many research results (see, for example, [5,10,11,13,17,18,28,30,39]). For the Stokes eigenvalue problem, Lovadina et al. [26] have presented a suitable error indicator based on residual type. The second objective of this paper is to study a posteriori error estimates of some mixed finite element approximations for the Stokes eigenvalue problem further. To fulfill these two goals, the theory of Babuˇska-Brezzi (see [7]) and the theory of spectral approxima- tions (see [4, 8]) are adopted in this paper. Firstly, several mixed finite element identities for the Stokes eigenvalue problem are proved. Based on these identities, the following results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P1-P1 element and Q1-Q1 element approxi- mate the exact eigenvalues from above. (2) As for the P1-P1, Q1-Q1 and Q1-P0 element eigenvalues, the asymptotically exact recovery type by post-processing interpolation a posteriori error indicators are presented, this is a new work, about which we have not seen the report until now. (3) The reliable and efficient a posteriori error estimator proposed by Verf¨urth [35] is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis. The rest of this paper is organized as follows. In the next section, some preliminaries needed in this paper are presented, and several mixed element identities are proved. In Sections 3 and 4, we prove the property of upper spectral bounds and present a posteriori error of mini-element and P1-P1 element, respectively. In Section 5, we analyze a posteriori error of Q1-P0 element. Finally, numerical experiments are presented to support the theoretical analysis. 2 The mixed finite element identities for the Stokes eigenvalue problem Consider the Stokes eigenvalue problem − Δu + ∇p = λu, in Ω, divu =0, in Ω, (2.1) u =0, on ∂Ω, ⊂ R2 where Ω is a polygonal domain, u =(u1,u2) is the velocity of the flow, and p is the pressure. 1 × 1 2 { ∈ 2 } 2 × 2 Denote U = H0 (Ω) H0 (Ω),V= L0(Ω) = v L (Ω) : Ω v =0 and H = L (Ω) L (Ω). In this paper, the above symbol denotes the element in product space U or H.SetD ⊂ Ω. For any function p in Hm(D), denote 1 2 α 2 pm,D = |∂ p| ,α=(α1,α2), |α| = α1 + α2. |α|m D For simplicity, write pm,Ω = pm. For the vector function u =(u1,u2), denote 1 1 2 2 2 2 2 2 um,D =(u1m,D + u2m,D) , |u|m,D =(|u1|m,D + |u2|m,D) .