Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations
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Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations Huipo Liu · Ningning Yan
Received: 22 November 2006 / Accepted: 28 September 2007 / Published online: 22 November 2007 © Springer Science + Business Media, LLC 2007
Abstract In this paper, the superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes Equations is discussed. The superclose property is proven for rectangular meshes; then global superconvergence is derived by applying a postprocessing technique. In addition, some numerical examples are presented to demonstrate our theoretical results. Keywords Stokes equations · Nonconforming finite element · Superconvergence Mathematics Subject Classification (2000) 65N30
1 Introduction For the velocity-pressure formulation of the Stokes equations, many finite element schemes which satisfy the Babuška–Brezzi condition have been introduced in Babuška [2], Brezzi and Fortin [4] and Girault and Raviart [7]. The first approach is the conforming finite element scheme that uses a continuous piecewise polynomial to approximate the velocity (the finite element space of velocity belongs to (H 1 ())2 ); for instance, the Hood–Taylor element [8], the Mini element [1], the Bernadi–Raugel
Communicated by Martin Stynes. The research was supported by National Natural Science Foundation of China (No. 60474027). H. Liu Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China N. Yan (B) Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China e-mail: [email protected]
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element [3], the bilinear–bilinear element [21], the bilinear-constant element [7]. The second approach is the nonconforming finite element scheme that uses a weak continuous piecewise polynomial to approximate the velocity (the finite element space of velocity does not belong to (H 1 ())2 ); for example, the nonconforming P1 − P0 element [6], the nonconforming rotated bilinear-constant element [20], and the nonconforming quadrilateral linear-constant scheme [9]. It is well known that the finite element schemes mentioned above are usually accompanied with large computational complexity, especially when higher-order finite elements are used. However, it has been found that there exists a potential for high accuracy or superconvergence for several finite element schemes when the exact solution is smooth enough and the mesh satisfies some special conditions. Using the element analysis and integral identity technique, Lin, Li, and Zhou in 3 [13] proved O h 2 -superconvergence for the bilinear-bilinear element. Moreover, Pan in [18] derived O h2 -superconvergence for the bilinear-constant element on uniform rectangular meshes. Similarly, the O h2 -superconvergence property has been proved for the Bernadi–Raugel element on nonuniform rectangular meshes (cf. [15]). In addition, using the surface fitting technique, Wang and le
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