Criteria for the approximation property for multigrid methods in nonnested spaces
نویسندگان
چکیده
We extend the abstract frameworks for the multigrid analysis for nonconforming finite elements to the case where the assumptions of the second Strang lemma are violated. The consistency error is studied in detail for finite element discretizations on domains with curved boundaries. This is applied to prove the approximation property for conforming elements, stabilized Q1/P0elements, and nonconforming elements for linear elasticity on nonpolygonal domains. Proving the approximation property for the multigrid analysis for nonconforming finite element discretizations is formalized in [7, 4, 15] for many cases: it suffices to verify criteria on the approximation quality and the consistency error. In these papers, it is required that a continuous bilinear form can be extended to a nonconforming finite element space, which is not valid for many interesting applications. The purpose of this paper is to establish a full set of criteria which guarantees the approximation property for a wide range of nonnested discretizations, where we do not assume that the discrete bilinear form coincides with the continuous bilinear form for all conforming functions. In the notation, we follow Bramble [5, Chap. 4], and our results can be applied directly to the multigrid theory described there. The results extend known results by Brenner [7] and Stevenson [15], and they provide a systematic and constructive way of studying nonnested multigrid algorithms for more general nonnested spaces and varying forms. The paper is organized as follows. First, we introduce an abstract setting describing a multigrid hierarchy for nonconforming discretizations of an elliptic partial differential equation without full regularity. As usual, the multigrid approximation property is derived by comparison with the finite element approximation property, which we formulate using an interpolation operator and its adjoint with respect to the energy scalar product. In a second step (Section 1.9), we derive the approximation property from consistency assumptions on a conforming comparison space, similar to the approach in [7]. In Section 2, we consider the case of conforming finite elements on a polygonal approximation of the computational domain. Here, we choose a comparison space consisting of curved finite elements. After introducing a suitable interpolation operator, we use the equivalence of the operator norm scale (used throughout Section 1) Received by the editor January 23, 2001 and, in revised form, March 21, 2003. 2000 Mathematics Subject Classification. Primary 65N55, 65F10.
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عنوان ژورنال:
- Math. Comput.
دوره 73 شماره
صفحات -
تاریخ انتشار 2004