A V-cycle mutigrid for quadrilateral rotated Q1 element with numerical integration

The rotated Q(1) nonconforming element first proposed and used to solve the Stokes problem by Rannacher and Turek in [12]. Kloucek, Li and Luskin have implemented it to simulate the martensitic crystals with microstructures [9], [10]. Recently, Shi and Ming [14] gave a detailed mathematics analysis for this element under the bi-section condition for mesh subdivisions, which was first introduced by Shi [13] for analyzing the quadrilateral Wilson element. Meanwhile they also proposed some effective numerical quadrature schemes for this element[14]. Moreover, they have succeeded in using this element for the Mindlin-Reissner plate problem [11]. Quasi-optimal maximum norm estimations for the quadrilateral rotated Q(1) element approximation of Navier-Stokes equations were established in [17]. In this paper, we will investigate multigrid methods for solving discrete algebraic equations obtained by use of the quadrilateral rotated Q(1) elements. An effective V-cycle multigrid algorithm is presented with numerical integrations. A uniform convergence factor is obtained. A similar idea has been exploited for the Wilson nonconforming element [15] and the TRUNC plate element [16]. We also mention that some nonconforming multigrid algorithms for the second order problem are studied in early papers, see [1], [6] for P-1 nonconforming element, and [8] for the rectangular rotated Q(1) element. The outline of the paper is as follows. In section 2, we introduce the quadrilateral rotated Q(1) element. In the last section an effective V-cycle multigrid algorithm is presented.