Multigrid and multilevel methods for nonconforming 𝑄₁ elements

Abstract

In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming rotated Q 1 Q_1 finite elements in two space dimensions. In particular, for the case of square partitions and the Laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the W \mathcal {W} -cycle with any number of smoothing steps and close-to-optimal condition number estimates for V \mathcal {V} -cycle preconditioners. This is in contrast to most of the other nonconforming finite element discretizations where only results for W \mathcal {W} -cycles with a sufficiently large number of smoothing steps and variable V \mathcal {V} -cycle multigrid preconditioners are available. Some numerical tests, including also a comparison with a preconditioner obtained by switching from the nonconforming rotated Q 1 Q_1 discretization to a discretization by conforming bilinear elements on the same partition, illustrate the theory.