An Application of the Abstract Multilevel Theory to Nonconforming Finite Element Methods

A multilevel product algorithm and its additive version (or a multilevel preconditioner) are proposed for the Crouzeix–Raviart nonconforming finite element methods for second order elliptic equations. The authors show that the preconditioned matrix has a condition number bounded uniformly from above if the domain can be partitioned into overlapping convex subdomains. A uniform convergence for the multilevel product algorithm is also established. The results are regularity-free for the continuous problem and can be applied to problems with rough solutions and local grid refinement. Numerical results that illustrate the theory are presented.