A Hermite Multi-grid Method And The Elliptic Eigenvalue Problem Of Non-conforming Element Approximation

This paper contains two parts. In the first part, we focus on a V-cycle multigrid method for a Hermite-type rectangular finite element. We define new mesh-dependent inner products (·,·) k and mesh-dependent norms ||| · ||| s,k . And we show that ||| · ||| 0,k is equivalent to the L2-norm. The convergent property of the V-cycle multigrid method is studied by using intergrid transfer operators and error operators and the optimal error estimate is obtained. In the second part, the convergent property of a nonconforming element with constraints for the 2nd order elliptic problem is analyzed by applying new techniques. And the optimal order is obtained. The element is of less degrees of freedom and simple structure.