On a hierarchical basis multilevel method with nonconforming P1 elements
The paper extends \textit{H. Yserentant}'s hierarchical basis multilevel method [ibid. 49, 379-412 (1986; Zbl 0608.65065)] to second order elliptic boundary value problems discretized by nonconforming \(P1\) triangular (Crouseiz-Raviart) elements. The method is based on the sequence of finite element subspaces \(S_ 0,\dots,S_ K=S\) which correspond to a successive refinement. On the contrary to the conforming case, these finite element subspaces are not nested. This difficulty is overcome by introducing intergrid transfer operators. Thorough analysis of these operators and their norms is important for deriving the main result: The condition number of the hierarchical discretization matrix is bounded by \(O(k)\) where \(k\) is the number of refinement levels. Note that this result is better than the corresponding result for conforming \(P1\) elements which is \(O(k^ 2)\). These results are supported by numerical experiments.
nonconforming \(P1\) triangular elements
second order elliptic boundary value problems
scientific article; zbMATH DE number 118306
scientific article