Optimality of Local Multilevel Methods for Adaptive Nonconforming P1 Finite Element Methods

Abstract

In this paper, a local multilevel product algorithm and its additive version are considered for linear systems arising from adaptive nonconforming P1 finite element approximations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jacobi or Gauss-Seidel smoothers performed on local nodes on coarse meshes and global nodes on the finest mesh. It is shown that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.