On the convergence of nonnested multigrid methods with nested spaces on coarse grids

Abstract

Multigrid methods for discretized partial differential problems using nonnested conforming and nonconforming finite elements are here defined in the general setting. The coarse-grid corrections of these multigrid methods make use of different finite element spaces from those on the finest grid. In general, the finite element spaces on the finest grid are nonnested, while the spaces are nested on the coarse grids. An abstract convergence theory is developed for these multigrid methods for differential problems without full elliptic regularity. This theory applies to multigrid methods of nonnested conforming and nonconforming finite elements with the coarse-grid corrections established on nested conforming finite element spaces. Uniform convergence rates (independent of the number of grid levels) are obtained for both the V and W-cycle methods with one smoothing on all coarse grids and with a sufficiently large number of smoothings solely on the finest grid. In some cases, these uniform rates are attained even with one smoothing on all grids. The present theory also applies to multigrid methods for discretized partial differential problems using mixed finite element methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 265–284, 2000