Multigrid Methods for the Biharmonic Problem Discretized by Conforming 1 Finite Elements on Nonnested Meshes

Abstract. We consider multigrid algorithms for the biharmonic problem discretized by conforming 1 finite elements. Most finite elements for the biharmonic equation are nonnested in the sense that the coarse finite element space is not a subspace of the space of similar elements defined on a refined mesh. To define multigrid methods, certain intergrid transfer operators have to be constructed. We construct intergrid transfer operators that satisfy a certain stable approximation property. The so-called regularity-approximation assumption is established by using this stable approximation property of the intergrid transfer operator. Optimal convergence properties of the W-cycle and a uniform condition number estimate for the variable V-cycle preconditioner are established by applying an abstract result of Bramble, Pasciak and Xu. Our theory covers the cases when the multilevel triangulations are nonnested and the spaces on different levels are defined by different finite elements.