Multigrid Methods for Hellan–Herrmann–Johnson Mixed Method of Kirchhoff Plate Bending Problems

Long Chen,Jun Hu,Xuehai Huang

doi:10.1007/s10915-017-0636-z

Abstract

A V-cycle multigrid method for the Hellan–Herrmann–Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ mixed method is bounded away from one uniformly with respect to the mesh size. The uniform convergence is achieved for the V-cycle multigrid method with only one smoothing step and without full elliptic regularity assumption. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ mixed method and the corresponding commutative diagram is of some interest independent of the current context.

Highlights

We consider multigrid methods for solving the saddle point system arising from the Hellan-Herrmann-Johnson (HHJ) mixed method discretization of a fourth order equation: the Kirchhoff plate bending problem.Linear systems arising from discretization of fourth order partial differential equations are difficult to solve due to the poor spectral properties
For C0 interior penalty methods of fourth order equations in [16, 27], it is proved in [17] that V-cycle, F-cycle and W-cycle multigrid algorithms are uniform contractions
An algebraic multigrid method by smooth aggregation is developed for the fourth order elliptic problems in [49]

Summary

Introduction

We consider multigrid methods for solving the saddle point system arising from the Hellan-Herrmann-Johnson (HHJ) mixed method discretization (cf. [32, 33, 39]) of a fourth order equation: the Kirchhoff plate bending problem.Linear systems arising from discretization of fourth order partial differential equations are difficult to solve due to the poor spectral properties. We consider multigrid methods for solving the saddle point system arising from the Hellan-Herrmann-Johnson (HHJ) mixed method discretization (cf [32, 33, 39]) of a fourth order equation: the Kirchhoff plate bending problem. Optimal-order nonconforming multigrid methods with the full regularity assumption are developed in [13, 45, 61, 57, 47]. An algebraic multigrid method by smooth aggregation is developed for the fourth order elliptic problems in [49]. In all of these work, special intergrid transfer operators are necessary for both these conforming and nonconforming multigrid methods, since either the underlying finite element spaces are non-nested or the quadratic forms are non-inherited.

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Results

Conclusion