Efficient Block Preconditioning for a $C^1$ Finite Element Discretization of the Dirichlet Biharmonic Problem

We present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretised by C1 bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a 2×2 block diagonal preconditioner (BD) and a block bordered diagonal (BBD) preconditioner. We prove mesh-independent bounds for the spectra of the BD-preconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite and is assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near-optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion and non-convex domains