Preconditioning a class of fourth order problems by operator splitting

Abstract

Abstract. We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H s -regularity, s > 1, of the continuous problem nor quasi-uniform grids.

Efficient preconditioning of variational time discretization methods for parabolic Partial Differential Equations

“…In [2], a preconditioner for a class of fourth order problems was introduced and analyzed. The main ingredient is an inexact factorization of the Schur complement S, for which uniform bounds can be proved.…”

“…In this case, the discrete Schur complement operator S = αβI + A 1 A 2 is no longer symmetric, and the analysis presented here for the symmetrically preconditioned operator does not apply. However, following [2], the application of a non-symmetric left-right preconditioner of the form…”

“…To derive the matrix-vector equation that corresponds to the operator equation (3.19) in V h we explain at first the isomorphism r m : L(V h , V h ) → R N ×N , N = n dof , that assigns to a linear operator A : V h → V h its matrix representation r m (A) ∈ R N ×N in the following way (see [2]): let r v : V h → R N denote the finite element isomorphism that assigns to a function U h ∈ V h its nodal vector representation U = r v (U h ) ∈ R N as described above. Then, the matrix representation r m (A) ∈ R N ×N is defined by the property…”

“…The time-discrete formulation for both approaches leads to a coupled 2 × 2 block system. Despite their similar structure, dG(1) and cGP (2) methods differ in the following properties: the dG(1) method is convergent of second order in time (with third order nodal superconvergence) while the cGP (2) is convergent of third order in time (with fourth order nodal superconvergence). However, dG(k) methods are known to be strongly Astable [7] while cGP(k) methods are A-stable only [9].…”

“…In order to fulfill requirements (1) to (3), for each time discretization method we employ a particular set of basis functions for which the Schur complement formulation in the essential unknown defines a symmetric, positive definite operator of fourth order. While at a first glance this approach seems unfavorable due to the inherent bad conditioning of discretized fourth order operators, we proceed by applying an efficient preconditioning technique introduced for a class of fourth order problems in [2]. This preconditioner consists of the concatenation of the (slightly modified) parabolic operators under consideration and consequently can be realized very efficiently.…”