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.

Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

“…By Lemma 6 and the relation betweenB −1 A and X, we obtain the following result. The following result can be proved by using the same proof for the Corollary 5.11 in [10] with…”

“…Hence, efficient preconditioners are necessary in order to speed up the convergence of GMRes method. In [10], Bänsch, Morin, and Nochetto proposed symmetric and non-symmetric preconditioners that work well for the Schur complement system (2.8). However, the convergence rates of these methods are not uniform with respect to h or τ .…”

“…However, the convergence rates of these methods are not uniform with respect to h or τ . Besides, the performance of those methods deteriorate for degenerate problems [10].…”

“…In [8,9], Bänsch, Morin and Nochetto solved a discrete fourth order parabolic equation by applying the conjugate gradient method on the two-bytwo block system in Schur complement form. In [10], the same authors proposed symmetric and nonsymmetric preconditioners based on operator splitting. Axelsson, Boyanova, Kronbichler, Neytcheva, Do-Quang, and Wu [2,3,18,19] proposed a block preconditioner by adding an extra small term to the (2, 2) block and then followed by a block LU factorization which results in a preconditioned system when eigenvalues have a uniform bound.…”

“…Bound on the -pseudospectrum[10]) If R is a square matrix of order n and 0 < ≤ 1, then,σ (R) ⊂ ∪ λ∈σ(R) B(λ, C R 1 m ), where C R := n(1 + √ n − p)κ(V ), with κ(V ) thecondition number of V and V is a nonsingular matrix transforming R into its Jordan canonical form J, i.e. V −1 RV = J, p is the number of Jordan blocks, and m is the size of the largest Jordan block of R.…”