Preconditioning for a Class of Spectral Differentiation Matrices

Abstract

We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton's method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator.

Equivalent operator preconditioning for elliptic problems

“…A general theory for such preconditioning has been developed in [55], see also [58,93], where the notion of equivalent operator has been introduced and used to give a rigorous Hilbert space background for the convergence properties. A similar preconditioning approach is used in [90,109] where preconditioning by S is replaced by a related solution operator, for first order problems in [37] and for second order nonlinear problems in e.g. [99] and the authors' works [9,14,56,79].…”