Preconditioned Legendre Spectral Galerkin Methods for the Non-separable Elliptic Equation

Abstract

The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned conjugate gradient (PCG) method where the preconditioner M is constructed by approximating the variable coefficients with a (T+1)-term Legendre series in each direction to a desired accuracy. A feature of the proposed PCG method is that the iteration step increases slightly with the size of the resulting matrix when reaching a certain approximation accuracy. The efficiency of the method lies in that the system with the preconditioner M is approximately solved by a one-step method based on the incomplete LU factorization technique with no fill-in, denoted by ILU(0). The ILU(0) factorization of $M\in {\mathbb {R}}^{(N-1)^d\times (N-1)^d}$ can be computed using ${\mathcal {O}}(T^{2d} N^d)$ operations, and the number of nonzeros in the factorization factors is of ${\mathcal {O}}(T^{d} N^d)$, $d=1,2,3$. A conclusion of the algorithm is to fast solve the resulting system from the Legendre Galerkin spectral method for Poisson equations with Dirichlet boundary conditions, which has a complexity of ${\mathcal {O}}(N^d)$. To further speed up the PCG method, an algorithm is developed for fast matrix-vector multiplications by the resulting matrix of Legendre-Galerkin spectral discretization, without the need to explicitly form it. The complexity of the fast matrix-vector multiplications is of ${\mathcal {O}}(N^d (\log _2 N)^{2})$. In view that T is independent of N in one dimension and is set to be of order ${\mathcal {O}}(\log _2 N)$ in two and three dimensions, the PCG method has a ${\mathcal {O}}(N^d (\log _2 N)^{2d})$ quasi-optimal complexity for a d dimensional domain with $(N-1)^d$ unknows, $d=1,2,3$. In addition, a fast direct solver for the three-dimensional Poisson equation is developed, which is of ${\mathcal {O}}(N^{3} (\log _2 N)^{2})$ and improves the existing results on the computational complexity. Numerical examples are given to demonstrate the efficiency of proposed preconditioners and the algorithm for fast matrix-vector multiplications.