Preconditioned Legendre spectral Galerkin methods for the non-separable elliptic equation

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 iterative method based on the ILU(0) factorization. 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 $\mathcal{O}(T^{d} N^d)$, $d=1,2,3$. 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 $\mathcal{O}(N^d (\log N)^2)$. As a result, the PCG method has a $\mathcal{O}(N^d (\log N)^2)$ total complexity for a $d$ dimensional domain with $(N-1)^d$ unknows, $d=1,2,3$. Numerical examples are given to demonstrate the efficiency of proposed preconditioners and the algorithm for fast matrix-vector multiplications.