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