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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 is
constructed by approximating the variable coefficients with a (+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 is
approximately solved by a one-step iterative method based on the ILU(0)
factorization. The ILU(0) factorization of can be computed using operations, and the number of nonzeros in the factorization factors is of
, . 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 . As a result, the PCG method has a
total complexity for a dimensional domain
with unknows, . Numerical examples are given to demonstrate
the efficiency of proposed preconditioners and the algorithm for fast
matrix-vector multiplications