Preconditioning Legendre Spectral Collocation Approximations to Elliptic Problems

This work deals with the H¹ condition numbers and the distribution of the ₍, ₌ -singular values of the preconditioned operators \ ₍, ₌^{ - 1 W₍, ₌ A₍, ₌ \}. A₍, ₌ is the matrix representation of the Legendre spectral collocation discretization of the elliptic operator A defined by Au: = - u + a₁ uₓ + a₂ uᵧ + a₀ u in (the unit square) with boundary conditions u = 0 on ₀, u{ A } = u on ₁. ₍, ₌ is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator B defined by Bv: = - v + b₀ v in with boundary conditions v = 0 on ₀, v{ B } = v on ₁. The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the Legendre–Gauss–Lobatto (LGL) points or the space of continuous functions which are linear on a triangulation of determined by the LGL points. W₍, ₌ is the matrix of quadrature weights. When A = B we obtain results on the eigenvalues of ₍, ₌^ - 1 W₍, ₌ B₍, ₌. In the general case we show that there is an integer N₀ and constants, with 0 < <, such that if (N, M) N₀, then all the ₍, ₌ -singular values of ₍, ₌^ - 1 W₍, ₌ A₍, ₌ lie in the interval,. Moreover, there is a smaller interval, ₀, ₀, independent of the operator A, such that if (N, M) N₀, then all but a fixed finite number of the ₍, ₌ -singular value lie in ₀, ₀. These results are related to results of Manteuffel and Parter SIAM J. Numer. Anal. , 27 (1990), pp. 656–694, Parter and Wong J. Sci. Comput. , 6 (1991), pp. 129–157 and Wong Numer. Math. , 62 (1992), pp. 391–411, 413–437 for finite element discretizations.