Preconditioning on high-order element methods using Chebyshev–Gauss–Lobatto nodes

Based on Chebyshev–Gauss–Lobatto points, the piecewise linear finite element preconditioner is analyzed in terms of condition numbers for the high-order element discretizations applied to a model elliptic operator. The optimality of such a preconditioner is proved for one-dimensional case and the scalability is shown for two-dimensional case. Further, we provide O(N1/3) growth of piecewise linear finite element preconditioner numerically.