Preconditioning on high-order element methods using Chebyshev--Gauss--Lobatto nodes
Applied Numerical Mathematics
Journal of Computational Physics
Semi-automatic sparse preconditioners for high-order finite element methods on non-uniform meshes
Journal of Computational Physics
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In 1979 Orszag proposed a finite-difference preconditioning of the Chebyshev collocation discretization of the Poisson equation. In 1984 Haldenwang, Labrosse, Abboudi, and DeVille gave analytic formulae for the eigenvalues of this preconditioned operator in the one-dimensional case. Experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions. However, there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions. In this work we consider the generalized field of values $(U^*\hat A_NU)/(U^*L_NU)$, where $\hat A_N$ is the matrix of the Chebyshev collocation scheme and $L_N$ is the matrix of the finite-difference operator. For the case of the Chebyshev collocation of the Helmoltz operator, $Au:=-\Delta u+au$, $a\ge 0$ preconditioned by the finite-difference operator associated with the Helmholtz operator $Bu:=-\Delta u+bu$, $b\ge 0$, we prove that there are two constants $00$ and $|(U^*\hat A_NU)/(U^*L_NU)| \le\Lambda_1$. These results extend to higher dimensions and to bounds on $\| L_N^{-1}\hat A_N\|_{1,w}$ and $\|\hat A_N^{-1}L_N\|_{1,w}$ in the general case where $Au:=-\Delta u+a_1u_x+a_2u_y+au$.