Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations

Abstract

The uniform bounds on eigenvalues of ${\widehat{\mathsf{B}}}_{h^2}^{-1}$${\hat{\mathsf{A}}}_{N^2}$ are shown both analytically and numerically bythe $\mathcal{P}$₁ finite element preconditioner ${\hat{\mathsf{B}}}_{h^2}^{-1}$ for the Legendre spectral element system ${\hat{\mathsf{A}}}_{N^2}$$\underset{¯}{\mathbf{u}}=\underset{¯}{\mathbf{f}}$ which is arisen from a coupled elliptic system occurred by an optimal control problem. The finite element preconditioner is corresponding to a leading part of the coupled elliptic system.