Fitted norm preconditioners for the Hodge Laplacian
We consider the Hodge-Laplace problem in mixed form: Find $(u, p) \in H\Lambda^{k - 1} \times H \Lambda^k$ such that
\[\begin{bmatrix} \alpha & -d^* \\ -d & - d^* d \end{bmatrix} \begin{bmatrix} u \\ p \end{bmatrix} = \begin{bmatrix} g \\ -f \end{bmatrix}\]in which $d$ is the differential and $\alpha > 0$ a constant.
We recognize this as a perturbed saddle point problem and use the framework of Hong et al. to derive the parameter-dependent norms in which the problem is well-posed. Two different strategies are adopted which both lead to the same preconditioner:
\[\mathcal{P} := \begin{bmatrix} \alpha I + (1 + \alpha) d^* d \\ & (1 + \alpha)^{-1} I + d^* d \end{bmatrix}^{-1}\]Numerical experiments in 2D and 3D confirm that the preconditioner is robust with respect to $h$ and $\alpha$.