An AMG saddle point preconditioner with application to mixed Poisson problems on adaptive quad/cube meshes

TL;DR

This work addresses robust preconditioning for saddle-point systems arising from $RT_0$ mixed Poisson discretizations on adaptively refined quad/cube meshes. By introducing SPAMG, a monolithic algebraic multigrid method with a stabilized, flux–pressure–coupled prolongation, the authors achieve near mesh-independent GMRES convergence even under strong coefficient variation and anisotropy. The approach outperforms standard diagonal and Schur-complement preconditioners, particularly on adaptive meshes in both 2D and 3D, demonstrating its practical relevance for multiscale and subsurface-type problems. The results suggest SPAMG as a viable, scalable preconditioner for mixed finite-element systems on nonuniform meshes, with potential extensions to higher-order RT discretizations and larger conductivity contrasts.

Abstract

We investigate various block preconditioners for a low-order Raviart-Thomas discretization of the mixed Poisson problem on adaptive quadrilateral meshes. In addition to standard diagonal and Schur complement preconditioners, we present a dedicated AMG solver for saddle point problems (SPAMG). A key element is a stabilized prolongation operator that couples the flux and scalar components. Our numerical experiments in 2D and 3D show that the SPAMG preconditioner displays nearly mesh-independent iteration counts for adaptive meshes and heterogeneous coefficients.

Figures (12)

• Figure 1: Dregrees of freedom for the rectangular $\mathcal{RT}_0$ element in two (a) and three (b) dimensions. For the velocity, the degrees of freedom are normal components at the edge (face) mid sides of an element. The pressure is located at the center of the element.
• Figure 2: Locally refined mesh with hanging nodes. To enforce continuity of the flux normals for a $\mathcal{RT}_0$ discretization, the velocity value on a hanging node is defined by the corresponding non-hanging node lying in the same edge. For the case in the picture we define $\tilde{u} _0 := - u_1$.
• Figure 3: Error plot for the numerical solution of a mixed Poisson system corresponding to the example specified in Section \ref{['sec:example1']} (homogeneous Dirichlet boundary conditions) in two (a) and three dimensions (b) for uniform meshes and identity conductivity tensor. The level $\ell$ is related to the mesh size $h$ via $h = 2^{-\ell}$. We confirm the expected convergence rates predicted by \ref{['eqn:conv_rate']}.
• Figure 4: Error plot in two (a) and three dimensions (b) for adaptive meshes and identity conductivity tensor. We choose to refine an element of side length $h$ by two additional levels whenever its centroid lies within the circle/sphere of radius $2^{d}h^{2}$ around the point $(\frac{1}{2},\frac{1}{2})$ for $d=2$ and $(\frac{1}{2},\frac{1}{2}, \frac{1}{2})$ for $d=3$. Because of the smoothness of the solution and the equations coefficients we do not expect that local refinement translates into an improvement of the approximation with respect to a uniform case mesh.
• Figure 5: Error plot (Section \ref{['sec:example2']}, inhomogeneous Dirichlet/Neumann boundary conditions) in two (a) and three (b) dimensions for uniform meshes.