Super-convergence analysis on two symmetric Poisson solvers in octree grids

The Hodge decomposition, that is an important feature of incompressible fluid flows, is orthogonal and the projection taking its incompressible component is therefore stable. The decomposition is implemented by solving the Poisson equation. In order to simulate incompressible fluid flows in a stable manner, it is desired to utilize a Poisson solver that attains the orthogonality of the Hodge decomposition in a discrete level. When a Poisson solver induces the orthogonality, its associated linear system is necessarily symmetric. With this regard, the symmetric Poisson solvers [9,8] by Losasso et al. are more advantageous not only to efficiently solving the linear system but also to stably simulating fluid flows than nonsymmetric ones. Their numerical solutions were empirically observed to be first and second order accurate, respectively. One may expect that each of their numerical gradients has convergence order that is one less than that of its numerical solution. However, we in this work show that super-convergence holds true with both Poisson solvers. Rigorous analysis is presented to prove that the difference is one half, not one between the convergence orders of numerical solution and gradient in both solvers. The analysis is then validated with numerical results. We furthermore show that both Poisson solvers, being symmetric, indeed satisfy the orthogonal property in the discrete level and yield stable implementations of the Hodge decomposition in octree grids.