The Hodge projection of a vector field is the divergence-free component of its Helmholtz decomposition. In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou–Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition.

In the decomposition by the Gibou–Min method, an important L2$L^2$-orthogonality holds between the gradient field and the solenoidal field, which is similar to the continuous Hodge decomposition.

Using the orthogonality, we present a novel analysis which shows that the convergence order is 1.5$1.5$ in the L2$L^2$-norm for approximating the divergence-free vector field. Numerical results are presented to validate our analyses.