Communications in Mathematical Sciences

Volume 15 (2017)

Number 5

Convergence analysis on the Gibou–Min method for the Hodge projection

Pages: 1211 – 1220



Gangjoon Yoon (National Institute of Mathematical Sciences, Korea)

Jea-Hyun Park (Department of Mathematics, Kunsan National University, Kunsan, Korea)

Chohong Min (Department of Mathematics, Ewha Womans University, Seoul, Korea)


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 $L^2$-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 $1.5$ in the $L^2$-norm for approximating the divergence-free vector field. Numerical results are presented to validate our analyses.


Hodge projection, finite volume method, Poisson equation, Gibou–Min

2010 Mathematics Subject Classification


This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF), funded by the Ministry of Education (2009-0093827).

Received 30 August 2014

Published 26 June 2017