# CONVERGENCE ANALYSIS ON GIBOU-MIN METHOD FOR THE SCALAR FIELD IN HODGE-HELMHOLTZ DECOMPOSITION

• Min, Chohong (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY) ;
• Yoon, Gangjoon (INSTITUTE OF MATHEMATICAL SCIENCES, EWHA WOMANS UNIVERSITY)
• Accepted : 2014.11.16
• Published : 2014.12.25

#### Abstract

The Hodge-Helmholtz decomposition splits a vector field into the unique sum of a divergence-free vector field (solenoidal part) and a gradient field (irrotational part). 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. Using the $L^2$-orthogonality between the error vector and the consistency, the convergence for approximating the divergence-free vector field was recently proved to be $O(h^{1.5})$ with step size h. In this work, we analyze the convergence of the irrotattional in the decomposition. To the end, we introduce a discrete version of the Poincare inequality, which leads to a proof of the O(h) convergence for the scalar variable of the gradient field in a domain with general intersection property.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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