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We are going to talk about axial Green's function methods (AGMs) on free grids called axial lines. These are novel approaches in numerical computations. AGMs that we have developed for elliptic boundary value problems \cite{IJNME2008} and the steady Stokes flows \cite{JCP2011} in complicated geometry use axial lines for discretization. These axial lines are parallel to axes and there is no restriction on their distribution. The salient feature of the methods is that not only one-dimensional Green's function for the axially split differential operators is sufficient to solve the multi-dimensional problems but also the free grids are available. In this talk, short introduction to AGMs is presented and then we show that the localization \cite{JCP2014} of axial lines enables us to enforce Neumann boundary condition, and refinement of axial lines on separated regions are readily available as well.

numerical methods, computation, modeling