# GfsMetricLaplace

### From Gerris

GfsMetricLaplace computes a numerical orthogonal mapping of the computational space `(rx,ry)`

into the physical space

by solving
* x*=(x,y,z)

*∇*^{2}= -2**x***H**n*

where *∇ ^{2}* is the Laplace-Beltrami operator,

**is the normal to the surface and**

*n**H*is the mean curvature.

The syntax in parameter files is

MetricLaplace NAME { spherical = 0 conformal = 0 }

The parameter block is optional. If `spherical`

is set to one, the mapping is from the plane in computational coordinates to the sphere in physical coordinates. If `conformal`

is set to one, a conformal Laplace-Beltrami operator is used (the existence of such a mapping depends on the boundary conditions).

In addition to the metric scaling factors, additional variables will be defined for each of the physical coordinates i.e. `(NAMEx,NAMEy,NAMEz)`

. Grid generation is controlled by imposing Dirichlet or Neumann boundary conditions on these variables.

Note that this does not work with adaptivity yet.

### References

*Eca, L.* - **2D orthogonal grid generation with boundary point distribution control**

- Journal of Computational Physics 125(2):440-453, 1996
- Bibtex