# sandbox/easystab/unknow-domain

This page puts together the different test cases that I wrote to progress towards the computation of steady states of the Navier-Stokes equation with a free surface. The final result is below in the code free_surface_navier_stokes.m.

Tests and validations in 1D:

- domain_derivative_1D.m In 1D with a method similar to the “flattening” used for stability of free-surface flows.
- domain_derivative_1D_adapt.m like the code above, but we adapt the computational domain at each Newton iteration
- domain_derivative_1D_mapping.m In this version, we use a mapping of the mesh instead of rebuilding the domain and interpolating the previous guess on the new domain.

Tests and validations in 2D:

- stretching_formula.m Where I test the mapping formula for the code above.
- jacobian_formula.m To validate the expression of the Jacobian of the mapping: how the diferentiation operators depend upon the stretching.
- domain_derivative_2D_adapt.m like the 1D version but in 2D.
- domain_derivative_2D_mapping.m like the 1D version but in 2D.

With a free surface and a velocity potential for the flow:

- free_surface_adapt.m To find the nonlinear shape of a free surface with a fluid flow
- free_surface_mapping.m Instead of using the
*flattening*, here the domain mapping is explicitely an unknown of the system.

With a free surface and Navier-Stokes for the flow

- free_surface_navier_stokes.m With the mapping as part of the unknowns of the system