# Arbitrary Choosing of the Solution Domain - Navier Stokes and Manufactured Solutions

I want to verify a finite-volume solver (SIMPLE-Algorithm) for the incompressible Navier-Stokes equations by using a manufactured solution. I use Dirichlet boundary conditions for the velocity at all boundaries. The manufactured solution for the velocities I use, is constructed as the curl of a vector field and thereby fulfills the continuity equation at any point.

Depending on how I choose the domain on which to solve the equations I run into convergence problems: The residual of the pressure correction equation stagnates after an initial decrease.

I am approximating the mass fluxes through boundary faces using the midpoint rule and apparently this leads to a net loss/gain of mass considering the entire problem domain.

Is this the normal behavior? Are there any remedies other than choosing a manufactured solution that identically vanishes at the boundaries? Would it be an option to provide the exact mass flux instead of the approximate (not sure if this is allowed, since the Dirichlet boundary condition only fixes the velocity)?

• The behavior you observe is normal if you don't approximate the boundary conditions right. For some domains the total defect might be alright or even vanishing due to symmetry, while for others it might be severe and prevent the solver from converging further. You could try a higher order integration at the boundaries to get rid of such effects. – Christian Waluga Jan 11 '15 at 17:58
• Midpoint rule should not produce any mass loss. If you write down the total fluxes in a cell and add them for all the cells , you end up with only the boundary terms, which shows midpoint rule conserves mass. – decltype_usr Jan 26 '17 at 9:21