  We show that an attempt to compute numerically a viscous flow in a domain
with a piece-wise smooth boundary by straightforwardly applying well-tested
numerical algorithms (and numerical codes based on their use, such as COMSOL
Multiphysics) can lead to spurious multivaluedness and nonintegrable
singularities in the distribution of the fluid's pressure. The origin of this
difficulty is that, near a corner formed by smooth parts of the piece-wise
smooth boundary, in addition to the solution of the inhomogeneous problem,
there is also an eigensolution. For obtuse corner angles this eigensolution (a)
becomes dominant and (b) has a singular radial derivative of velocity at the
corner. A method is developed that uses the knowledge about the eigensolution
to remove multivaluedness and nonintegrability of the pressure. The method is
first explained in the simple case of a Stokes flow in a corner region and then
generalised for the full-scale unsteady Navier-Stokes flow in a domain with a
free surface.
