In the Cartesian case, this value is used only for the pure Neumann boundary conditions (

BCtype

="NNNN" in the 2D case;

BCtype

="NNNNNN" in the 3D case). This is a special case, because the right-hand side of the problem cannot be arbitrary if the coefficient

q

is zero. The Poisson Solver verifies that the classical solution exists (up to rounding errors) using this tolerance. In any case, the Poisson Solver computes the normal solution, that is, the solution that has the minimal Euclidean norm of residual. Nevertheless, the

?_Helmholtz_2D/?_Helmholtz_3D

routine informs you that the solution may not exist in a classical sense (up to rounding errors).

In the spherical case, the value is used for the special case of a periodic problem on the entire sphere. This special case is similar to the Cartesian case with pure Neumann boundary conditions. Here the Poisson Solver computes the normal solution as well. The parameter is also used to detect whether the problem is periodic up to rounding errors.