On 6/18/19 2:35 PM, Alexander wrote: > Wolfgang, > thanks, I'm inclined towards using FE_Nothing. Since this becomes quite > technical, I'm thinking of a following test. Can you please confirm if this > is > a sensible way of validation. > > (i) Construct a single cell quad mesh. Possibly refine n times. > (ii) At the top (arbitrary choice) boundary set no-flux BCs. Set (smooth) > Dirichlet BCs on all other sides. > (iii) Solve laplace problem. > > (iv) Construct a two cells quad mesh such that the top boundary of the mesh > from (i) became an internal face shared by the two cells. > (v) Set cell #1 to FE_Q and cell #2 to FE_Nothing. Refine n times. > (vi) Impose no-flux face integral at the shared faces. Set same Dirichlet BCs > as in (ii) on all other sides of cell #1. > (vii) Solve laplace problem. > > (viii) Compare solution in cell #1 to the one in step (iii). They should be > equal.
Yes, this seems eminently sensible. > I'll add another thought: If you have non-zero Neumann boundary > conditions, > and you approximate the original domain by one that approximates the > boundary > by a stair-step function as you seem to do, then the numerical solution > does > not converge to the exact one. That's because the stair-step boundary has > a > different length than the exact boundary (and doesn't converge to the > exact > length), and the total flux into the domain equals the average flux on the > boundary times the length of the boundary. > > > An interesting point, although I guess in a row of all approximations I have > already done in this problem, this one won't be the first to hit me back. It should only affect you if you had non-zero boundary fluxes. But it's an O(1) error, and it's not small because it is proportional to the ratio of the lengths of the exact boundary to the length of the approximated stair-step boundary. If your exact domain is a triangle, then that ratio is sqrt(2)=1.4! Best W. -- ------------------------------------------------------------------------ Wolfgang Bangerth email: [email protected] www: http://www.math.colostate.edu/~bangerth/ -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/65f9a7cc-011b-ac94-aef2-c8539f90d5ad%40colostate.edu. For more options, visit https://groups.google.com/d/optout.
