Hi Wolfgang, thanks for looking! > * In the formula above, P(.) is a functional, I assume, i.e., it takes a > function and returns a number, right? > * If so, what exactly does > f(u) . v > actually mean? How do you compute this? > * Same for the second derivatives? >
Sorry if that was unclear -- if only we could write LaTeX in a newsgroup :) Yes, P is a functional, so P : H^1 to R; f : H^1 to (H^1)^*; and df : H^1 to the space of linear operators from (H^1) to (H^1)^*. Maybe a better way to write it would be P(u + h * v) = P(u) + h * <f(u), v> + h^2 * <df(u)v, v> / 2 + O(h^3) where < * , * > is the duality pairing. I assume you mean by "error" the size of the second and third term? > Yes exactly, I checked that the errors in the linear approximation looked quadratic and that the errors in the quadratic approximation looked cubic as a function of h. > > In addition, the > > value of the action is roughly the same for each h for both the > adaptively > > refined and uniform meshes. > > I don't think I understand this statement. > By that, I meant that, if u and v are the velocity fields on the uniform mesh and u_a, v_a are the velocity fields on the adaptively-refined mesh, then P(u + h * v) = P(u_a + h * v_a) to within some negligible error. > The only other clue I have is that the error of > > the numerical solution against the analytic solution is actually > somewhat > > worse on the adaptively-refined mesh than for the uniform mesh, but > before I > > assumed this just had something to do with the linear solver. > > How do you measure "worse"? As the error as a function of the number of > unknowns? > If u_true is the projection of the exact solution onto the uniform mesh and u_a_true is the projection of the exact solution onto the adaptively-refined mesh, then |u - u_true| is less than |u_a - u_a_true|, where |*| is the L^2 norm of the functions (not the l^2 norm of the vector of coefficients). This is one of the weirder things to me; I would think that the error would be lower for the adaptive case since overall the cells are finer. But that could just be a quirk of my hand-rolled Newton solver. Thanks for taking the time to check this out, I really appreciate it! Let me know if any more information would be helpful in diagnosing this, Daniel -- 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 dealii+unsubscr...@googlegroups.com. For more options, visit https://groups.google.com/d/optout.
