Hi Dan,
* 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.
...which you compute via quadrature? Or do you compute a vector F that
corresponds to f(u) somehow?
If you go the route via vectors you have to pay attention to *what kind
of vector you have*, namely one that does or does not incorporate
constraints. Dealing with dual space vectors (e.g., rhs vectors) is
tricky because they obey different rules than primal space vectors
(e.g., solution vectors).
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.
So u_a, v_a are the same as u, v in the pointwise sense, just
interpolated from a uniform mesh to an adaptively refined one obtained
from the uniform one?
> 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.
Are you using the L2 projection? For example, for the Laplace equation,
I think you can only guarantee that the H1 seminorm error is smaller for
a finer mesh, but not necessarily the L2 error. It all depends on your
equation.
And how do the two meshes differ?
Best
W.
--
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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