I'm assembling a vector F that represent f(u); the action of f(u) on a
vector field v with FE expansion coefficients V can be represented as
F*V.
OK, so F_i would then be (f(u),\varphi_i), right?
I think the question is whether you do or do not apply
ConstraintMatrix::condense to F. I never really know whether you do or
do not have to do that when you want to use F that way, so I typically
just assemble the (scalar number)
(f(u),v)
directly.
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.
I did not know that! I shouldn't have said projection, really I
interpolated an analytic formula for the solution.
That, too, is not clear to me. We know that for elliptic problems, the
Galerkin approximation is the projection in the energy norm, and
consequently optimal in the energy norm (i.e., the one that minimizes
the energy norm error). But the interpolant is not optimal, either in
the L2 or energy norm, and so it's not clear to me that the interpolant
should be better (in either of these norms) on any one mesh compared to
another.
The domain is a rectangle; the grid spacing is half as large in the
right half of the mesh.
OK, so they're nested and the finite element space is strictly larger in
the adaptive mesh. Then indeed the energy norm error of the respective
solutions should be better, but we don't know whether that's true for
the L2 norm.
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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