Dear all,
I want to solve a coupled PDE for two variables u and v. The
uncoupling of PDE is not linear. I want to use two meshes mesh_u and
mesh_v for these two variables u and v, respectively. Generally we
want to calculate the integration of
U(u) phi_i
in the whole domain, where u is finite element function on mesh_u and
the basis function phi_i is defined on mesh_v. If the function U is
linear, as explained in step-28, it can be calcuated by finding the
common finest mesh {K} between mesh_u and mesh_v and then expressing u
to be the combination of basis function defined on mesh_u. Now The
problem seems that we can not calculate this integration in K, where K
is active in mesh_u but not active in mesh_v. This is because we can
not obtain the value of u on the integration points in the children of
K. The only way seems that we had to approximate U(u) by the finite
element function u_h defined on the mesh_u.
I looked through the function assemble_cross_group_rhs_recursive. The
variable prolongation_matrix should present
B_cB_c^\prime… B_c^{(k)}
as given this notation in the introduction of step-28.
In case of
cell_g->level() > cell_g_prime->level()),
this means that K is active in mesh g^\prime, but not active in mesh
g, which corresponding to the case 3. We should multiply the transpose
of the prolongation matrix (prolongation_matrix). But in the code, the
prolongation matrix, not its transpose is multiplied!
Some writting errors seems occurs in the introduction of step-28. I
try to modify these errors.
In case 2, the notation
$B_cM_{K_c}^{il}$
should changed to be
$B_cM_{K_c}^{ij}$.
In case 3, the notation
$\int_{K_c} f(x)\varphi_{g^\prime}^i(x)B_c^{jl}\varphi_{g^\prime}^l(x)dx$
should changed to be
$\int_{K_c} f(x)\varphi_{g }^i(x)B_c^{jl}\varphi_{g }^l(x)dx$.
Best regards
Daming
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