James,
> 1. Using Dirichlet boundary conditions using
> VectorTools::interpolate_boundary_values. This causes Deal.II to
> abort with an exception complaining that the fixed DOFs are not on
> a boundary surface - which is obviously true, but is that
> necessary?
I'm not exactly sure what functions you are calling here, but
VectorTools::interpolate_boundary_values really only walks over boundary
faces, so if you want to constrain interior ones this isn't your function.
> 2. For each fixed degree of freedom i with value V_i, set the ith row
> of the Laplace matrix to T_{ij} = \delta_{ij} and rhs_i = V_i.
This will also lead to wrong solutions because you neglect that these degrees
of freedom also couple to their neighbors.
> 3. My third idea was to use two meshes. I.e., I would build the full
> mesh, doing all my refinement, but before solving Poisson's
> equation, cut out the fixed regions and apply Dirichlet boundary
> conditions to the surfaces of the "holes". After solving, embed
> the solution back into the full mesh.
This could work, but it would appear to me to be too complicated.
Andrew has already shown a solution that should work. Alternatively, instead
of using a std::map<uint,double> and then MatrixTools::apply_boundary_values,
you could also use a ConstraintMatrix, add non-zero constraints for your
constrained degrees of freedom to it, and use ConstraintMatrix::condense on
your matrix (or condense the matrix while copying the local contributions
into the global matrix, to make things simpler).
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.tamu.edu/~bangerth/
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