coordinate-dependent Dirichlet data for other than P1

The specification of inhomogeneous Dirichlet data for the Laplace equation is demonstrated in ex14.py for ElementTriP1: https://github.com/kinnala/scikit-fem/blob/9c0cf416745941abd72f70865351ef0e095f83b2/docs/examples/ex14.py#L37

however, Mesh.p only contains the nodal coordinates, so this doesn’t work if the element is changed to say ElementTriP2.

How might one go about specifying inhomogeneous Dirichlet data for elements with degrees of freedom that aren’t all located at nodes of the mesh? Does this relate to #84? The P2 degrees of freedom on the boundary are available through basis.get_dofs().all().

I suppose one could fall back on other techniques like:

• penalization, i.e. approximating the Dirichlet condition with a Robin condition (Ern & Guermond 2002, §9.5.5)
• Lagrange multiplier (Stenberg 1995)

[kinnala]

Yes this is L^2 projection on the boundary. Consistency error in the sense that in this case we would know exactly the value for the degree-of-freedom but when using L^2 projection you are unlikely to get the exact correct value. This is probably just fine for any practical applications and has only some theoretical implications.

[kinnala]

Have you tried doing something like

fb = FacetBasis(m, ElementTriP2())

@linear_form
def fv(v, dv, w):
    x, y = w[0]
    return 4.0*y*(1-y)

u[I] = solve(asm(mass, fb), asm(fv, fb), I=I)

where I contains the DOF indices belonging to the respective boundary. I didn’t try it but I think it should work fine. This must be of course adapted to the vectorial case.