I am currently trying to solve the Biharmonic equation with non-homogeneous Dirichlet boundary conditions:
$$ \Delta^2 u = f \text{ in } \Omega, \\ u = g \text{ on } \Gamma, \\ \partial_\nu u = 0 \text{ on } \Gamma $$ where $ \Gamma $ is a polygonal boundary. I am using a C^1 Hermite element that is specified by values at vertices of triangles, gradients at vertices of triangles, and the outward facing normal derivatives at each edge midpoint.
I decompose my function $ u = u_0 + u_g $ where $u_0$ vanishes on the boundary, and $ u_g $ coincides with $g$ on the boundary. The weak formulation is given by the linear and bilinear operators
$$ a(u, v) = \int_\Omega \Delta u \Delta v dx \\ L(v) = \int_\Omega f v dx $$ and using the decomposition of $u$, I can solve the homogeneous system
$$ a(u_0, v) = L(v) - a(u_g, v) $$
My two questions are:
How do I find $u_g$? Can I interpolate it into my function space? If so, do I need to interpolate just values, or gradients as well? Or normal derivatives? Or, can I solve the weak form: $$ \int_\Gamma uv dx = \int_\Gamma gv dx $$ in order to find $u_g$.
How do I implement the modified system, given that I already have procedures for solving the homogeneous case? I haven't been able to find a single reference where this is done, If someone has any tips that would be great.
Thanks in advance for all help!