Setting inhomogeneous Dirichlet boundary conditions

A mesh stores boundary elements, which know the bc index given in the geometry. The Dirichlet boundaries are given as a list of boundary condition indices to the finite element space:

V = FESpace(mesh,order=3,dirichlet=[2,5])
u = GridFunction(V)

If bc-labels are used instead of numbers, the list of Dirichlet bc numbers can be generated as follows. Note that bc-nums are 1-based:

bcnums = [ i+1 for i,bcname in enumerate(mesh.GetBoundaries()) if bcname in ["dir1", "dir2"] ]

The BitArray of free (i.e. unconstrained) dof numbers can be obtained via

freedofs = V.FreeDofs()
print (freedofs)

Inhomogeneous Dirichlet values are set via

u.Set(x*y, BND)

This function performs an element-wise L2 projection combined with arithmetic averaging of coupling dofs.

As usual we define biform a and liform f. Here, the full Neumann matrix and non-modified right hand sides are stored.

Boundary constraints are treated by the preconditioner. For example, a Jacobi preconditioner created via

c = Preconditioner(a, "local")

inverts only the unconstrained diagonal elements, and fills the remaining values with zeros.

The boundary value solver keeps the Dirichlet-values unchanged, and solves only for the free values

BVP(bf=a,lf=f,gf=u, pre=c).Do()

A do-it-yoursolve version of homogenization is:

res = f.vec.CreateVector()
res.data = f.vec - a.mat * u.vec
u.vec.data += a.mat.Inverse(v.FreeDofs()) * res