# Setting inhomogeneous Dirichlet boundary conditions¶

A mesh stores boundary elements, which know the bc name given in the geometry. The Dirichlet boundaries are given as a regular expression of these names to the finite element space:

```V = FESpace(mesh,order=3,dirichlet="top|back")
u = GridFunction(V)
```

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, definedon=mesh.Boundaries("top|back"))
```

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
```