BVPs with inhomogeneous Dirichlet boundary conditions
Let Ω ⊂ Rd be an open and bounded domain with sufficiently smooth boundary ∂Ω = Ω \ Ω, and let f
and g be sufficiently smooth functions in Ω and ∂Ω, respectively. We consider the BVP
−∇ · a∇u + cu = f
u=g
in Ω ⊂ Rd
on ∂Ω.
In order to derive the variational formulation of this BVP, we introducing the Sobolev space
Hg1 (Ω) := {v ∈ H 1 (Ω) : v = g on ∂Ω} ⊂ H 1 (Ω)
for the space of trial functions, i. e., the inhomogeneous Dirichlet boundary condition is — analogously
to the homogeneous Dirichlet boundary condition — an essential boundary condition, that we have to
incorporate into the space of trial functions. The space of test functions, however, is chosen to be H01 (Ω),
such that the boundary integral that appears after integration by parts vanishes. Thus, the variational
formulation reads: find u ∈ Hg1 (Ω) such that
Z
Z
a∇u · ∇v + cuv dx =
f v dx
∀v ∈ H01 (Ω).
Ω
Ω
Now let us write u = u0 + ug with an extension ug ∈ Hg1 (Ω) of g into Ω and the new unknown function
u0 ∈ H01 (Ω). Due to linearity, the variational formulation is equivalent to: find u0 ∈ H01 (Ω) such that
Z
Z
Z
a∇u0 · ∇v + cu0 v dx =
f v dx −
a∇ug · ∇v + cug v dx
∀v ∈ H01 (Ω).
Ω
Ω
Ω
Note that we can choose ug to be any of the functions v for which v = g on ∂Ω. This procedure to handle
inhomogeneous Dirichlet boundary conditions is called Dirichlet lift ansatz with the Dirichlet lift ug .
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