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Numerical Analysis of a Finite Element/Volume Penalty Method 181
Proof. The first part is a consequence of Proposition 4 (we established in
the proof of Proposition 9 that the range of B is closed), which ensures the
existence of λ ∈ Λ its uniqueness in B(V ).
Let us now describe λ. Wehave
∫
∫ ∫
∇u ·∇v + λ ·∇v = fv,
Ω
O
so that, by taking tests functions in D(O), we get λ ∈ H div (O) with ∇·λ =0.
Taking now test functions which do not vanish on the boundary of O, we
identify the normal trace of λ as ∂u/∂n ∈ H 1/2 (∂O). Therefore, λ is defined
as the unique divergence free vector field in O, with normal derivative equal
to ∂u/∂n on ∂O, which, in addition, is a gradient. In other words: λ = ∇Φ,
with
⎧
⎨ △Φ =0 inO,
⎩
∂Φ
∂n = ∂u on ∂O.
∂n
As O is smooth, Φ ∈ H 2 (O), so that λ = ∇Φ ∈ H 1 (O) 2 .
Now we consider again the family of Cartesian triangulations (T h )ofthe
square Ω (see Fig. 1), and we denote by V h the standard Finite Element space
of continuous, piecewise affine functions with respect to T h . As indicated in
the beginning of this section, we suppress the small areas in the computation
of the penalty term by introducing a reduced obstacle O h :
Definition 2. The reduced obstacle O h ⊂Ois defined as the union of the
sets T ∩O, where T runs over triangles of T h such that their intersection with
O compares reasonably with their own size, in the following sense: given η>0
a fixed parameter, we set
⋃
O h = (T ∩O) . (17)
|T ∩O|≥η|T |
Definition 3. We recall that V = H 1 0 (Ω), Λ is L 2 (O) 2 ,andB ∈L(V,Λ) is
the gradient operator (see Proposition 12). We define B h ∈L(V,Λ) as
v ∈ V ↦−→ µ = B h v = χ Oh ∇v,
where χ Oh is the characteristic function of O h (see Definition 2). Finally, the
discretization space Λ h ⊂ Λ = L 2 (O) 2 is the set of all those vector fields µ h
such that their restriction to O h is the gradient of a scalar field v h ∈ V h ,and
which vanish almost everywhere in O\O h , which we can express
Λ h = {µ h ∈ Λ |∃v h ∈ V h , µ h = B h v h } = B h (V h ).
The fully discretized problem reads
Ω