Chapter 7 Mixed finite element methods 7 . 1 The membrane problem revisited

We recall from Chapter 1 that the computation of the equilibrium state of a clamped membrane amounts to the solution of the convex minimization problem (7.1) J(u) = inf v∈H 1 0 (Ω) J(v) , 1 0 (Ω) → lR stands for the convex functional (7.2) J(v) := 1 2 Ω |a grad v| 2 dx − Ω f v dx , v ∈ H 1 0 (Ω). As we observed in Chapter 1, (7.1) is a particular example of a more general convex optimization problem of the form: Given a Hilbert space V and a convex functional J : V → lR, find u ∈ V such that (7.3) J(u) = inf v∈V J(v). The optimization problem (7.3) can be given a dual formulation by means of the Fenchel conjugate of the functional J. Definition 7.1 Fenchel conjugate Let V be a Hilbert space with the dual space V * and assume that J : V → lR is a convex functional. Then, the Fenchel conjugate J * : V * → lR of J is given by (7.4) J * (v *) := sup v∈V < v * , v > − J(v) , where < ·, · > denotes the dual pairing between V and V *. Remark 7.1 The Fenchel conjugate in case V = lR If V = lR, then J * (v *) is the intercept with the v-axis of the tangent to J of slope v *. Theorem 7.1 Characterization of the Fenchel conjugate Let V be a Hilbert space with the dual space V * and assume that J : V → lR is a convex functional with J