Partial Differential Equations - Modelling and ... - ResearchGate

56 E.J. Dean and R. Glowinski
and ⎧
⎪⎨
⎪ ⎩
ψ n+1 ∈ V gh ,
where we have
(△ h [(ψ n+1 − ψ n )/τ], △ h ϕ) h +((D 2 h ψn+1 , D 2 h ϕ)) h
=((p n+1 , D 2 h ϕ)) h, ∀ϕ ∈ V 0h ,
(54)
(1) △ h ϕ = Dh11(ϕ)+D 2 h22(ϕ), 2 ∀ϕ ∈ V h , (55)
(2) (ϕ 1 ,ϕ 2 ) h = 1 ∑N 0h
A k ϕ 1 (P k )ϕ 2 (P k ), ∀ϕ 1 ,ϕ 2 ∈ V 0h , (56)
3
k=1
the associated norm being still denoted by ‖·‖ h .
The constrained minimization sub-problems (53) decompose into N 0h
three-dimensional minimization problems (one per internal vertex of T h )
similar to those encountered in Section 4, concerning the solution of the problem
(10). The various solution methods (briefly) discussed in Section 4 still
apply here. For the solution of the linear sub-problems (54), we advocate
the following discrete variant of the conjugate gradient algorithm (23)–(29)
(Algorithm 1):
Algorithm 2
Step 1. u 0 is given in V gh .
Step 2. Solve
⎧
⎪⎨ gh 0 ∈ V 0h,
(△ h g
⎪⎩
0 , △ h ϕ) h =(△ h u 0 , △ h ϕ) h + τ((D 2 h u0 , D 2 h ϕ)) h − L h (ϕ),
∀ϕ ∈ V 0h ,
(57)
and set
w 0 = g 0 . (58)
Step 3. Then, for k ≥ 0, assuming that u k ,g k and w k are known with the last
two different from 0, solve
⎧
⎪⎨
ḡ k ∈ V 0h ,
(△ h ḡ
⎪⎩
k , △ h ϕ) h =(△ h w k , △ h ϕ) h + τ((D 2 hw k , D 2 hϕ)) h , (59)
∀ϕ ∈ V 0h ,
and compute
ρ k =(△ h g k , △ h g k ) h /(△ h ḡ k , △ h w k ) h , (60)
u k+1 = u k − ρ k w k , (61)
g k+1 = g k − ρ k ḡ k . (62)