Partial Differential Equations - Modelling and ... - ResearchGate

Algorithm 1
Step 1. u 0 is given in V g .
Step 2. Solve then
⎧
g 0 ∈ V 0 ,
⎪⎨ ∫
∫
△g 0 △vdx=
Ω
⎪⎩
∀v ∈ V 0 ,
Elliptic Monge–Ampère Equation in Dimension Two 51
Ω
∫
△u 0 △vdx+ τ
Ω
D 2 u 0 :D 2 vdx− L(v),
(23)
and set w 0 = g 0 .
Step 3. Then, for k ≥ 0, u k ,g k , w k being known, the last two different from
0, we compute u k+1 , g k+1 , and if necessary w k+1 , as follows:
Solve
⎧
ḡ k ∈ V 0 ,
⎪⎨ ∫
∫
△ḡ k △vdx=
⎪⎩
Ω
∀v ∈ V 0 ,
Ω
∫
△w k △vdx+ τ
Ω
D 2 w k :D 2 vdx,
(24)
and compute
ρ k =
∫
Ω |△gk | 2 dx
∫Ω △ḡk △w k dx , (25)
u k+1 = u k − ρ k w k , (26)
g k+1 = g k − ρ k ḡ k . (27)
Step 4. If ∫ Ω |△gk+1 | 2 dx/ ∫ Ω |△g0 | 2 dx ≤ tol take u = u k+1 ; else, compute
∫
Ω
γ k =
|△gk+1 | 2 dx
∫
Ω |△gk | 2 dx
(28)
and
w k+1 = g k+1 + γ k w k . (29)
Step 5. Do k = k +1andreturntoStep3.
Numerical experiments have shown that Algorithm 1 (in fact, its discrete
variants) has excellent convergence properties when applied to the solution of
(E-MA-D). Combined with an appropriate mixed finite element approximation
of (E-MA-D) it requires the solution of two discrete Poisson problems at
each iteration.