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Elliptic Monge–Ampère Equation in Dimension Two 55
this direction is to approximate the least-squares problem (LSQ). To achieve
this goal, we approximate the sets Q and Q f by
and
Q h = {q | q =(q ij ) 1≤i,j≤2 , q 21 = q 12 , q ij ∈ V 0h } (45)
Q fh = {q | q ∈ Q h ,q 11 (P k )q 22 (P k ) −|q 12 (P k )| 2 = f h (P k ),
∀k =1, 2,...,N 0h }, (46)
respectively, the function f h in (46) (and in (E-MA-D) h ) being a continuous
approximation of f. Next, we approximate the least-squares functional j(·, ·)
(defined by (3) in Section 2) by j h (·, ·) defined as follows:
with
j h (ϕ, q) = 1 2 ‖D2 hϕ − q‖ 2 h, ∀ϕ ∈ V h , q ∈ Q h , (47)
D 2 hϕ =(Dhij(ϕ)) 2 1≤i,j≤2 , (48)
((S, T)) h = 1 ∑N 0h
A k S(P k ) :T(P k )
3
k=1
(
= 1 ∑N 0h
)
A k (s 11 t 11 + s 22 t 22 +2s 12 t 12 )(P k ) , ∀S, T ∈ Q h , (49)
3
k=1
and then
‖S‖ h =((S, S)) 1/2
h , ∀S ∈ Q h. (50)
From the above relations, we approximate the problem (LSQ) by the following
discrete least-squares problem:
{
{ψh , p h }∈V gh × Q fh ,
(51)
j h (ψ h , p h ) ≤ j h (ϕ, q), ∀{ϕ, q} ∈V gh × Q fh .
6.4 On the Solution of the Problem (51)
To solve the minimization problem (51), we shall use the following discrete
variant of the algorithm (9)–(11):
{ψ 0 , p 0 } = {ψ 0 , p 0 }. (52)
Then, for n ≥ 0, {ψ n , p n } being known, compute {ψ n+1 , p n+1 } via the solution
of
p n+1 =argmin
[ 1
2 (1 + τ)‖q‖2 h − ((p n + τD 2 hψ n ]
, q)) h , (53)
q∈Q fh