Transcript On the Numerical Solution of some Fully Nonlinear Elliptic

MADRID LECTURE # 6
On the Numerical Solution of the Elliptic
Monge-Ampère Equation in 2-D
A Least Squares Approach
1. Introduction
Our goal here is to discuss the least-squares solution in
H2(Ω) of some fully nonlinear elliptic equations of the
Monge-Ampère type in 2-D.
Why H2(Ω) and why least-squares?
● Because from a computational point of view there is
always advantage at solving a given problem in a Hilbert
space and, here H2(Ω) is a natural choice.
● Least-squares methods are well suited to Hilbert
spaces and provide apparently an alternative to viscosity
solution based methods.
Introduction (2)
We will focus only on the solution of the Dirichlet problem
for the canonical Monge-Ampère equation
(E-MA-D) det D2ψ = f in Ω, ψ = g on Г,
with Ω  R2 and f > 0, but “our” methodology applies also
(among other problems) to the Pucci-Dirichlet problem
(PUC-D)  λ+ + λ– = 0 in Ω, ψ = g on Г,
Introduction (3)
with λ+ (resp., λ–) the largest (resp., the smallest )
eigenvalue of the matrix-valued function (Hessian)
D2ψ and   (1, +∞) (if  = 1, one recovers the
linear Poisson-Dirichlet problem).
The Gaussian curvature equation
det D2ψ = f (1 + |ψ|2)2 in Ω
is also in our agenda.
Introduction (4)
The Mathematics of Monge-Ampère type equations has generated a
large literature (Th. Aubin, L.A.Caffarelli, …). On the other hand (cf. Google
Scholar) one can not say the same of their Numerics, with some notable
exceptions such as Olicker- Prussner and Benamou-Brenier, and more
recently A. Oberman; indeed (from B-B):
“ It follows from this theoretical result that a natural computational solution of the L2 MKP is the
numerical resolution of the Monge-Ampère equation (6). Unfortunately, this fully nonlinear
second-order elliptic equation has not received much attention from numerical analysts and, to
the best of our knowledge, there is no efficient finite-difference or finite-element methods,
comparable to those developed for linear second-order elliptic equations (such as fast Poisson
solvers, multigrid methods, preconditioned conjugate gradient methods,….).”
Our goal is to show that several of the tools mentioned in the above statement
concerning the solution of linear second order elliptic problems still apply for
these fully nonlinear elliptic equations.
2. A least-squares method for the elliptic
Monge-Ampère equation in dimension 2
The Dirichlet problem for the prototypical Monge-Ampère
equation reads as follows:
detD2ψ = f in , ψ = g on .
(MA-D)
If f is positive the above equation is elliptic (E-MA-D).
This equation is somewhat tricky. Take  = (0, 1)2 and
consider the particular case of (E-MA-D) defined by
2ψ/x12 2ψ/x22 – |2ψ/x1x2|2 = 1 in , ψ = 0 on .
(2.1)
Clearly, (2.1) can not have smooth solutions, despite the smoothness
of its data. Trouble lies with the non-strict convexity of .
Section 2 (2)
From now on, we suppose that f > 0 and that
{f, g}  {L1(), H3/2()}, implying that the following space
and set are non empty:
Vg = { |   H2(),  = g on },
Qf = {q| q  Q, det q = f },
with Q = {q| q  (L2())22, q = qt }.
Solving the Monge-Ampère equation in H2() is equivalent
to looking at the intersection in Q of D2Vg and Qf .
Section 2 (3)
(E-MA-D) has a solution in H2()
Section 2 (3)
(E-MA-D) has no solution in H2()
Section 2 (4)
In order to handle those situations where (E-MA-D)
has no solution in H2() despite the fact that
neither Vg nor Qf are empty we suggest to solve
the above problem via the following least squares
formulation
Min{ ,q}{ ½ ∫|D2 – q|2dx}, {, q}  Vg×Qf (LSQ)
with |q| = (q112 + q222 +2q122)½.
Section 2 (5)
In order to solve (LSQ) by operator-splitting techniques
we observe that (LSQ) is equivalent to
Min{ ,q}{ ½ ∫|D2 – q|2dx + If (q)}, {, q}  Vg × Q (LSQ-P)
0 if q  Qf ,
with If(q) =
of Qf .
i.e., If is the indicator functional
+ if q  Q\Qf ,
Section 2(6)
We can now solve (LSQ-P) by a block relaxation method
operating alternatively between Vg and Qf . A closely
related algorithm is obtained as follows:
(i) Derive the Euler-Lagrange equation of (LSQ-P),
namely
{ψ, p} Vg  Q,
∫ΩD2ψ:D2φ dx = ∫Ωp:D2φ dx,  φ  V0,
∫Ωp:q dx + <∂If(p), q> = ∫Ω D2ψ:q dx,  q  Q,
Section 2(7)
with
V0 = H2()H01()
and ∂If(p) a generalized differential of If at p.
(ii) Associate to this E-L equation an initial value problem
(flow) in Vg×Q.
(iii) Use operator-splitting to time discretize the above
flow problem.
Section 2(8)
The above program leads to the following
algorithm:
Section 2 (9)
(1) {ψ0, p0} = {ψ0, p0};
for n  0, {ψn, pn} being known, solve for {ψn+1, pn+1}
(2) (pn+1 – pn)/ + pn+1 + If(pn+1)  D2ψn,
ψn+1  Vg ,
(3) ∫ Δ[(ψn+1– ψn)/] Δ dx + ∫ D2ψn+1: D2 dx =
∫ pn+1: D2 dx,    V0,
Section 2 (10)
with r:s = r11s11 + r22s22+ 2r12s12 if r = rt, s = st.
Problem (2) can be solved point-wise, while problem (3)
can be solved by a conjugate gradient algorithm
operating in Vg and V0 equipped with the scalar product
{v, w} → ∫ Δv Δw dx.
Each iteration of the above c.g. algorithm requires
the solution of 2 Poisson-Dirichlet problems.
3. Finite Element Approximation of
(E-MA-D). Numerical Experiments
3.1. Finite Element Approximation
Suppose that T h is a finite element triangulation of Ω; we
approximate H2(Ω),Vg, V0(=H2(Ω)H10(Ω)), Q and Qf by:
(3.1) Vh = {φ|φ  C0(Γ), φ|T  P1,  T  T h},
(3.2) Vgh= {φ|φ  Vh, φ(P) = g(P), P  Г and vertex of T h},
(3.3) V0h= {φ|φ  Vh, φ = 0 on Г },
(3.4) Qh = {q|q  (V0h)4, q = qt},
(3.5) Qfh = {q|q  Qh, (q11 q22 – q212)(P) = fh(P), P vertex of
T h,P  Г },
with fh a continuous approximation of f.
Section 3 (2)
Next, we approximate ∂2φ/∂xi∂xj by Dijh(φ) defined as
follows for 1 ≤ i, j ≤ 2:
Dijh(φ)  V0h,
∫ΩDijh(φ) v dx = – ½ ∫Ω [∂φ/∂xi ∂v/∂xj + ∂φ/∂xj ∂v/∂xi]dx,
 v V0h,  φ Vh.
This is a mixed finite element approximation of the second
order derivatives, classically used for solving linear and
nonlinear bi-harmonic problems (Cahn-Hilliard, Von
Kármán equations for plates, Navier-Stokes equations in
their {ψ, ω} formulation, etc…).
Section 3 (3)
Deriving a discrete analogue of the above least squares formulation of
(E-MA-D) is pretty obvious now.
3.2. Numerical Experiments
The first test problem is defined as follows:
(i)
Ω = (0, 1) × (0, 1).
(ii)
f(x) = 1/|x|,  x  Ω.
(iii)
g(x) = (2|x|)3/2/3,  x  Γ.
With these data one can easily show that the function ψ defined by
ψ(x) = (2|x|)3/2/3,  x  Ω, is solution of the corresponding (E-MA-D)
problem. The above function does not belong to C2(ΩГ) but belongs
to W2, p(Ω) for p  [1, 4); it has in principle enough regularity to be
handled by our approach. We have used a uniform mesh like the one
below.
Section 3 (4)
A uniform triangulation of Ω (h= 1/4).
Section 3 (5)
h

nit
||D2hψch – pch ||Q
|| ψch – ψ||L2(Ω)
_____________________________________________________________________
1/ 32
1
145
0.9381 × 10– 6
0.556 × 10– 4
1/ 32
10
56
0.8290 × 10 – 6
0.556 × 10– 4
1/ 32
100
46
0.9285 × 10– 6
0.556 × 10– 4
1/ 32
1,000
45
0.9405 × 10– 6
0.556 × 10– 4
1/ 64
1/ 64
1/ 64
1/ 64
1
10
100
1,000
151
58
49
48
0.9500 × 10– 6
0.9974 × 10– 6
0.9531 × 10– 6
0.9884 × 10– 6
0.145 × 10– 4
0.145 × 10– 4
0.145 × 10– 4
0.145 × 10– 4
First Test Problem
The above results suggest an approximation error in O(h2) for the L2 (Ω)norm.
First test problem:
Graph of f
First test problem:
Graph of Ψhc
Section 3(8)
Second Test Problem
Data:  = (0,1)×(0,1), f = 1, g = 0.
Results:
Section 3 (9)
Second Test Problem
Section 3 (10)
Second Test Problem
4. Other Fully Nonlinear Elliptic
Equations
With some subtle differences the methodology we
applied to the solution of the Monge-Ampère
equation applies also to the solution of the
following Pucci’s Equation
αλ+ + λ- = 0 in Ω, Ψ = g on ∂Ω,
where: (i) α  (1, + ∞). (ii) λ+ and λ- are the
(PE)
largest and smallest eigenvalues of the Hessian
matrix D2Ψ of the function Ψ. (iii) Ω  R2.
Section 4 (2)
(PE) is equivalent to the following system
α|Ψ|2 + (α – 1)2 detD2Ψ = 0 in Ω, Ψ = g on ∂Ω,
(PE)’
Ψ ≤ 0 in Ω.
A note which appeared in the CRAS, Paris (2005)
describes the LS/OS solution of (PE)’.
5. Final Observations
Solving (E-MA-D) by a mixed method is really solving it via its
equivalent PFAFF System, namely
d  u1dx1  u2 dx2  0,
du  p dx  p dx  0,
 1
11
1
12
2

du2  p12 dx1  p22 dx2  0,
 p p  p2  f ,
 11 22
12
completed by:
ψ = g on Г.
The “burden of nonlinearity” has been transferred from ψ to p.
Section 5 (2)
A natural question is the following: Is our approach a (kind of) viscosity
method ? The answer is “yes” as shown below. Let us show it: The
Flow associated to the Least-Squares optimality conditions reads as
follows:
Find {ψ(t), p(t)} Vg  Q,  t > 0, such that
∫ ∂(Δψ)/∂t Δφ dx + ∫ D2ψ:D2φ dx = ∫Ωp:D2φ dx,  φ  V0,
Ω
Ω
∫ ∂p/∂t : q dx + ∫ p:q dx + <∂I (p), q> = ∫
Ω
Ω
{ψ(0), p (0)} = {ψ0, p0}.
f
Ω
D2ψ:q dx,  q  Q, (FE)
Section 5 (3)
Assuming that Ω is simply connected, introduce:
u = {u1, u2} = {∂ψ/∂x2, – ∂ψ/∂x1},
v = {v1, v2} = {∂φ/∂x2, – ∂φ/∂x1},
ω = ∂u2/∂x1– ∂u1/∂x2,
θ = ∂v2/∂x1– ∂v1/∂x2,
Vg = {v| v  (H1(Ω))2, .v = 0, v.n = dg/ds on Γ},
V0 = {v| v  (H1(Ω))2, .v = 0, v.n = 0 on Γ},
L = (– 10 01).
The formulation (FE) is equivalent to
Section 5 (3)
Find u(t) Vg,  t > 0, such that
∫ ∂ω/∂t θ dx + ∫ u:v dx = ∫Ω Lp:v dx,  v  V0,
Ω
Ω
∂p/∂t + p + ∂If(p) + Lu = 0,
(FE)*
{u(0), p (0), ω(0)} = {u0, p0, ω0}.
.
Problem (FE)* has a visco-elasticity flavor, – L p playing here the
role of the so-called extra-stress tensor. As t → +∞, we obtain thus at
the limit a viscosity solution, but in a sense different from M.CrandallP.L. Lions’.