Galerkin methods for the MongeAmpre problem with transport boundary - - PowerPoint PPT Presentation

Galerkin methods for the Monge–Ampère problem with transport boundary conditions

Omar Lakkis

Computer Science and Technology — Free University of Bozen-Bolzano IT and Mathematics — University of Sussex — Brighton, England GB

with the precious contribution of Tristan Pryer University of Reading, GB and Ellya Kawecki University of Oxford, GB talk given 24 November 2016 at

Numerical methods for Hamilton–Jacobi equations in optimal control and related fields Special Semester on Computational Methods in Science and Engineering – Workshop 3

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 1 / 53

Outline

1

From Monge–Kantorovitch to Monge–Ampère

2

PDE Background: fully nonlinear elliptic PDE’s

3

History and competing approaches Finite differences Finite elements

4

Iterative nonlinear solvers Fixed point Newton Hessian recovery

5

A non-variational FEM (NVFEM) solver

6

Convergence

7

Experiments

8

Transport (second) boundary conditions

9

Conclusions

10 References

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 2 / 53

Monge’s mass transportation problem

ρ σ t

ρ(x)dx σ(y)dy

Consider ρ,σ > 0,

∫ ρ = ∫ σ, find t field such that

for any Lebesgue-measurable subset A of d d ∈ (3.1)

• A

ρ(x)dx =

• t(A)

σ(y)dy ∀ A (Borel) ⊆ Ω.

(3.2)

• t(A)

σ(y)dy =

• A

σ(t(x))|detDt(x)|dx.

This works for any positive measure densities ρ,σ. If ρ,σ are functions, then by du Bois-Reymond’s lemma (3.3)

σ(t(x))|detDt(x)| = ρ(x) ∀ x ∈ d.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 3 / 53

Monge’s mass transportation problem

Optimise the cost

Find t such that (4.1)

(detDt)σ ◦ t = ρ

which minimises the transportation cost functional (4.2)

• |t(x) − x|2ρ(x)dx.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 4 / 53

From Monge to Monge–Ampère

Following Brenier, (1991)

Under convexity and regularity assumptions, the Monge-Ampere equation

detD2u(x) = f (x,∇u(x))

coupled to the second boundary condition second boundary condition (5.1)

∇u(Ω) = Υ,

and the right-hand side (5.2)

f (x,∇u(x)) = ρ(x) g(∇u(x))

provides a solution to the Monge problem (5.3)

t = ∇u.

Full analysis with viscosity solutions was performed by Urbas, (1997). For regularity and related issues see L. A. Caffarelli, (1990), L. Caffarelli, Nirenberg and Spruck, (1985) and Luis A. Caffarelli, (1990) .

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 5 / 53

Fully nonlinear elliptic PDE’s

Definition and notation

Given a real-valued nonlinear function F of matrices (FNFun)

F : Sym(d) → .

Consider the equation

[u] := F(D2u) − f = 0 and u|∂ Ω = 0

(FNE) Conditional ellipticity condition, i.e.,

λ(M) sup |ξ|=1

• Nξ
• ≤ F(M + N) − F(M) ≤ Λ sup

|ξ|=1

• Nξ
• ∀ M ∈ ⊆ Sym(d),N ∈ Sym(d).

(NL-Ellip) for some ellipticity domain and “constants” λ(·),Λ > 0.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 6 / 53

Fully nonlinear elliptic PDE’s

The ellipticity fauna

[u] := F(D2u) − f = 0

Conditionally elliptic

∃ ⊆ Sym(d),λ(·),Λ > 0 : λ(M) sup |ξ|=1

• Nξ
• ≤ F(M + N) − F(M) ≤ Λ sup

|ξ|=1

• Nξ
• ∀ M ∈ ⊆ Sym(d),N ∈ Sym(d).

Unconditionally elliptic if = Sym(d). Uniformly elliptic infλ > 0.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 7 / 53

Characterisation of the ellipticity condition

in the smooth case

Ellipticity condition, i.e.,

λ(M) sup |ξ|=1

• Nξ
• ≤ F(M + N) − F(M) ≤ Λ sup

|ξ|=1

• Nξ
• ∀ M ∈ ⊆ Sym(d),N ∈ Sym(d).

(NL-Ellip) for some ellipticity “constants” λ(·),Λ > 0. If F is differentiable then (NL-Ellip) is satisfied if and only if for each M ∈ there exists µ > 0 such that (8.1)

ξ⊺F′(M)ξ ≥ µ

• ξ
• 2

∀ ξ ∈ d.

Furthermore = Sym(d) and µ is independent of M if and only if F is uniformly elliptic.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 8 / 53

The Monge–Ampère–Dirichlet problem

A classical fully nonlinear elliptic PDE

Boundary value problem

detD2u = f

in Ω

u = 0

• n ∂ Ω

(MAD) admits a unique solution in the cone of convex functions when

f > 0.[Luis A. Caffarelli and Cabré, 1995]

Derivative of nonlinear function F(X) = detX yields

F′(X) = CofX.

Problem elliptic if and only if

ξ⊺ CofD2uξ ≥ λ

• ξ
• 2

∀ ξ ∈ d

for some λ > 0.

Conotonic constraint

Restriction on unknown functions u: they must be globally either convex or concave (conotonic).

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 9 / 53

A simple fully nonlinear elliptic PDE

Consider problem

[u] := sin(∆u) + 2∆u − f = 0 in Ω, u = 0 on ∂Ω.

Differentiating, we see that

D [v]w = (cos(∆v) + 2)I:D2w = (cos(∆v) + 2)∆w.

Hence problem uniformly elliptic.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 10 / 53

A Krylov-type cubic elliptic PDE

The problem is for d = 2

[u] := (∂11u)3 + (∂22u)3 + ∂11u + ∂22u − f = 0

in Ω

u = 0

• n ∂ Ω.

(Krylov) Problem is uniformly elliptic since its differentiation gives:

F′(X) =

• 3x2

22 + 1

3x2

11 + 1

• .

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 11 / 53

Pucci’s equation

Consider F : Sym(d) → to be the extremal function

F(N) =

d

• i=1

αiλi(N) where λi(N) eigenvalues of N

for some given α1,...,αd ∈ . (Pucci) Special case when d = 2, α1 = α ≥ 1 and α2 = 1 yields equation (2 Pucci)

0 = (α + 1)∆u + (α − 1)

• (∆u)2 − 4detD2u

1/2 .

The problem is unconditionally elliptic.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 12 / 53

Classes of fully nonlinear equations

A rough guide

See Luis A. Caffarelli and Cabré, 1995 for a more systematic classification. Isaacs form: infβ supα Lαβu = 0.

Bellman type: Isaacs with only one β ( ⇔ no inf). Related to Hamilton–Jacobi–Bellman, stochastic control and differential game theory.

Isaacs form is very general: “non-algebraic” and harder to treat

• numerically. We don’t,

yet.[M. Jensen and Iain Smears, 2012; Lio and Ley, 2010, e.g.].

Hessian invariants (algebraic): Monge–Ampère, Pucci, Laplace (!). Subdivided into unconeditionally elliptic (Pucci, Laplace) and coneditionally elliptic (Monge–Ampère). Other algebraic FNE’s (Krylov, algebraic nonlinearities, etc.) Aronson equations and infinite-harmonic functions, nicely reviewed in Barron, L. C. Evans and R. Jensen, 2008. (These aren’t proper FNE’s, as they are quasilinear, nevertheless, Hessian recovery applies well.)

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 13 / 53

Finite difference approaches

1 Earliest known provided approximations of the Monge–Ampère (and

• ther equations) by Oliker and Prussner, 1988.

2 Kuo and Trudinger, 1992 gave mostly theoretical work introduced

the concept of wide stencils and proving convergence for wide enough stencils.

3 Benamou and Brenier, 2000 proposed an approach based on the

Brenier-solution concept related to fluid-dynamics and mass-trasportation.

4 Oberman, 2008 introduced more practically effective work working

• ut the details, proiding a bound on the wide stencil’s width. See

also Froese, 2011 and Benamou, Froese and Oberman, 2012 for second boundary conditions.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 14 / 53

Galerkin (mainly finite element) methods

Dean and Glowinski, 2006 (and earlier work) introduced a FE least square method to solve Monge–Ampère equation. Böhmer, 2010 (and earlier papers) advocates (mostly theoretically, proving convergence) the use of C1/spline finite elements to directly compute the Hessian. Practical schemes have been constructed recently by Davydov and Saeed, 2013. Feng and M. Neilan, 2009 introduce the vanishing moment method a fourth order semilinear approximation: ε∆2u + F[u] = 0 and take

ε → 0. (Similar to early vanishing viscosity methods for first order

PDE’s.)

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 15 / 53

Galerkin (mainly finite element) methods

Brenner et al., 2011 introduces a penalty method to deal with the convexity.

• M. J. Neilan, 2012 considers a generalization of scheme in Lakkis

and Pryer, 2011 and proves convergence rates for MAD in 2d. Awanou, 2011 uses a pseudo time [sic] approach.

• M. Jensen and Iain Smears, 2012 provide and analyze a FEM for a

special class of Hamilton–Jacobi–Bellman equation.

• I. Smears and E. Süli, 2013; Iain Smears and Endre Süli, 2014 for a

DGFEM approach

• X. Feng and M. Jensen use a semi-Lagrangian method based on a

finite element mesh.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 16 / 53

A fixed-point solution

Nonlinear PDE

[u] := F(D2u) − f = 0

can be rewritten as follows

[u] = 1 F′(t D2u)dt

• :D2u + F(0) − f = 0.

Define

(D2u) := 1 F′(t D2u)dt, g := f − F(0),

then if u solves (FNE), it also solves

(D2u):D2u = g.

Fixed point iteration: given u0 find

(D2un):D2un+1 = g, for n = 1,2,... .

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 17 / 53

Crucial remark

Note that solving

(D2un):D2un+1 = g

involves a linear elliptic equation in non-divergence form.

Big fat note

Standard variational FEM’s do not apply.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 18 / 53

Newton’s method

Given an initial guess u0, let

D [un]

• un+1 − un

= − [un], for n = 0,1,2,...,

where

D [u]v = F′(D2u) : D2v.

I.e.,

F′(D2un) : D2 un+1 − un = f − F(D2un).

Big fat note (repeated)

Equation in nondivergence form, standard FEM’s will not apply.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 19 / 53

The need for Hessian recovery

Detailed by Lakkis and Pryer, (2013)

Fixed point iteration

(D2un):D2un+1 = g

and Newton’s iteration

F′(D2un) : D2 un+1 − un = f − F(D2un).

besides being nonvariational, like fixed-point, requires the suitable approximation of a Hessian’s function.

Big fat note (a variation)

Hence the use of the recovered Hessian introduced by Lakkis and Pryer, (2011).

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 20 / 53

Hessian recovery

Introduce Galerkin finite element spaces

:=

• Φ ∈ H1(Ω) : Φ|K ∈ p ∀ K ∈ and Φ ∈ C0(Ω)
• ,

0 := ∩ H1

0(Ω),

Unbalanced mixed problem:

Find (U,H) ∈ 0 × d×d satisfying

〈H,Φ〉 +

• Ω

∇U ⊗ ∇Φ −

• ∂ Ω

∇U ⊗ n Φ = 0 〈A:H,Ψ〉 =

• f ,Ψ
• ∀ (Φ,Ψ) ∈ × 0.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 21 / 53

A (sometimes) commutative diagram

discretization are often possible (e.g., when the nonlinearity is algebraic in the Hessian): (exact) nonlinear PDE (exact) nonvariational linear PDE’s FE fully nonlinear PDE discrete linear 1 discrete linear 2 Newton NVFEM FNFEM Newton

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 22 / 53

Convergence analysis

Available for the linear nondivergence case so far

A priori estimates for the error

• A : (D2u − H[U])
• H−1(Ω) .

A posteriori error estimate for the error

u − uh2

L2(Ω) ≤

• K∈
• h2

Kf − A:D2U2 L2(K) + hK A:∇U⊗2 L2(∂ K)

• where the tensor jump of a field v across an edge E = K ∩ K ′ is given by

v⊗ E := lim

ε→0(v(x + εnK) ⊗ nK + v(x − εnK ′) ⊗ nK ′)

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 23 / 53

A nonlinear function of ∆u

[u] := sin(∆u) + 2∆u − f = 0 in Ω, u = 0 on ∂Ω.

P1 elements (left) and P2 elements (right)

1e-05 0.0001 0.001 0.01 0.1 1 10 100 100 1000 10000 100000 1e+06 1e+07 error dim(W) EOC = 1.82 EOC = 1.01 EOC = 0.63 EOC = 1.95 EOC = 1.01 EOC = 0.58 EOC = 1.93 EOC = 1.00 EOC = 0.55 EOC = 1.90 EOC = 1.00 EOC = 0.53 EOC = 1.91 EOC = 1.00 EOC = 0.52 ||u-uh|| ||u-uh||1 ||D2 u-H[uh]|| 1e-08 1e-06 0.0001 0.01 1 100 100 1000 10000 100000 1e+06 1e+07 error dim(W) EOC = 2.68 EOC = 1.98 EOC = 1.49 EOC = 3.31 EOC = 2.01 EOC = 1.50 EOC = 3.00 EOC = 2.00 EOC = 1.50 EOC = 3.00 EOC = 2.00 EOC = 1.50 ||u-uh|| ||u-uh||1 ||D2 u-H[uh]||

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 24 / 53

Krylov’s equation

[u] := (∂11u)3 + (∂22u)3 + ∂11u + ∂22u − f = 0

in Ω

u = 0

• n ∂ Ω.

(25.1)

P1 elements (left) and P2 elements (right)

1e-05 0.0001 0.001 0.01 0.1 1 10 100 100 1000 10000 100000 1e+06 1e+07 error dim(W) EOC = 1.53 EOC = 0.93 EOC = 1.99 EOC = 2.51 EOC = 0.98 EOC = 2.04 EOC = 2.01 EOC = 0.99 EOC = 1.99 EOC = 2.00 EOC = 1.00 EOC = 1.94 EOC = 2.00 EOC = 1.00 EOC = 1.32 ||u-uh|| ||u-uh||1 ||D2 u-H[uh]|| 1e-08 1e-06 0.0001 0.01 1 100 100 1000 10000 100000 1e+06 1e+07 error dim(W) EOC = 2.27 EOC = 1.91 EOC = 0.88 EOC = 3.19 EOC = 1.98 EOC = 1.20 EOC = 2.95 EOC = 2.01 EOC = 1.30 EOC = 2.08 EOC = 2.01 EOC = 1.31 ||u-uh|| ||u-uh||1 ||D2 u-H[uh]||

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 25 / 53

Pucci’s equation

0 = (α + 1)∆u + (α − 1)

• (∆u)2 − 4detD2u

1/2 .

2,α = 2 (left) and 2,α = 5 (right)

1e-05 0.0001 0.001 0.01 0.1 1 10 100 100 1000 10000 100000 1e+06 1e+07 error dim(V x (W)dxd)

EOC = 0.48 EOC = 0.45 EOC = 0.25 EOC = 2.20 EOC = 1.22 EOC = 0.38 EOC = 2.57 EOC = 1.62 EOC = 0.74 EOC = 2.81 EOC = 1.88 EOC = 1.08 EOC = 2.86 EOC = 1.95 EOC = 1.26

||u-UN|| ||u-UN||1 ||D2 u-H[UN]|| 0.1 1 10 100 1000 10000 100000 100 1000 10000 100000 1e+06 1e+07 error dim(V x (W)dxd)

EOC = -0.40 EOC = 0.07 EOC = 0.03 EOC = 1.11 EOC = 0.59 EOC = 0.09 EOC = 3.37 EOC = 1.17 EOC = 0.33 EOC = 2.86 EOC = 1.60 EOC = 0.73 EOC = 2.76 EOC = 1.91 EOC = 1.12

||u-UN|| ||u-UN||1 ||D2 u-H[UN]||

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 26 / 53

MAD stuff I

reminder: MAD = Monge–Ampère–Dirichlet

FE-convexity check inspired from Aguilera and Morin, 2009.

Exact solution and EOC’s for 2 elements (suboptimal for 1)

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000 1e+06 1e+07 error dim(V x (W)dxd) 2.06 1.60 0.90 2.56 1.84 1.15 2.81 1.93 1.29 2.91 1.97 1.39 2.95 1.98 1.44 2.98 1.99 1.47 ||u-U|| ||u-U||1 ||D2 u-H[U]||

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 27 / 53

MAD stuff II

reminder: MAD = Monge–Ampère–Dirichlet

principal minor and determinant instances

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 28 / 53

Nonclassical solutions

Viscosity or Alexandrov[Lawrence C. Evans, 2001]

Singular solution u(x) = |x|2α

0.001 0.01 0.1 1 100 1000 10000 100000 1e+06 1e+07 error dim(V x (W)dxd) 1.04 1.25 0.84 1.22 0.80 1.12 0.80 1.00 0.80 0.91 ||u-U|| ||u-U||1

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 29 / 53

Nonclassical solutions

Viscosity or Alexandrov[Lawrence C. Evans, 2001]

More singular, α = 0.6, α = 0.55,...

0.01 0.1 1 100 1000 10000 100000 1e+06 1e+07 error dim(V x (W)dxd) 0.41 0.62 0.41 0.51 0.41 0.45 0.41 0.42 0.40 0.41 ||u-U|| ||u-U||1 0.01 0.1 1 100 1000 10000 100000 1e+06 1e+07 error dim(V x (W)dxd) 0.28 0.23 0.26 0.28 0.21 0.22 0.20 0.20 0.20 0.20 0.20 0.20 ||u-U|| ||u-U||1

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 30 / 53

Adaptive approximation of nonclassical solutions

Viscosity or Alexandrov

Singular solution u(x) = |x|1.1 (empirical ZZ-estimators)

1e-06 1e-05 0.0001 0.001 0.01 0.1 100 1000 10000 100000 1e+06 error dim(V x (W)dxd) ||u-U|| ||u-U||1 O(N(-.75)) O(N(-1))

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 31 / 53

The transport boundary conditions

the true ones actually

Recall the Monge–Ampère with transport boundary condition (MAT) is to find u such that (32.1)

detD2u(x) = f (x,∇u(x))

for x ∈ Ω

∇u(∂ Ω) ⊆ ∂ Υ

To turn this inclusion into a workable algebraic equation we use an idea proposed by Urbas, (1997) (to prove problem is well posed): (32.2)

∂ Υ =

• y ∈ d : b(y) = 0
• ,

where b is a convex (or concave) level set function, e.g., the signed distance function to ∂ Υ. Hence (32.3)

b(∇u(x)) = 0 for all x ∈ ∂ Ω.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 32 / 53

Netwon iteration for MAT

detD2u(x) = ρ(x) σ(∇u(x))

for

x ∈ Ω, b(∇u(x)) = 0

for

x ∈ ∂ Ω.

(33.1) Newton is derived as before, plus an extra one involving the gradient. Gradient recovery used to build G[U] as a better approximation of the gradient ∇u on the boundary than ∇U itself. Optimal convergence observed for 1 elements on curved domains,

• r k elements on convex polygonal domains.

Work in progress of higher order methods on curved convex domains.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 33 / 53

Monge–Ampère with transport

Operators

[u(x)] = 0, x ∈ Ω, [u(x)] = 0, x ∈ ∂ Ω.

(34.1) where the Monge–Ampère with transport operator is (34.2)

[v] := F(x,∇v,D2v) with F(x,p,M) := detM − ρ(x) σ(p)

defined on the cone of convex smooth functions (34.3)

:=

• v ∈ C2(Ω) : D2v(x) ∈ SPD(d)

∀ x ∈ Ω and 〈v〉Ω = 0

• with boundary integral

(34.4)

[u] := b(∇u).

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 34 / 53

To approximate the solution of (34.1) we will apply the Newton–Raphson method. For each n ∈ 0, assuming un ∈ is given, the Newton–Raphson iteration consists in finding un+1 ∈ satisfying (35.1)

D[un(x)](un+1(x) − un(x)) + [un(x)] = 0, for x ∈ Ω, D[un(x)](un+1(x) − un(x)) + [un(x)] = 0, for x ∈ ∂ Ω,

where the D and D are the (infinite dimensional) directional derivatives, explicity calculated as (35.2)

D[v]w = Cof(D2v):D2w + ρ σ(∇v)2 ∇σ(∇v) · ∇w.

and (35.3)

D[v]w = Db(∇v)∇w = ∇b(∇v) · ∇w.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 35 / 53

It follows that at each given Newton iteration n, we have to solve for the unknown θn+1 := un+1 − un an oblique derivative elliptic problem in nondivergence form (36.1)

A(x) : D2u(x) + b(x) · ∇u(x) + c(x)u(x) = r(x) for x ∈ Ω, β(x) · ∇u(x) = s(x) for x ∈ ∂ Ω.

with the following substitutions

A(x) := CofD2un(x),

(36.2)

b(x) := ρ(x) σ(∇un(x))2 ∇σ(∇un(x)),

(36.3)

c(x) := 0,

(36.4)

r(x) := −detD2un(x) + ρ(x) σ(∇un(x)),

(36.5)

β(x) := ∇b(∇un(x)),

(36.6) and

s(x) := −b(∇un(x)).

(36.7)

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 36 / 53

NVFEM–Newton–Raphson with plain finite element gradient

A first attempt to discretise the Newton–Raphson iteration (35.1) can be derived, as follows, for each n ∈ 0, assuming (Un,Hn) ∈ h ×h is given, find (Un+1,Hn+1) ∈ h × h such that

〈Hn+1,Φ〉Ωh + 〈∇Un+1∇Φ⊺〉Ωh − 〈∇Un+1 nΩh

⊺Φ〉∂ Ωh = O

∀ Φ ∈ h,

(37.1)

• CofHn : (Hn+1 − Hn) + ρ Dσ(∇Un)∇[Un+1 − Un]

σ(∇un) + F(·,∇Un,Hn),Φ

• Ω

−〈Db(∇Un)∇[Un+1 − Un] + b(∇Un),Φ〉∂ Ωh = 0 ∀ Φ ∈ h.

(37.2) The role of the nil sum constraint on u, needed to ensure its uniqueness is enforced with a Lagrange multiplier equation (not on space) (37.3)

〈Un+1,λ〉Ωh + 〈ln+1,Φ〉Ωh = 0 ∀ Φ ∈ h,λ ∈ .

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 37 / 53

NVFEM–Newton–Raphson with finite element gradient recovery I

We incorporate the gradient recovery operator into our system, by replacing ∇Un+1 with GhUn+1 in (37.2), which rewritten in incremental form reads for each n ∈ 0:

1 given (Un,Gn,Hn) ∈ h × h × h, satisfying

(38.1)

Gn = GhUn, Hn = HhUn, Un is strictly finite element convex,

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 38 / 53

NVFEM–Newton–Raphson with finite element gradient recovery II

2 find the Newton–Raphson increment Θn+1 ∈ h (along with its

recovered gradient GhΘn+1 =: Γ n+1 ∈ h and its recovered Hessian HhΘn+1 =: ∆n+1 ∈ h) and a scalar ln+1 such that:

〈Γ n+1,Φ〉Ωh − 〈∇Θn+1,Φ〉Ωh = 0 ∀ Φ ∈ h,

(38.2)

〈∆n+1,Φ〉Ωh + 〈Γ n+1∇Φ⊺〉Ωh − 〈Γ n+1 nΩ

⊺Φ〉∂ Ωh = O

∀ Φ ∈ h,

(38.3)

〈Θn+1,λ〉Ωh + 〈ln+1,Φ〉Ωh = 0 ∀ (Φ,λ) ∈ h × ,

(38.4)

• CofHn:∆n+1 + ρ Dσ(Gn)Γ n+1

σ(Gn) ,Φ

• Ωh

+ 〈F(·,Gn,Hn),Φ〉Ωh −〈Db(Gn)Γ n+1,Φ〉∂ Ωh + 〈b(Gn),Φ〉∂ Ωh = 0 ∀ Φ ∈ h.

(38.5)

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 39 / 53

NVFEM–Newton–Raphson with finite element gradient recovery III

3 define the next Newton–Raphson iterate

(38.6)

(Un+1,Gn+1,Hn+1) := (Θn+1,Γ n+1,∆n+1) + (Un,Gn,Hn).

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 40 / 53

convergence

0.0001 0.001 0.01 0.1 1 100 1000 10000 100000 1x106 1x107 1.27 1.24 0.76 1.97 1.46 0.32 2.22 2.51 1.93 1.72 1.52 0.82 1.88 1.75 0.98 1.92 1.95 1.36 1.98 1.97 1.34 error dim(V x (W)dxd) ||u-U||L2 ||u-U||H1 ||D2U-D2u||L2

Convergence curves for 2 elements

• n a circular domains Ω,Υ, without

gradient recovery.

1.5 1.0 0.5 0.0 0.5 1.0 1.5 du/dx 1.5 1.0 0.5 0.0 0.5 1.0 1.5 du/dy

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 41 / 53

convergence

0.0001 0.001 0.01 0.1 100 1000 10000 100000 1x106 1x107 2.08 0.99 2.15 1.69 1.84 1.48 1.72 0.77 1.99 1.49 1.91 1.00 1.88 1.46 1.80 0.96 1.99 1.59 1.90 0.86 error dim(V x (W)dxd) ||u-U||L2 ||u-U||H1 ||Du-G[U]||L2 ||D2U-D2u||L2

Convergence curves for 2 elements

• n a circular domains Ω,Υ with gradi-

ent recovery.

1.5 1.0 0.5 0.0 0.5 1.0 1.5 G[u]_1 1.5 1.0 0.5 0.0 0.5 1.0 1.5 G[u]_2

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 42 / 53

Conclusions and outlook

Obtained and tested a practical and “easy” Netwon scheme based on nonvariational FEM (NVFEM) via Hessian recovery. Convergence rates optimal in all examples. A posteriori error esimates for very weak norms in the linear problem, provide an elementary way to do adaptivity. In progress: prove apriori convergence for stronger norms in linear problems. In progress: embed second boundary condition (∇u(Ω) = Υ with prescribed Υ). (This was achieved for wide-stencils but on structured grids by Benamou, Froese and Oberman, 2012.) Open problem: prove conservation of conotonicity for MAD/MAS. Open problem: apriori and aposteriori analysis for nonlinear problem.

Omar Lakkis (Unibz IT & Sussex GB) Galerkin for Transport Monge–Ampère Linz AT 2016-11-24 43 / 53

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