Discontinuous Galerkin finite element methods for Hamilton … · 2017-06-20 · Talk outline 1....

Discontinuous Galerkin finite element methodsfor Hamilton–Jacobi–Bellman equations

with Cordes coefficients

Iain Smears

INRIA Paris

LJLL Seminar, June 2017

joint work with

Endre Suli, University of Oxford

Overview

Talk outline

1. Introduction: Hamilton–Jacobi–Bellman (HJB) equations.

2. Analysis: Analysis of HJB equations with Cordes coefficients.

3. Numerical methods: High-order discontinuous Galerkin methods for HJBequations with Cordes coefficients.

Overview

Talk outline




1. Stochastic optimal control

Stochastic differential equation

dXt = b(Xt , αt)dt + σ(Xt , αt)dBt , X0 = x ,

Find a control α(·) : t 7→ αt ∈ Λ that minimises

J(x , α(·)) = E[∫ τexit

0

f (Xt , αt) e−∫ t

0c(Xs ,αs ) dsdt

]

• b(x , α) ∈ Rd drift, σ(x , α) ∈ Rd×m volatility

• scalar f and c : running cost and discount

• αt control variable

• Λ controls set (assumed to be a compact metric space).

• stopping time τexit: first exit from bounded domain Ω ⊂ Rd

Example applications: energy, engineering, finance . . .

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1. Dynamic programming principle

The dynamic programming principle (DPP) is a solution process for astochastic control problem.

Overview: stages of DPP

1. Define the value function of theoptimal control problem.

2. DPP: the value function is thesolution of an HJB equation.

3. The optimal controls can becomputed once the value functionis available.

Richard Bellman (1920–1984)

Feedback control map αfeedback : Ω→ Λ =⇒ α(t) := αfeedback(Xt).

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Details in Fleming & Soner 2006

• Define the value function V defined by

V (x) := infJ(x , α(·)) | α(·) : t ∈ [0,∞) 7→ αt ∈ Λ, α(·) ∈ A.A = set of admissible controls: progressively measurable w.r.t. filtration.

• The function u := −V solves the HJB equation

supα∈Λ

[Lαu − f α] = 0 in Ω,

u = 0 on ∂Ω,(HJB)

where Lαu := aα(x) : D2u + bα(x) · ∇u − cα(x) u, with

aα(x) := 12σ(x , α)σ>(x , α) ∈ Rd×d ,

Notation: aα : D2u =d∑

i,j=1

aαij (x)uxixj ,

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Details in Fleming & Soner 2006

• Define the value function V defined by

V (x) := infJ(x , α(·)) | α(·) : t ∈ [0,∞) 7→ αt ∈ Λ, α(·) ∈ A.A = set of admissible controls: progressively measurable w.r.t. filtration.

• The function u := −V solves the HJB equation

supα∈Λ

[Lαu − f α] = 0 in Ω,

u = 0 on ∂Ω,(HJB)

where Lαu := aα(x) : D2u + bα(x) · ∇u − cα(x) u, with

aα(x) := 12σ(x , α)σ>(x , α) ∈ Rd×d ,

Notation: aα : D2u =d∑

i,j=1

aαij (x)uxixj ,

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1. Hamilton–Jacobi–Bellman Equation

Elliptic Dirichlet problem:

supα∈Λ

[Lαu − f α] = 0 in Ω,

u = 0 on ∂Ω,(Elliptic HJB)

where Lαu := aα(x) : D2u + bα(x) · ∇u − cα(x) u.

Notation: aα(x) : D2u =d∑

i,j=1

aαij (x)uxixj , bα(x) · ∇u =d∑

i=1

bαi (x)uxi .

Assumptions in this talk:

• Ω ⊂ Rd is bounded and convex, Λ a compact metric space.

• a, b, c and f are continuous functions in x ∈ Ω, α ∈ Λ.

• aα are symmetric positive definite, uniformly on Ω× Λ, and cα ≥ 0.

• Cordes coefficients: the coefficient functions a, b, c satisfy the Cordescondition (coming soon!)

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1. Examples

How do HJB equations relate to other PDEs?

supα∈Λ

[Lαu − f α] = 0

The HJB equation generalises many other equations:

• Linear nondivergence form elliptic equations

a : D2u + b · ∇u − cu = f , (assume that Λ is a singleton set).

• Hamilton–Jacobi: e.g. eikonal equation

supα∈Sd

[α · ∇u − 1] = |∇u| − 1 = 0.

• Monge–Ampere equation [Krylov 1987, Jensen & Feng 2017]detD2u − f = 0

u convex⇐⇒ inf

a∈Rd×dsym,+

Tr a=1

[a : D2u − d (f det a)1/d

]= 0.

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1. Examples


supα∈Λ

[Lαu − f α] = 0





supα∈Sd

[α · ∇u − 1] = |∇u| − 1 = 0.


u convex⇐⇒ inf

a∈Rd×dsym,+

Tr a=1


]= 0.

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1. Examples


supα∈Λ

[Lαu − f α] = 0





supα∈Sd

[α · ∇u − 1] = |∇u| − 1 = 0.


u convex⇐⇒ inf

a∈Rd×dsym,+

Tr a=1


]= 0.

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1. Approaches

What are the available approachesto PDE theory and numerical discretization?

• Weak solutions in H1(Ω): not applicable!

→ most existing finite element techniques cannot be used!

• Viscosity solutions: generally applicable, even to degenerate ellipticproblems. Solution space u ∈ C (Ω).

This leads to monotone numerical schemes (c.f. next few slides). . .

• Strong solutions in H2(Ω): under the Cordes condition . . .

• Classical solutions in C 2(Ω): Evans–Krylov Theorem and itsdevelopments guarantee interior regularity estimates for the viscositysolution under uniform ellipticity and data regularity assumptions.

Some references:

• Weak, strong and classical solutions Gilbarg & Trudinger, 1998.

• Viscosity solutions for fully nonlinear PDE Crandall, Lions & Ishii, 1992.

• Regularity theory of viscosity solutions Caffarelli & Cabre, 1995.

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1. Approaches








Some references:




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1. Approaches








Some references:




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1. Approaches








Some references:




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1. Literature

Pre-existing numerical methods:

• Monotone methods: low-order methods with discrete maximumprinciples:

Convergence theory to the viscosity solution: Barles & Souganidis1991.

mostly finite difference methods: Motzkin & Wasow, Kuo &Trudinger, Kocan, Camilli & Falcone, Barles & Jakobsen, Jakobsen &Debrabant, Fleming & Soner, Bonnans & Zidani,. . .

a few on monotone FEM: Jensen & S., SINUM 2013 and Nochetto &Zhang FOCM 2016.

limitations for anisotropic problems: Motzkin & Wasow 1953, Kocan1995, Bonnans & Zidani 2003, Crandall & Lions 2012.

• Other pre-existing non-monotone methods without discrete maximumprinciples, without convergence theory: Feng, Neilan, Glowinski, Brenner,Lakkis, Pryer, . . . [Feng et al. SIAM Rev. 2013].

Is it possible to have stable, consistent and convergent methods for fullynonlinear PDEs without discrete maximum principles?

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1. Literature

Pre-existing numerical methods:

• Monotone methods: low-order methods with discrete maximumprinciples:

Convergence theory to the viscosity solution: Barles & Souganidis1991.

mostly finite difference methods: Motzkin & Wasow, Kuo &Trudinger, Kocan, Camilli & Falcone, Barles & Jakobsen, Jakobsen &Debrabant, Fleming & Soner, Bonnans & Zidani,. . .

a few on monotone FEM: Jensen & S., SINUM 2013 and Nochetto &Zhang FOCM 2016.

limitations for anisotropic problems: Motzkin & Wasow 1953, Kocan1995, Bonnans & Zidani 2003, Crandall & Lions 2012.

• Other pre-existing non-monotone methods without discrete maximumprinciples, without convergence theory: Feng, Neilan, Glowinski, Brenner,Lakkis, Pryer, . . . [Feng et al. SIAM Rev. 2013].


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Overview

Talk outline



Motivation of Cordes coefficients Existence, Uniqueness, Well-posedness


2. PDE Theory: motivation

Cordes introduced his condition in the context of nondivergence formequations with discontinuous coefficients.

There is a link between HJB equations and nondivergence form equationswith discontinuous coefficients:

Classical solution algorithm: policy iteration, due to Howard and Bellman:

1. Choose an initial guess u0.

2. For k ∈ N, given a current guess uk , choose αk : Ω→ Λ, aLebesgue-measurable selection

αk(x) ∈ argmaxα∈Λ (Lαuk − f α)(x), ∀ x ∈ Ω.

3. Then, find uk+1 as a solution of the PDE

Lαkuk+1 = f αk in Ω, with uk+1 = 0 on ∂Ω,

where f αk : x 7→ f αk (x)(x), etc.

Howard 1960, Puterman & Brumelle 1979, Bokanowski et al. 2009, . . .8/37


Cordes introduced his condition in the context of nondivergence formequations with discontinuous coefficients.

There is a link between HJB equations and nondivergence form equationswith discontinuous coefficients:

Classical solution algorithm: policy iteration, due to Howard and Bellman:

1. Choose an initial guess u0.

2. For k ∈ N, given a current guess uk , choose αk : Ω→ Λ, aLebesgue-measurable selection

αk(x) ∈ argmaxα∈Λ (Lαuk − f α)(x), ∀ x ∈ Ω.

3. Then, find uk+1 as a solution of the PDE

Lαkuk+1 = f αk in Ω, with uk+1 = 0 on ∂Ω,

where f αk : x 7→ f αk (x)(x), etc.

Howard 1960, Puterman & Brumelle 1979, Bokanowski et al. 2009, . . .8/37


Linearized equation:

Lαkuk+1 = f αk in Ω, with uk+1 = 0 on ∂Ω, (1)

Question: is Eq. (1) a well-posed PDE?

In general, the answer is no: the linearization process (esp. the argmax)leads to discontinuous diffusion coefficients: aαk ∈ L∞(Ω)d×d andaαk /∈ C (Ω)d×d .

• Calderon–Zygmund: If a ∈ C (Ω)d×d and ∂Ω ∈ C 1,1, then existence anduniqueness in W 2,p [Gilbarg & Trudinger, 1998]

• If a ∈ L∞(Ω)d×d , a /∈ C (Ω)d×d , then there are counter-examplesshowing non-uniqueness in general (Maugeri et al, 2000):

∆u+ρd∑

i,j=1

xixj|x |2 uxi xj = 0 in B unit ball, ρ = −1+

d − 1

1− θ , 0 < θ < 1,

If d ≥ 3 and d > 2(2− θ) > 2, two solutions in H2(B) ∩ H10 (B)

u1(x) = 0 and u2(x) = |x |θ − 1

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Linearized equation:

Lαkuk+1 = f αk in Ω, with uk+1 = 0 on ∂Ω, (1)

Question: is Eq. (1) a well-posed PDE?

In general, the answer is no: the linearization process (esp. the argmax)leads to discontinuous diffusion coefficients: aαk ∈ L∞(Ω)d×d andaαk /∈ C (Ω)d×d .

• Calderon–Zygmund: If a ∈ C (Ω)d×d and ∂Ω ∈ C 1,1, then existence anduniqueness in W 2,p [Gilbarg & Trudinger, 1998]

• If a ∈ L∞(Ω)d×d , a /∈ C (Ω)d×d , then there are counter-examplesshowing non-uniqueness in general (Maugeri et al, 2000):

∆u+ρd∑

i,j=1

xixj|x |2 uxi xj = 0 in B unit ball, ρ = −1+

d − 1

1− θ , 0 < θ < 1,

If d ≥ 3 and d > 2(2− θ) > 2, two solutions in H2(B) ∩ H10 (B)

u1(x) = 0 and u2(x) = |x |θ − 1

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2. PDE theory: Cordes condition

Cordes condition: Case 1: without advection and reaction

Assume that there exists ε ∈ (0, 1] s. t.

|a(x)|2(Tr a(x))2 ≤

1

d − 1 + εa.e. x ∈ Ω, (Cordes0)

Theorem (Cordes, 1956)If Ω is convex, and if a ∈ L∞(Ω)d×d unif. ellipt. satisfies (Cordes0), then forany f ∈ L2(Ω) there exists a unique u ∈ H2(Ω) ∩ H1

0 (Ω) solving

a : D2u = f in Ω, with u = 0 on ∂Ω,

ExampleIf dimension d = 2, (Cordes0) ⇐⇒ uniform ellipticity.

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Cordes condition: Case 1: without advection and reaction

Assume that there exists ε ∈ (0, 1] s. t.

|aα(x)|2(Tr aα(x))2 ≤

1

d − 1 + εa.e. x ∈ Ω, α ∈ Λ (2)

Theorem (Cordes, 1956)If Ω is convex, and if aα ∈ L∞(Ω)d×d unif. ellipt. satisfies (Cordes0), thenfor any f ∈ L2(Ω) there exists a unique u ∈ H2(Ω) ∩ H1

0 (Ω) solving

aα : D2u = f in Ω, with u = 0 on ∂Ω,

ExampleIf dimension d = 2, (Cordes0) ⇐⇒ uniform ellipticity.

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Cordes condition: Case 2: extension to bα 6= 0 and cα 6= 0

Assume that there exist λ > 0 and ε ∈ (0, 1] s. t.

|aα|2 + |bα|2/2λ+ (cα/λ)2

(Tr aα + cα/λ)2≤ 1

d + εin Ω, ∀α ∈ Λ. (Cordes1)

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2. PDE theory: well-posedness

Theorem (Strong solutions of HJB equations with Cordescoefficients)Let Ω be a bounded convex open subset of Rd , and let Λ be a compactmetric space.

Let the data be continuous on Ω× Λ, and satisfy (Cordes1) with uniformlyelliptic aα and cα ≥ 0 for all α ∈ Λ.

Then, there exists a unique u ∈ H2(Ω) ∩ H10 (Ω) that solves (Elliptic HJB)

pointwise a.e. in Ω.

I. S. & E. Suli, SIAM J. Numer. Anal. 2014:

Discontinuous Galerkin finite element approximation of

Hamilton–Jacobi–Bellman equations with Cordes coefficients.

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2. PDE theory: proof of well-posedness

Define

γα :=Tr aα + cα/λ

|aα|2 + |bα|2/2λ+ (cα/λ)2

Fγ [u] := supα∈Λ

[γα(Lαu − f α)]

Because γα > 0, we can renormalize the operator:

Fγ [u] = supα∈Λ

[γα(Lαu − f α)] = 0 ⇐⇒ supα∈Λ

[Lαu − f α] = 0. (3)

The problem (Elliptic HJB) for u ∈ H2(Ω) ∩ H10 (Ω) is equivalent to

A(u; v) :=

∫Ω

Fγ [u] Lλv dx = 0 ∀ v ∈ H2(Ω) ∩ H10 (Ω), (4)

where Lλv := ∆v − λv .

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Let H := H2(Ω) ∩ H10 (Ω), ‖v‖2

H := ‖D2u‖2 + 2λ‖∇u‖2 + λ2‖u‖2

1. A : H × H → R is linear in its second argument (only).

2. A is Lipschitz continuous: there is C > 0 such that

|A(u; v)−A(w , v)| ≤ C‖u − w‖H‖v‖H ∀u, v , w ∈ H,

3. We now show that A is strongly monotone1 : there exists a positiveconstant c = c(ε) > 0 such that

1

c‖u − v‖2

H ≤ A(u; u − v)−A(v ; u − v) ∀ u, v ∈ H. (5)

On verifying these conditions, we conclude that there exists a unique u ∈ Hthat solves A(u; v) = 0 for all v ∈ H and thus solves (Elliptic HJB).

1Remark: must not confuse monotone schemes (i.e. discrete max principle)with strongly monotone operators in functional analytic sense.

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Notation: H := H2(Ω) ∩ H10 (Ω), ‖v‖2

H := ‖D2u‖2 + 2λ‖∇u‖2 + λ2‖u‖2

Key ingredients

1. The Cordes condition, which implies by direct calculation that

‖Fγ [u]− Fγ [v ]− Lλ(u − v)‖L2 ≤√

1− ε‖u − v‖H (6)

2. Miranda–Talenti: for convex Ω,

‖w‖H ≤ ‖Lλw‖L2 ∀w ∈ H2(Ω) ∩ H10 (Ω) (7)

where Lλv := ∆v − λv (recall λ > 0)

Maugeri, Palagachev & Softova, 2000, and Grisvard 1985

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‖Fγ [u]− Fγ [v ]− Lλ(u − v)‖L2 ≤√

1− ε‖u − v‖H , ‖v‖H ≤ ‖Lλv‖L2

Strong monotonicity:Recall A(u; v) =

∫ΩFγ [u]Lλvdx .

A(u; u − v)−A(v ; u − v) =

∫Ω

(Fγ [u]− Fγ [v ]) Lλ(u − v)dx .

Addition–subtraction of ‖Lλ(u − v)‖2L2 gives

A(u; u − v)−A(v ; u − v) = ‖Lλ(u − v)‖2L2

+

∫Ω

(Fγ [u]− Fγ [v ]− Lλ(u − v)) Lλ(u − v)dx︸︷︷︸≥−√1−ε‖u−v‖H‖Lλ(u−v)‖L2≥−

√1−ε‖Lλ(u−v)‖2

L2

Therefore

A(u; u−v)−A(v ; u−v) ≥ (1−√

1− ε)‖Lλ(u−v)‖2L2 ≥ (1−

√1− ε)‖u−v‖2

H

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‖Fγ [u]− Fγ [v ]− Lλ(u − v)‖L2 ≤√

1− ε‖u − v‖H , ‖v‖H ≤ ‖Lλv‖L2



A(u; u − v)−A(v ; u − v) =

∫Ω

(Fγ [u]− Fγ [v ]) Lλ(u − v)dx .


A(u; u − v)−A(v ; u − v) = ‖Lλ(u − v)‖2L2

+

∫Ω


√1−ε‖Lλ(u−v)‖2

L2

Therefore

A(u; u−v)−A(v ; u−v) ≥ (1−√

1− ε)‖Lλ(u−v)‖2L2 ≥ (1−

√1− ε)‖u−v‖2

H

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‖Fγ [u]− Fγ [v ]− Lλ(u − v)‖L2 ≤√

1− ε‖u − v‖H , ‖v‖H ≤ ‖Lλv‖L2



A(u; u − v)−A(v ; u − v) =

∫Ω

(Fγ [u]− Fγ [v ]) Lλ(u − v)dx .


A(u; u − v)−A(v ; u − v) = ‖Lλ(u − v)‖2L2

+

∫Ω


√1−ε‖Lλ(u−v)‖2

L2

Therefore

A(u; u−v)−A(v ; u−v) ≥ (1−√

1− ε)‖Lλ(u−v)‖2L2 ≥ (1−

√1− ε)‖u−v‖2

H

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2. PDE theory

Approach to numerical analysis:

Since the proof of well-posedness hinges on the strong monotonicity of

A(u; v) =

∫Ω

Fγ [u]Lλvdx ,

we will attempt to discretise the operator A and conserve its strongmonotonicity.

• The Cordes condition carries over straightforwardly to discrete setting

• The Miranda–Talenti inequality does not carry over if the approximationspace is not inside H2(Ω) ∩ H1

0 (Ω).

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Overview

Talk outline




Design of a consistent, stable and convergent method Error bounds Extension to parabolic problems Numerical experiments

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3. Numerics: design of the method

Let Thh a shape-regular sequence of meshes on Ω.

• Elements composing the mesh can be parallelepipeds, simplices, or moregenerally any combination of standard elements.

• The mesh is not assumed to be quasi-uniform (very useful for hp-refinement).

• Hanging nodes allowed.

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Construction of the discontinuous finite element space

Discontinuous finite element space:

Vh,p := vh ∈ L2(Ω): vh|K ∈ PpK (K ) ∀K ∈ Th.

Polynomial degrees p = (pK )K∈Th

Approximation in H2 requires pK ≥ 2 for all elements K .

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Notation of discontinuous Galerkin methods:

F nFKintKext

Distinguish interior and boundary faces

F ih interior faces of Th, Fb

h boundary faces of Th,

F i,bh := F i

h ∪ Fbh .

Jump operators over faces:

JφK := τF (φ|Kext)− τF (φ|Kint) , φ := 12τF (φ|Kext) + 1

2τF (φ|Kint) , if F ∈ F i

h,

JφK := τF (φ|Kext) , φ := τF (φ|Kext) , if F ∈ Fbh ,

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Notation of discontinuous Galerkin methods:

F nFKintKext

Let tid−1i=1 ⊂ Rd be an orthonormal coordinate system on F . Define the

tangential gradient and divergence

∇T u :=d−1∑i=1

ti∂u

∂ti, divT v :=

d−1∑i=1

∂vi∂ti

. (8)

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The goal is to discretise

A(u; v) =

∫Ω

Fγ [u] Lλv dx ,

whilst conserving the strong monotonicity bound.

Recall main ingredients:

1. The Cordes condition → remains unchanged in discrete setting.

2. Miranda–Talenti inequality: not conserved when replacingH2(Ω) ∩ H1

0 (Ω) by Vh,p.

Our approach:

• Miranda–Talenti inequality was derived from an integration by partsidentity (Maugeri et al 2000, Grisvard 1984)

• We will include a discrete weak form of this identity in the scheme (nextslide)

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Numerical scheme: find uh ∈ Vh,p such that

Ah(uh; vh) = 0 ∀ vh ∈ Vh,p. (scheme)

Ah(uh; vh) :=∑K∈Th

〈Fγ [uh], Lλvh〉K + Jh(uh, vh)

+1

2

(Bh(uh, vh)−

∑K∈Th

〈Lλuh, Lλvh〉K).

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+1

2

(Bh(uh, vh)−

∑K∈Th


〈Fγ [uh], Lλvh〉K :=

∫K

supα∈Λ

[γα(Lαuh − f α)] (∆vh − λvh) dx .

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+1

2

(Bh(uh, vh)−

∑K∈Th


Jump penalisation with µF ' p2K/hK and ηF ' p4

K/h3K for F ⊂ ∂K :

Jh(uh, vh) :=∑

F∈F i,bh

[µF 〈J∇T uhK, J∇T vhK〉F + ηF 〈JuhK, JvhK〉F

]+∑F∈F i

h

µF 〈J∇uh · nF K, J∇vh · nF K〉F .

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+1

2

(Bh(uh, vh)−

∑K∈Th


〈Lλuh, Lλvh〉K :=

∫K

(∆uh − λuh) (∆vh − λvh) dx .

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+1

2

(Bh(uh, vh)−

∑K∈Th


Bh(uh, vh) :=∑K∈Th

[〈D2uh,D

2vh〉K + 2λ〈∇uh,∇vh〉K + λ2〈uh, vh〉K]

+∑F∈F i

h

[〈divT∇Tuh, J∇vh · nF K〉F + 〈divT∇Tvh, J∇uh · nF K〉F

]−∑

F∈F i,bh

[〈∇T∇uh · nF, J∇T vhK〉F + 〈∇T∇vh · nF, J∇T uhK〉F

]−λ∑

F∈F i,bh

[〈∇uh·nF ,JvhK〉F +〈∇vh·nF ,JuhK〉F ]−λ∑F∈F i

h[〈uh,J∇vh·nF K〉F +〈vh,J∇uh·nF K〉F ]

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+1

2

(Bh(uh, vh)−

∑K∈Th


Key consistency result: If u ∈ H2(Ω) ∩ H10 (Ω) has well-defined second

derivative traces on faces F of the mesh, then

Bh(u, vh) =∑K

〈Lλu, Lλvh〉K , Jh(u, vh) = 0 ∀ vh ∈ Vh,p.

Technical point: a sufficient condition is that u ∈ Hs(K) with s > 5/2 for every

K ∈ Th.

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3. Numerics: consistency, stability and error bounds



Full theoretical justification given in [S. & Suli, SINUM 2014]:

• Consistency Theorem: sufficiently regular solution of (Elliptic HJB)solves:

Ah(u; vh) = 0 ∀ vh ∈ Vh,p.

• Discrete Stability Theorem: Existence & uniqueness of numerical solutionsince the nonlinear form Ah is strongly monotone: provided µF & p2/hand ηF & p2/h

‖uh − vh‖2h . Ah(uh; uh − vh)− Ah(vh; uh − vh) ∀ uh, vh ∈ Vh,p,

where

‖vh‖2h :=

∑K∈Th

[|vh|2H2(K) + 2λ|vh|2H1(K) + λ2‖vh‖2

L2(K)

]+ Jh(vh, vh)

• Consistency+Stability =⇒ error bounds and convergence.

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3. Numerics: error bounds

‖vh‖2h :=

∑K∈Th

[|vh|2H2(K) + 2λ|vh|2H1(K) + λ2‖vh‖2

L2(K)

]+ Jh(vh, vh).

Theorem (High-order convergence rates)(Under previous assumptions & standard assumptions for DG meshes...)

Assume that u ∈ Hs(Ω; Th), with sK > 5/2 for all K ∈ Th.

‖u − uh‖2h .

∑K∈Th

[htK−2K

psK−5/2K

‖u‖HsK (K)

]2

,

where tK = min(pK + 1, sK ) for each K ∈ Th.

Simplified form:

‖u−uh‖h .hmin(s,p+1)−2

ps−5/2‖u‖Hs (Ω).

• Optimal in h, half-order subopt. in p

• High-order convergence rates.

• Higher efficiency on well-chosen meshesand hp-refinement.

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3. Numerics: error bounds

If u has only minimal regularity, then we have the following quasi-optimalapproximation property with respect to the H2-conforming subspace:

Theorem (Minimal regularity error bound)Under previous assumptions. . .

Let u ∈ H2(Ω) ∩ H10 (Ω) be the solution of (Elliptic HJB). Then

‖u − uh‖h ≤ infzh∈Vh,p∩H2(Ω)∩H1

0 (Ω)‖u − zh‖h.

Note however that DG method requires only quadratic polynomials, whereasH2-conforming methods may require higher (e.g. Argyris elements requirequintic polynomials).

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3. Numerics: extensions to parabolic problems

S. & Suli, Num. Math. 2016: extension to parabolic HJB equations

• Generalisation of the Cordes condition and the PDE theory: existenceand uniqueness of the strong solutionu ∈ L2(0,T ;H2(Ω) ∩ H1

0 (Ω)) ∩ H1(0,T ; L2(Ω)).

• Numerical scheme: hp-τq-version space-time DGFEM using tensorproduct of Vh,p with piecewise polynomials in time.

• Stability, consistency and convergence rates that are:

h-optimal,

p-suboptimal by p3/2,

τ -optimal,

q-optimal.

• Exponential convergence rates under hp-τq refinement verifiedexperimentally.

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3. Numerics: experiment 1: h-refinement

Experiment 1 : Test of high order convergence rates under h-refinement,fixed p.

Example (Control of correlated diffusions)

aα :=1

2R>(

1 + sin2 θ sin θ cos θsin θ cos θ cos2 θ

)R

α := (θ,R) ∈ [0, π3 ]× SO(2) =: Λ.

Remark: aα becomes increasingly anisotropic as θ → π/3; rotation matricesR ∈ SO(2) prevent monotone schemes from aligning the grid with theanisotropy.

28/37

3. Numerics: experiment 1: h-refinement

Example (Control of correlated diffusions)Uniform h-refinement on smooth solution u(x , y) = exp(xy) sin(πx) sin(πy):

1/21/41/81/161/321/6410−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

h

h2

h3

h4

Mesh size

‖u−

uh‖ H

2(Ω

;Th

)

p = 2

p = 3

p = 4

p = 5

29/37

3. Numerics: experiment 2: hp-refinement

Experiment 2: test of exponential convergence rates under hp-refinement

Example (Strong anisotropy + boundary layer)Let Ω = (0, 1)2, bα ≡ (0, 1), cα ≡ 10 and define

aα := α>(

20 11 0.1

)α, α ∈ Λ := SO(2), λ =

1

2.

(Cordes1) holds with ε ≈ 0.0024 and λ = 1/2. Choose solution:

u(x , y) = (2x − 1)(e1−|2x−1| − 1

)(y +

1− ey/δ

e1/δ − 1

), δ := 0.005 = O(ε)

• Near-degenerate and anisotropic diffusion.

• Sharp boundary layer.

• Non-smooth solution.

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3. Numerics: experiment 2: hp-refinement

Example (Strong anisotropy + boundary layer)We use boundary layer adapted meshes with p-refinement: 2 ≤ pK ≤ 10,from 100 to 1320 DoFs.

Boundary layer adaptedmesh. 5 6 7 8 9 10 11

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

3√

DoF

Rel

ativ

eer

ror

Broken H2 norm

Exponential rate: ‖u − uh‖h . exp(−c 3√DoF).

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3. Numerics: Experiment 3: linearization and algebraic solvers

Solution of nonlinear equation by a superlinearly convergent semismoothNewton method. [S. & Suli, SINUM 2014, Sect. 8]

1 2 3 4 5 6 7Converged

10−12

10−8

10−4

1

Iteration number k

‖uh−uk h‖ H

2(Ω

;Th

)

h = 1/4

h = 1/8

h = 1/16

h = 1/32

h = 1/64

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3. Numerics: experiment 3: linearization and algebraic solvers

Nonoverlapping domain decomposition preconditioners with GMRES: (alltolerances 10−6 in discrete H2-type norm)

Average GMRES iterations (Newton steps)

4 Subdomains 16 SubdomainsDoF h H = 2h H = 4h H = 8h H = 2h H = 4h H = 8h

144 1/4 14.3 (6)576 1/8 15.2 (5) 18.8 (5) 17.8 (5)

2304 1/16 15.4 (5) 20.0 (5) 26.8 (5) 18.0 (5) 25.0 (5)9216 1/32 16.3 (6) 19.7 (6) 29.5 (6) 17.3 (6) 24.0 (6) 36.5 (6)

36864 1/64 16.0 (6) 18.3 (6) 26.3 (6) 17.2 (6) 22.0 (6) 32.8 (6)147456 1/128 16.3 (6) 18.3 (6) 23.0 (6) 17.0 (6) 19.8 (6) 28.0 (6)

h p = 2 p = 3 p = 4 p = 5

1/4 18 21 21 221/8 19 20 19 20

1/16 19 19 19 191/32 18 19 17 181/64 17 19 16 17

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3. Numerics: experiment 4: Parabolic

Example (Strongly anisotropic parabolic problem)Let Ω = (0, 1)2, I = (0, 1), Λ = SO(2),

aα := α

(1 1/40

1/40 1/800

)α>, α ∈ Λ.

For ω = 1, Cords condition holds with ε ≈ 1.25× 10−3.

Solution u = (1− e−t) exp(xy) sin(πx) sin(πy).

Uniform refinement with q = p − 1, h ' τ .

Remark (Monotone FDM)Consistency requires (at least) stencil width ≥ 20, with more than 1529stencil points.

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3. Numerics: experiment 4: Parabolic

1/21/41/81/161/32

10−6

10−5

10−4

10−3

10−2

10−1 h

h2

h3

Mesh size h ' τ

|||u−uh|||

h|||u|||

h

p = 2

p = 3

p = 4

1/21/41/81/161/3210−8

10−6

10−4

10−2 h2

h3

h4

Mesh size h ' τ

‖u(T

)−uh(T

)‖H

1(Ω

;Th)

‖u(T

)‖H

1(Ω

;Th)

p = 2

p = 3

p = 4

|||v |||2h :=N∑

n=1

∫In

∑K∈Th

[ω2‖∂tv‖2

L2(K) + ‖v‖2H2(K)

]dt.

35/37

Summary and outlook


• For equations with Cordes coefficients as presented here

Consistency & Stability of non-conforming discretisations Convergence rates for sufficiently regular solutions Non-structured meshes, varying polynomial degrees, etc.

36/37

References

Linear nondivergence form PDE: Discontinuous Galerkin finite elementapproximation of nondivergence form elliptic equations with Cordescoefficients,

I. S. & E. Suli, SIAM J. Numer. Anal. 2013.

Elliptic HJB: Discontinuous Galerkin finite element approximation ofHamilton–Jacobi–Bellman equations with Cordes coefficients,

I. S. & E. Suli, SIAM J. Numer. Anal. 2014.

Parabolic HJB: Discontinuous Galerkin finite element methods for

time-dependent Hamilton–Jacobi–Bellman equations with Cordes coefficients,

I. S. & E. Suli, Numerische Mathematik 2016.

Solvers: Nonoverlapping Domain Decomposition Preconditioners forDiscontinuous Galerkin Approximations of Hamilton–Jacobi–BellmanEquations,

I. S., Journal of Scientific Computing, 2017.

Thank you!

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