A DPG method for the Schr odinger equation Jay Gopalakrishnan - - PowerPoint PPT Presentation

A DPG method for the Schr¨

• dinger equation

Jay Gopalakrishnan

Portland State University

Collaborators: L. Demkowicz, S. Nagaraj, P. Sep´ ulveda

RICAM Workshop on spacetime methods November 2016 Thanks:

AFOSR, NSF, RICAM

Jay Gopalakrishnan 1/28

“Petrov-Galerkin” schemes (PG)

PG schemes are distinguished by different trial and test (Hilbert) spaces. The problem:

• P.D.E.+

boundary conditions. ↓ Variational form:    Find x in a trial space X satisfying b(x, y) = ℓ(y) for all y in a test space Y. ↓ Discretization:    Find xh in a discrete trial space Xh ⊂ X satisfying b(xh, yh) = ℓ(yh) for all yh in a discrete test space Yh ⊂ Y . For PG schemes, Xh = Yh in general.

Jay Gopalakrishnan 2/28

Elements of theory

Variational formulation:    Exact inf-sup condition CxX ≤ sup

y∈Y

|b(x, y)| yY   +

• a uniqueness

condition

• =

⇒ wellposedness Babuˇ ska’s theorem:    Discrete inf-sup condition CxhX ≤ sup

yh∈Yh

|b(xh, yh)| yhY    = ⇒ x − xhX ≤ C inf

wh∈Xh

x − whX. Difficulty: Exact inf-sup condition

• =

⇒ Discrete inf-sup condition

Jay Gopalakrishnan 3/28

Elements of theory

Variational formulation:    Exact inf-sup condition CxX ≤ sup

y∈Y

|b(x, y)| yY   +

• a uniqueness

condition

• =

⇒ wellposedness Babuˇ ska’s theorem:    Discrete inf-sup condition CxhX ≤ sup

yh∈Yh

|b(xh, yh)| yhY    = ⇒ x − xhX ≤ C inf

wh∈Xh

x − whX. Difficulty: Exact inf-sup condition

• =

⇒ Discrete inf-sup condition Is there a way to find a stable test space for any given trial space?

Jay Gopalakrishnan 3/28

The ideal DPG method

Pick any Xh ⊆ X. The ideal DPG method finds xh ∈ Xh such that b(xh, y) = ℓ(y), ∀y ∈ Y opt

h def

= T(Xh), where T : X → Y is defined by (Tw, y)Y = b(w, y), ∀w ∈ X, y ∈ Y . [Demkowicz+G 2011] Rationale:

Jay Gopalakrishnan 4/28

The ideal DPG method

Pick any Xh ⊆ X. The ideal DPG method finds xh ∈ Xh such that b(xh, y) = ℓ(y), ∀y ∈ Y opt

h def

= T(Xh), where T : X → Y is defined by (Tw, y)Y = b(w, y), ∀w ∈ X, y ∈ Y . [Demkowicz+G 2011] Rationale: Q: Which function y maximizes |b(x, y)| yY for any given x ? A: y = Tx is the maximizer. ← Optimal test function. DPG Idea: If the discrete test space contains the optimal test functions, exact inf-sup condition = ⇒ discrete inf-sup condition.

Jay Gopalakrishnan 4/28

The ideal DPG method

Pick any Xh ⊆ X. The ideal DPG method finds xh ∈ Xh such that b(xh, y) = ℓ(y), ∀y ∈ Y opt

h def

= T(Xh), where T : X → Y is defined by (Tw, y)Y = b(w, y), ∀w ∈ X, y ∈ Y . It is important to (re)formulate the problem so that Y admits DG functions.

Jay Gopalakrishnan 4/28

Quasi-Optimality of DPG Methods

Assumption [U] (Uniqueness) {w ∈ X : b(w, y) = 0 ∀y ∈ Y } = {0}. Assumption [I] (Inf-Sup & Continuity) ∃ C1, C2 > 0 such that C1yY ≤ sup

w∈X

|b(w, y)| wX ≤ C2yY . Theorem [QO] (Quasi-Optimality) Assumptions [U+I] = ⇒ x − xhX ≤ C2 C1 inf

wh∈Xh

x − whX. [Demkowicz+G 2011]

Jay Gopalakrishnan 5/28

Next

Quick introduction to DPG methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ Unbounded operator equations Broken weak form Heat equation vs. Schr¨

• dinger equation

A density result DPG method for the Schr¨

• dinger equation

Jay Gopalakrishnan 6/28

An unbounded operator setting

We are interested in numerically solving operator equations Au = f where A : L2(Ω)m → L2(Ω)l is an (unbounded) operator satisfying [Au]i = ∂α(aijαuj), aijα : Ω → C are bounded functions ∀i = 1, . . . l, ∀j = 1, . . . , m, ∀multi-indices |α| ≤ k, D(Ω)m ⊆ dom A.

Jay Gopalakrishnan 7/28

Example: Helmholtz equation

Helmholtz equation: −∆φ − ω2φ = ˆ ıωf , in Ω ⊆ RN. First order reformulation: ˆ ıωv + grad φ = 0, in Ω ˆ ıωφ + div v = f , in Ω.

Jay Gopalakrishnan 8/28

Example: Helmholtz equation

Helmholtz equation: −∆φ − ω2φ = ˆ ıωf , in Ω ⊆ RN. First order reformulation: ˆ ıωv + grad φ = 0, in Ω ˆ ıωφ + div v = f , in Ω. Fit to the general structure: Au = A v φ

• =

ˆ ıωv + grad φ ˆ ıωφ + div v

• = ˆ

ıω v φ

• +

grad div v φ

• = ˆ

ıωu + ∂1 e1 et

1

• u
• + · · · + ∂N

eN et

N

• u
• = ∂α(aijαuj)

Jay Gopalakrishnan 8/28

The Adjoint

The adjoint of A is a closed (unbounded) operator A∗ : L2(Ω)l → L2(Ω)m whose domain is dom A∗ =

• s ∈ L2(Ω)l

: ∃ ℓ ∈ L2(Ω)m such that (Av, s)Ω = (v, ℓ)Ω ∀v ∈ dom(A)

• ,

and satisfies (Av, v∗)Ω = (v, A∗v∗)Ω, ∀v ∈ dom A, v∗ ∈ dom A∗. Assume that A∗u is a distribution for all u ∈ L2(Ω)l.

Jay Gopalakrishnan 9/28

Graph spaces

These are the graph spaces of A and A∗: W (Ω) = {u ∈ L2(Ω)m : Au ∈ L2(Ω)l}, W ∗(Ω) = {u ∈ L2(Ω)l : A∗u ∈ L2(Ω)m} When solving a PDE Au = f in the graph space, we incorporate boundary conditions in dom A. V = dom A in W (Ω)-topology, V ∗ = dom A∗ in W ∗(Ω)-topology. Elements of the modern theory of Friedrichs systems [Ern+Guermond+Caplain 2007] can be generalized to this setting.

Jay Gopalakrishnan 10/28

Boundary Operator

Define D : W (Ω) → W ∗(Ω)′ and D∗ : W ∗(Ω) → W (Ω)′ by Dw, w∗W ∗(Ω) = (Aw, w∗)Ω − (w, A∗w∗)Ω, D∗w∗, wW (Ω) = (A∗w∗, w)Ω − (w∗, Aw)Ω, for all w ∈ W (Ω) and w∗ ∈ W ∗(Ω). Lemma [DA] (Domain of the Adjoint) V ∗ = ⊥D(V ) = {w∗ ∈ W ∗(Ω) : Dv, w∗ = 0 ∀v ∈ V }.

Jay Gopalakrishnan 11/28

Example: Helmholtz equation

Recall the operator: Au = A v φ

• =

ˆ ıωv + grad φ ˆ ıωφ + div v

• Add impedance boundary condition:

v · n − φ = 0,

• n ∂Ω.

Incorporating the boundary condition into domain, dom A = V = v φ

• ∈ H(div, Ω) × H1(Ω) :

(v.n − φ)

• ∂Ω = 0
• .

Jay Gopalakrishnan 12/28

Example: Helmholtz equation

Recall the operator: Au = A v φ

• =

ˆ ıωv + grad φ ˆ ıωφ + div v

• Add impedance boundary condition:

v · n − φ = 0,

• n ∂Ω.

Incorporating the boundary condition into domain, dom A = V = v φ

• ∈ H(div, Ω) × H1(Ω) :

(v.n − φ)

• ∂Ω = 0
• .

What is D? For smooth functions w, v, ψ, φ, D w ψ

• ,

v φ

• =
• ∂Ω

w · n ¯ φ + ψ v · n.

Jay Gopalakrishnan 12/28

Example: Helmholtz equation

Recall the operator: Au = A v φ

• =

ˆ ıωv + grad φ ˆ ıωφ + div v

• Add impedance boundary condition:

v · n − φ = 0,

• n ∂Ω.

Incorporating the boundary condition into domain, dom A = V = v φ

• ∈ H(div, Ω) × H1(Ω) :

(v.n − φ)

• ∂Ω = 0
• .

What is D? For smooth functions w, v, ψ, φ, D w ψ

• ,

v φ

• =
• ∂Ω

w · n ¯ φ + ψ v · n. What is dom A∗, or V ∗? Lemma [DA] = ⇒ V ∗ = ⊥D(V ) = w ψ

• ∈ H(div, Ω) × H1(Ω) :

(w.n + ψ)

• ∂Ω = 0
• .

Jay Gopalakrishnan 12/28

Next

Quick introduction to DPG methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ Unbounded operator equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✦ Broken weak form Heat equation vs. Schr¨

• dinger equation

A density result DPG method for the Schr¨

• dinger equation

Jay Gopalakrishnan 13/28

Strong & Weak forms

Classical operator form Given f ∈ L2(Ω)l, find u ∈ V satisfying Au = f . Strong Petrov-Galerkin form Find u ∈ V satisfying (Au, v)Ω = (f , v) ∀v ∈ L2(Ω)l. Weak Petrov-Galerkin form (Unbroken) Find u ∈ L2(Ω)m satisfying (u, A∗v)Ω = (f , v)Ω ∀v ∈ V ∗. Au = f = ⇒ (u, A∗v)Ω + Du, v

Lemma [DA] = ⇒ 0

= (f , v)Ω

Jay Gopalakrishnan 14/28

Strong & Weak forms

Classical operator form Given f ∈ L2(Ω)l, find u ∈ V satisfying Au = f . Strong Petrov-Galerkin form Find u ∈ V satisfying (Au, v)Ω = (f , v) ∀v ∈ L2(Ω)l. Weak Petrov-Galerkin form (Unbroken) Find u ∈ L2(Ω)m satisfying (u, A∗v)Ω = (f , v)Ω ∀v ∈ V ∗. This form is not amenable to DPG discretization!

Jay Gopalakrishnan 14/28

Broken spaces

We use a mesh Ωh of Ω to “break” the spaces and weak form. Element-by-element graph space: Wh =

• K∈Ωh

W (K), W ∗

h =

• K∈Ωh

W ∗(K) Element-by-element differential operator: Ah : Wh → L2(Ω)m, A∗

h : W ∗ h → L2(Ω)l.

Element interface operator: Dh : Wh → (W ∗

h )′ is defined by

Dhu, vh =

• K∈Ωh

K

Au · ¯ v −

• K

u · A∗v

• ≡ (Ahu, v)h − (u, A∗

hv)h

Restriction of Dh to V : Dh,V = Dh|V : V → (W ∗

h )′

Jay Gopalakrishnan 15/28

Broken weak form

Derivation: Au = f = ⇒ (Au, v)Ω = (Ahu, v)h = (u, A∗

hv)h + Dhu

• q

, vh Space of interface variables Q: Define Q = ran(Dh,V ) normed by qQ = inf

v∈D−1

h,V {q}

vW (Ω). Broken Weak Formulation Find u ∈ L2(Ω)m and q ∈ Q satisfying (u, A∗

hv)h + q, vh = F(v)

∀v ∈ W ∗

h .

Jay Gopalakrishnan 16/28

Wellposedness

Theorem [WB] (Wellposedness of Broken Weak Form) Suppose V = ⊥D∗(V ∗), and A : V → L2(Ω)l is a bijection. Then, for any F ∈ (W ∗

h )′, there is a unique u ∈ L2(Ω)m and q ∈ Q,

depending continuously on F and satisfying the Broken Weak Formulation (u, A∗

hv)h + q, vh = F(v)

∀v ∈ W ∗

h .

Jay Gopalakrishnan 17/28

Wellposedness

Theorem [WB] (Wellposedness of Broken Weak Form) Suppose V = ⊥D∗(V ∗), and A : V → L2(Ω)l is a bijection. Then, for any F ∈ (W ∗

h )′, there is a unique u ∈ L2(Ω)m and q ∈ Q,

depending continuously on F and satisfying the Broken Weak Formulation (u, A∗

hv)h + q, vh = F(v)

∀v ∈ W ∗

h .

The choice of the domain of A is constrained by the requirement V = ⊥D∗(V ∗) .

Jay Gopalakrishnan 17/28

Wellposedness

Theorem [WB] (Wellposedness of Broken Weak Form) Suppose V = ⊥D∗(V ∗), and A : V → L2(Ω)l is a bijection. Then, for any F ∈ (W ∗

h )′, there is a unique u ∈ L2(Ω)m and q ∈ Q,

depending continuously on F and satisfying the Broken Weak Formulation (u, A∗

hv)h + q, vh = F(v)

∀v ∈ W ∗

h .

The choice of the domain of A is constrained by the requirement V = ⊥D∗(V ∗) . Wellposedness of Theorem [WB] ⇐ ⇒ [U+I] holds. Hence, the DPG method can now be applied to the Broken Weak Form b( (u, q), v) = (u, A∗

hv)h + q, vh.

Jay Gopalakrishnan 17/28

Next

Quick introduction to DPG methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ Unbounded operator equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✦ Broken weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✦ Heat equation vs. Schr¨

• dinger equation

A density result DPG method for the Schr¨

• dinger equation

Jay Gopalakrishnan 18/28

Example: Heat equation

Heat equation: ∂tφ − ∆xφ = f φ|Γ = 0 First order reformulation: ∂tφ − divxq = f q − gradxφ = 0 L T Γ Γ Γ Ω x t

Jay Gopalakrishnan 19/28

Example: Heat equation

Heat equation: ∂tφ − ∆xφ = f φ|Γ = 0 First order reformulation: ∂tφ − divxq = f q − gradxφ = 0 L T Γ Γ Γ Ω x t Fit to abstract structure: Au = A q φ

• =

q − gradxφ ∂tφ − divxq

• =

q ∂tφ

• −

gradx divx q φ

• =

I

• u − ∂x1

e1 et

1

• u
• · · · − ∂xN

eN et

N

• u
• + ∂t

1

• u
• = ∂α(aijαuj)

Jay Gopalakrishnan 19/28

Example: The Schr¨

• dinger equation

Schr¨

• dinger equation:

ˆ ı∂tφ − ∆xφ = f φΓ = 0 First order reformulation ? ˆ ı∂tφ − divxq = f q − gradxφ = 0 L T Γ Γ Γ Ω x t This first order system is not equivalent to the original Schr¨

• dinger system!

Jay Gopalakrishnan 20/28

Irregular spacetime solutions

There are spacetime Schr¨

• dinger solutions φ with gradxφΩ = ∞ :

1 On the one-space-dimensional domain Ω0 = (0, 1), let ϕk denote the

L2-normalized Dirichlet eigenfunction of eigenvalue ωk = kπ.

2 Put

f (x, t) =

∞

• k=1

1 k eiω2

ktϕk(x)

in L2(Ω).

Jay Gopalakrishnan 21/28

Irregular spacetime solutions

There are spacetime Schr¨

• dinger solutions φ with gradxφΩ = ∞ :

1 On the one-space-dimensional domain Ω0 = (0, 1), let ϕk denote the

L2-normalized Dirichlet eigenfunction of eigenvalue ωk = kπ.

2 Put

f (x, t) =

∞

• k=1

1 k eiω2

ktϕk(x)

in L2(Ω).

3 Solve ˆ

ı∂tφ − ∆xφ = f in terms of ϕk: φ =

∞

• k=1

−it k eiω2

ktϕk(x), Jay Gopalakrishnan 21/28

Irregular spacetime solutions

There are spacetime Schr¨

• dinger solutions φ with gradxφΩ = ∞ :

1 On the one-space-dimensional domain Ω0 = (0, 1), let ϕk denote the

L2-normalized Dirichlet eigenfunction of eigenvalue ωk = kπ.

2 Put

f (x, t) =

∞

• k=1

1 k eiω2

ktϕk(x)

in L2(Ω).

3 Solve ˆ

ı∂tφ − ∆xφ = f in terms of ϕk: φ =

∞

• k=1

−it k eiω2

ktϕk(x), 4 Observe that the Mth partial sum of this series

• gradx

M

• k=1

−iteiω2

kt

k ϕk(x)

• 2

Ω

= T 3

M

• k=1

ω2

k

3k2 = π2 3 T 3M → ∞.

Jay Gopalakrishnan 21/28

Proceed in second order form

Schr¨

• dinger equation:

ˆ ı∂tφ − ∆xφ = f φ|Γ = 0 2nd order form fits general setting: Au = ˆ ı∂tu − ∆xu = ∂α(aijαuj) L T Γ Γ Γ Ω x t

Jay Gopalakrishnan 22/28

Proceed in second order form

Schr¨

• dinger equation:

ˆ ı∂tφ − ∆xφ = f φ|Γ = 0 2nd order form fits general setting: Au = ˆ ı∂tu − ∆xu = ∂α(aijαuj) L T Γ Γ Γ Ω x t How to incorporate the boundary condition into dom A? Unlike the Helmholtz example, here the dom A cannot be immediately specified using standard Sobolev spaces and known trace theory . . .

Jay Gopalakrishnan 22/28

Specifying the domain

1 Define

Γ ∗ = ∂Ω0 × [0, T] ∪ Ω0 × {T} Ω0 Γ ∗ Γ ∗ Γ ∗ x t Γ Γ Γ Ω x t

2 Set

V∗ = {ϕ ∈ D( ¯ Ω) : ϕ|Γ ∗ = 0}.

3 Define

dom A = V = ⊥D(V∗) .

Jay Gopalakrishnan 23/28

Summary of the Schr¨

• dinger example’s settings

A = ˆ ı∂t − ∆x. dom A = V = ⊥D(V∗) . A∗ = A. Lemma [DA] = ⇒ dom A∗ = V ∗ = ⊥D(V ) . D∗ = D. Moreover for smooth φ, ψ, Dφ, ψW =

• ∂Ω

intφ ¯ ψ +

• ∂Ω

φ(nx · ∇x ¯ ψ) −

• ∂Ω

(nx · ∇xφ) ¯ ψ.

Jay Gopalakrishnan 24/28

Summary of the Schr¨

• dinger example’s settings

A = ˆ ı∂t − ∆x. dom A = V = ⊥D(V∗) . A∗ = A. Lemma [DA] = ⇒ dom A∗ = V ∗ = ⊥D(V ) . D∗ = D. Moreover for smooth φ, ψ, Dφ, ψW =

• ∂Ω

intφ ¯ ψ +

• ∂Ω

φ(nx · ∇x ¯ ψ) −

• ∂Ω

(nx · ∇xφ) ¯ ψ. Recall: Theorem [WB] needs V = ⊥D∗(V ∗) . This follows if V∗ is dense in V ∗.

Jay Gopalakrishnan 24/28

Density result

Theorem [DS] (Density of Smooth functions) Let Ω = (0, L) × (0, T). Then V∗ is dense in V ∗. Sketch of the proof: Approximate a given w ∈ V ∗ in 3 steps:

t x

Ω

−L L 2L

• w(x, t)

Jay Gopalakrishnan 25/28

Density result

Theorem [DS] (Density of Smooth functions) Let Ω = (0, L) × (0, T). Then V∗ is dense in V ∗. Sketch of the proof: Approximate a given w ∈ V ∗ in 3 steps:

1 Extension: The odd extension Gw of w satisfies

AGw = GAw.

t x

Ω

−L L 2L

• −w(−x, t)

−w(2L − x, t)

w(x, t)

Jay Gopalakrishnan 25/28

Density result

Theorem [DS] (Density of Smooth functions) Let Ω = (0, L) × (0, T). Then V∗ is dense in V ∗. Sketch of the proof: Approximate a given w ∈ V ∗ in 3 steps:

1 Extension: The odd extension Gw of w satisfies

AGw = GAw.

2 Translate Gw in time by some small δ and extend by 0 everywhere.

t x

−L L 2L

• Jay Gopalakrishnan

25/28

Density result

Theorem [DS] (Density of Smooth functions) Let Ω = (0, L) × (0, T). Then V∗ is dense in V ∗. Sketch of the proof: Approximate a given w ∈ V ∗ in 3 steps:

1 Extension: The odd extension Gw of w satisfies

AGw = GAw.

2 Translate Gw in time by some small δ and extend by 0 everywhere. 3 Mollify the resulting function to get an approximation in V∗.

t x

−L L 2L

• Jay Gopalakrishnan

25/28

The method

Broken Weak Form: b( (u, q), v) = (u, A∗

hv)h + q, vh.

The interface term for piecewise polynomials takes the form q, vh =

• K∈Ωh
• ∂K

q+ (int¯ v) +

• ∂K

q+ (nx∂x¯ v) −

• ∂K

q (nx¯ v).

continuous piecewise polynomials piecewise polynomials (discont.)

The DPG method for the Schr¨

• dinger equation finds

(uh, qh) ∈ Uh × Qh ⊂ L2(Ω) × Q satisfying b( (uh, qh), v) = ℓ(v) ∀v ∈ Y opt

h

. It only remains to specify Uh × Qh.

Jay Gopalakrishnan 26/28

Trial spaces and convergence rate

p = 3 case − → uh, degree ≤ p − 1 in each variable, discontinuous − → q+

h , degree ≤ p on each edge, continuous

− → q

h, degree ≤ p on each edge, discontinuous.

Theorem (DPG error estimates for Schrodinger eq.) Suppose p ≥ 3 and Ωh is a mesh of square elements of size h. Then there is a constant C independent of h such that the DPG solution uh ∈ Uh and qh ∈ Qh satisfies u − uhΩ + q − qhQ ≤ Chr|u|Hr+2(Ωh) for 2 ≤ r ≤ p − 1 .

Jay Gopalakrishnan 27/28

Conclusion

Quick introduction to DPG methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ Unbounded operator equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✦ Broken weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .✦ Heat equation vs. Schr¨

• dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . ✦

A density result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✦ DPG method for the Schr¨

• dinger equation . . . . . . . . . . . . . . . . . . . . . . .✦

Reference: [Demkowicz+G+Nagaraj+Sep´

ulveda 2016]

ArXiV:1610.04678 A spacetime DPG method for the Schrodinger equation

Jay Gopalakrishnan 28/28