Discontinuous Petrov Galerkin (DPG)

Discontinuous Petrov Galerkin (DPG) Method

Leszek DemkowiczICES, The University of Texas at Austin

ETH Summer School on Wave PropagationZurich, August 22-26, 2016

Relevant Collaboration:

ICES: S. Nagaraj, S. Petrides

Portland State: J. Gopalakrishnan, N. Olivares, P. Sepulveda

Humboldt U: C. Carstensen

Vienna TH: J. Schoeberl

KIT: Ch. Wieners

ETH Zurich, August 2016 DPG Method 2 / 180

Outline

I Petrov-Galerkin Method with Optimal Test Functions.

I Variational formulations.

I “Breaking test functions”. Localization.

I Approximation of optimal test functions.

I Implementation. Numerical examples.

I Preconditioning.

I Space-time discretizations. Schrodinger equation.

I Relation with Barret-Morton optimal test functions and weakly conformingleast squares.


Outline






I Preconditioning.




Three interpretations


Abstract variational problem

U, V - Hilbert spaces,b(u, v) - bilinear (sesquilinear) continuous form on U × V ,

|b(u, v)| ≤ ‖b‖︸︷︷︸=:M

‖u‖U ‖v‖V ,

l(v) - linear (antilinear) continuous functional on V ,

|l(v)| ≤ ‖l‖V ′ ‖v‖V

The abstract variational problem:u ∈ Ub(u, v) = l(v) ∀v ∈ V ⇔ Bu = l B : U → V ′

< Bu, v >= b(u, v) v ∈ V


Banach - Babuska - Necas Theorem

If b satisfies the inf-sup condition (⇔ B is bounded below),

inf‖u‖U=1

sup‖v‖V =1

|b(u, v)| =: γ > 0 ⇔ supv∈V

|b(u, v)|‖v‖V

≥ γ‖u‖U

and l satisfies the compatibility condition:

l(v) = 0 ∀v ∈ V0

whereV0 := N (B′) = v ∈ V : b(u, v) = 0 ∀u ∈ U

then the variational problem has a unique solution u that satisfies the stabilityestimate:

‖u‖ ≤ 1

γ‖l‖V ′ .

Proof: Direct reinterpretation of Banach Closed Range Theorem ∗.

∗see e.g. Oden, D, Functional Analysis, Chapman & Hall, 2nd ed., 2010, p.518ETH Zurich, August 2016 DPG Method 7 / 180

Petrov-Galerkin Method and (improved) Babuska Theorem

Uh ⊂ U, Vh ⊂ V, dimUh = dimVh - finite-dimensional trial and test (sub)spacesuh ∈ Uhb(uh, vh) = l(vh), ∀vh ∈ Vh

Theorem (Babuska†).The discrete inf-sup condition

supvh∈Vh

|b(uh, vh)|‖vh‖V

≥ γh‖uh‖U

implies existence, uniqueness and discrete stability

‖uh‖U ≤ γ−1h ‖l‖V ′h

†I. Babuska, “Error-bounds for Finite Element Method.”, Numer. Math, 16, 1970/1971.ETH Zurich, August 2016 DPG Method 8 / 180

Petrov-Galerkin Method and (improved) Babuska Theorem

Uh ⊂ U, Vh ⊂ V, dimUh = dimVh - finite-dimensional trial and test (sub)spacesuh ∈ Uhb(uh, vh) = l(vh), ∀vh ∈ Vh

Theorem (Babuska†).The discrete inf-sup condition

supvh∈Vh


≥ γh‖uh‖U

implies existence, uniqueness and discrete stability

‖uh‖U ≤ γ−1h ‖l‖V ′h

and convergence

‖u− uh‖U ≤M

γhinf

wh∈Uh

‖u− wh‖U

(Uniform) discrete stability and approximability imply convergence.

†I. Babuska, “Error-bounds for Finite Element Method.”, Numer. Math, 16, 1970/1971.ETH Zurich, August 2016 DPG Method 8 / 180

Lemma (Del Pasqua, Ljance, Kato, Szyld) ‡

Let U, (·, ·) be a Hilbert space and P : U → U a linear projection, i.e. P 2 = P .Then

‖I − P‖ = ‖P‖Proof: Let X = R(P ) and Y = N (P ). It is well known that U = X ⊕ Y . Pickan arbitrary unit vector u ∈ U . Let u = x+ y, x ∈ X, y ∈ Y be the uniquedecomposition of u. By the properties of a scalar product,

1 = ‖u‖2 = (x+ y, x+ y) = ‖x‖2 + ‖y‖2 + 2Re(x, y) .

‡D.B. Szyld. The many proofs of an identity on the norm of oblique projections. Numer.Algor., 42:309–323, 2006.


Lemma (Del Pasqua, Ljance, Kato), cont.

Consider now a “symmetric image” w of u,

w = x+ y, x = ‖y‖ x

‖x‖ , y = ‖x‖ y

‖y‖ .

Vector w has a unit length as well. Indeed,

‖w‖2 = (x+y, x+y) = ‖x‖2+‖y‖2+2Re(x, y) = ‖y‖2+‖x‖2+2Re

(‖y‖‖x‖‖x‖‖y‖ (x, y)

)= 1 .

We have now,

‖Pu‖ = ‖x‖ = ‖y‖ = ‖(I − P )w‖ ≤ ‖I − P‖ ‖w‖ = ‖I − P‖

Taking supremum over ‖u‖ = 1 finishes the proof.ETH Zurich, August 2016 DPG Method 10 / 180

Proof of Babuska Theorem

Observe that the Petrov–Galerkin discretization executes a linear projectionPh : U → Uh, Phu = uh,

b(Phu− u, vh) = 0 ∀vh ∈ Vh .The stability estimate implies an estimate on the norm of the projection,

‖Phu‖U = ‖uh‖ ≤1

γh‖l‖V ′h =

1

γhsup‖vh‖=1

|b(u, vh)| ≤ M

γh‖u‖U .

We have then:

‖u− uh‖U = ‖(I − Ph)u‖U (definition of Ph)

= ‖(I − Ph)(u− wh)‖U (Phwh = wh ∀wh ∈ Vh)

≤ ‖I − Ph‖ ‖u− wh‖

= ‖Ph‖ ‖u− wh‖ (Lemma)

≤ Mγh‖u− wh‖ (‖Ph‖ ≤ M

γh) ,

and we conclude the proof by taking infimum over wh ∈ Uh.ETH Zurich, August 2016 DPG Method 11 / 180

Babuska Theorem, cont.

If γh admit a positive lower bound, i.e. a uniform discrete inf–sup condition holds,

infhγh =: γ0 > 0 ,

then,

‖u− uh‖︸︷︷︸approximation error

≤ M

γ0︸︷︷︸stability constant

infwh∈Uh

‖u− wh‖U︸︷︷︸best approximation error

.

i.e. the actual and the best approximation errors must converge at the same rate.The result has coined the famous phrase:

(Uniform) discrete stability and approximability imply convergence.


Optimal test functions

The main trouble:

continuous inf-sup condtion =⇒/ discrete inf-sup condition

supv∈V

|b(uh, v)|‖v‖V

≥ γ‖uh‖U =⇒/ supvh∈Vh


≥ γ‖uh‖U



The main trouble:


supv∈V

|b(uh, v)|‖v‖V



≥ γ‖uh‖U

unless §

§L.D., J. Gopalakrishnan. “A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions.”Numer. Meth. Part. D. E., 27,

70-105, 2011.



The main trouble:


supv∈V

|b(uh, v)|‖v‖V



≥ γ‖uh‖U

unless § we employ special test functions that realize the supremum:

vh = arg maxv∈V|b(uh, v)|‖v‖


70-105, 2011.



The main trouble:


supv∈V

|b(uh, v)|‖v‖V



≥ γ‖uh‖U

unless § we employ special test functions that realize the supremum:

vh = arg maxv∈V|b(uh, v)|‖v‖

Recall that the Riesz operator RV : V → V ′ is an isometry. Then:

supv|b(uh,v)|‖v‖ = ‖Buh‖V ′ = ‖R−1

V Buh︸︷︷︸=vh

‖V =(R−1

VBuh,vh)V‖vh‖V

= 〈Buh,vh〉‖vh‖V

= b(uh,vh)‖vh‖V

Variational definition of vh:

vh ∈ V(v, δv)V = b(uh, δv) ∀δv ∈ V .

The operator T := R−1V B : Uh → V will be called the trial to test operator.


70-105, 2011.


DPG is a Minimum Residual Method

With the optimal test functions in place, γh ≥ γ, and the Galerkin method isautomatically stable. Trade now the original norm in U for an energy norm¶:

‖u‖E := ‖R−1V Bu‖V = ‖Bu‖V ′ = sup

v∈V

|b(u, v)|‖v‖V

Two points:

I With respect to the new, energy norm, both continuity constant M and inf-supconstant γ are unity.

I The use of optimal test functions (their construction is independent of the choice oftrial norm) implies that γh ≥ γ = 1.

Thus, by the Babuska Theorem,

‖u− uh‖E ≤M

γh︸︷︷︸=1

infwh∈Uh

‖u− wh‖E .

In other words, FE solution uh is the best approximation of the exact solution u in theenergy norm. We have arrived through a back door at a Minimum Residual Method.

¶Residual norm really...ETH Zurich, August 2016 DPG Method 14 / 180

Moral of the story

The minimum residual method,with the residual measured in the dual test norm,

is the most stable Petrov-Galerkin methodyou can have.


DPG is a minimum residual method ‖

u ∈ Ub(u, v) = l(v) v ∈ V ⇔ Bu = l B : U → V ′

〈Bu, v〉 = b(u, v)

I Minimum residual method: Uh ⊂ U ,

12‖Buh − l‖2V ′ → min

uh∈Uh

I Riesz operator:

RV : V → V ′, 〈RV v, δv〉 = (v, δv)V

is an isometry, ‖RV v‖V ′ = ‖v‖V .I Minimum residual method reformulated:

12‖Buh − l‖2V ′ = 1

2‖R−1V (Buh − l)‖2V → min

uh∈Uh

‖I J.H. Bramble, R.D. Lazarov, J.E. Pasciak, “A Least-squares Approach Based on a Discrete Minus One Inner Product for First Order

Systems”Math. Comp, 66, 935-955, 1997.

I L.D., J. Gopalakrishnan. “A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions.”Numer. Meth. Part. D. E., 27,70-105, 2011.




〈Bu, v〉 = b(u, v)


12‖Buh − l‖2V ′ → min

uh∈Uh

I Riesz operator:

RV : V → V ′, 〈RV v, δv〉 = (v, δv)V


12‖Buh − l‖2V ′ = 1


uh∈Uh







〈Bu, v〉 = b(u, v)


12‖Buh − l‖2V ′ → min

uh∈Uh

I Riesz operator:

RV : V → V ′, 〈RV v, δv〉 = (v, δv)V

is an isometry, ‖RV v‖V ′ = ‖v‖V .

I Minimum residual method reformulated:

12‖Buh − l‖2V ′ = 1


uh∈Uh







〈Bu, v〉 = b(u, v)


12‖Buh − l‖2V ′ → min

uh∈Uh

I Riesz operator:

RV : V → V ′, 〈RV v, δv〉 = (v, δv)V


12‖Buh − l‖2V ′ = 1


uh∈Uh





DPG is a minimum residual method

Taking Gateaux derivative,

(R−1V (Buh − l), R−1

V Bδuh)V = 0 δuh ∈ Uh




(R−1V (Buh − l), R−1


or〈Buh − l, R−1

V Bδuh〉 = 0 δuh ∈ Uh




(R−1V (Buh − l), R−1


or〈Buh − l, R−1

V Bδuh︸︷︷︸vh

〉 = 0 δuh ∈ Uh




(R−1V (Buh − l), R−1


or〈Buh − l, vh〉 = 0 vh = R−1

V Bδuh




(R−1V (Buh − l), R−1


or〈Buh, vh〉 = 〈l, vh〉 vh = R−1

V Bδuh




(R−1V (Buh − l), R−1


orb(uh, vh) = l(vh)

where vh ∈ V(vh, δv)V = b(δuh, δv) δv ∈ V


DPG is a mixed method

An alternate route ∗∗,

( R−1V (Buh − l)︸︷︷︸

=:ψ(error representation function)

, R−1V Bδuh)V = 0 δuh ∈ Uh

∗∗W. Dahmen, Ch. Huang, Ch. Schwab, and G. Welper. “Adaptive Petrov Galerkin methodsfor first order transport equations”, SIAM J. Num. Anal. 50(5): 242-2445, 2012




( R−1V (Buh − l)︸︷︷︸



or ψ = R−1

V (Buh − l)

(ψ,R−1V Bδuh)V = 0 δuh ∈ Uh





( R−1V (Buh − l)︸︷︷︸



or (ψ, δv)V − b(uh, δv) = −l(δv) ∀δv ∈ V

b(δuh, ψ) = 0 ∀δuh ∈ Uh



DPG method, a summary so far

I Stiffness matrix is always hermitian and positive-definite (it is ageneralization of the least squares method...).

I The method delivers the best approximation error (BAE) in the “energynorm”:

‖u‖E := ‖Bu‖V ′ = supv∈V

|b(u, v)|‖v‖V

I The energy norm of the FE error u− uh equals the residual and can becomputed,

‖u− uh‖E = ‖Bu−Buh‖V ′ = ‖l −Buh‖V ′ = ‖R−1V (l −Buh)‖V = ‖ψ‖V

where the error representation function ψ comes fromψ ∈ V(ψ, δv)V = 〈l −Buh, δv〉 = l(δv)− b(uh, δv), δv ∈ V

No need for a-posteriori error estimation, note the connection with implicita-posteriori error estimation techniques ††

††J.T.Oden, L.D., T.Strouboulis and Ph. Devloo, “Adaptive Methods for Problems in Solid and Fluid Mechanics”, in Accuracy Estimates and

Adaptive Refinements in Finite Element Computations, Wiley & Sons, London 1986






|b(u, v)|‖v‖V












|b(u, v)|‖v‖V








DPG method, a summary

I A lot depends upon the choice of the test norm ‖ · ‖V ; for different testnorms, we get get different methods.

I How to choose the test norm in a systematic way ?

I Is the inversion of Riesz operator (computation of the optimal test functions,energy error) feasible ?

I Being a Ritz method, DPG does not experience any preasymptoticlimitations.




















Digression: Trial to Test Operator of Barret and Morton ‡‡

UB−→ V ′

↓ RU ↑ RVU ′

B′←− V

DPG trial-to-test operator: T−1V B

Barret-Morton trial-to-test operator: (B′)−1RU

‡‡J.W. Barret and K. W. Morton, Comp. Meth. Appl. Mech and Engng., 46, 97 (1984).


Outline






I Preconditioning.




Time-harmonic linear acoustics

“Mathematician’s version”: iωp +div u = f in Ωiωu +∇p = g in Ω

p = u · n on Γ

(Strong) operator form:p ∈ H1(Ω), u ∈ H(div,Ω), p = u · n on ΓA(p, u) = (f, g)

where A(p, u) = (iωp+ div u, iωu+ ∇p), f ∈ L2(Ω), g ∈ (L2(Ω))N .


Trivial and ultraweak variational formulations

Trivial formulation:p ∈ H1(Ω), u ∈ H(div,Ω), p = u · n on Γiω(p, q) + (div u, q) = (f, q) q ∈ L2(Ω)iω(u, v) + (∇p, v) = (g, v) v ∈ (L2(Ω))N

or, in operator form:p ∈ H1(Ω), u ∈ H(div,Ω), p = u · n on Γ(A(p, u), (q, v)) = (f, q) + (g, v)

q ∈ L2(Ω), v ∈ (L2(Ω))N

Ultraweak formulation:p ∈ L2(Ω), v ∈ (L2(Ω))N

((p, u), A∗(q, v)) = (f, q) + (g, v)q ∈ H1(Ω), v ∈ H(div,Ω), q = −v · n on Γ

where A∗ = −A.


Mixed and reduced formulations I

Relaxing∗ conservation of mass eqn, p ∈ H1(Ω), u ∈ (L2(Ω))N

iω(p, q)− (u,∇q) + 〈p, q〉Γ = (f, q) q ∈ H1(Ω)iω(u, v) + (∇p, v) = (g, v) v ∈ (L2(Ω))N

Multiplying the first equation by iω and eliminating velocity, we obtain thestandard formulation for Helmholtz eqn,

u ∈ H1(Ω)(∇p,∇q)− ω2(p, q) + iω〈p, q〉Γ = (iωf + g, v) v ∈ H1(Ω)

∗I.e. integrating by parts and building the BC in...ETH Zurich, August 2016 DPG Method 25 / 180

Mixed and reduced formulations II

Relaxing conservation of momentum eqn, we get: p ∈ L2(Ω), u ∈ Viω(p, q) + (div u, q) = (f, q) q ∈ L2(Ω)iω(u, v)− (p, div v) + 〈u · n, v · n〉Γ = (g, v) v ∈ V

The energy space for the velocity has now to incorporate an extra regularityassumption resulting from building in the impedance BC,

V := v ∈ H(div,Ω) : v · n ∈ H1/2(Γ)

As before, we can eliminate now the pressure to obtain a variational formulation interms of the velocity only,

u ∈ V(div u, div v)− ω2(u, v) + iω〈u · n, v · n〉Γ = (f + iωg, v), v ∈ V


Stability and well posedness

All formulations are simultaneously † well- or ill-posed.

†L.D. “Various Variational Formulations and Closed Range Theorem”, ICES Report15/03


Time-harmonic Maxwell equations

∇× E + iωµH = 0 Faraday Law

∇×H − iωεE − σE = J imp Ampere Law

∇ · (µH) = 0 Gauss Magnetic Law

−∇ · (εE) = ρimp + ρ Gauss Electric Law

−iωρ+ ∇(σE) = 0 conservation of charge

where:EHρ

electric fieldmagnetic fieldfree charge

the unknowns

µεσ

permeabilitypermittivityconductivity

material constants

J imp

ρimpimpressed currentimpressed charge

load data


A bit of history

Electrostatics:

Coulomb’s Law (1775) → electric fieldpolarization, dielectrics, ε

→Gauss Electric Law (1835)

Magnetostatics:

Ampere Force Law (1820)(steady currents)

→ magnetic fieldmagnetic polarization, µ

→ Ampere LawGauss Magnetic Law

Faraday Law (1831)Maxwell equations (1856) including correction to Ampere Law.Current formalism due to Heaviside (1884)Inspired Einstein on his way to Special Relativity.


Maxwell cavity problem

Eliminating the free charge, we get:

∇× E + iωµH = 0 Faraday Law

∇×H − iωεE − σE = J imp Ampere Law

∇ · (µH) = 0 Gauss Magnetic Law

−∇ · ((σ + iωε)E) = −iωρimp continuity equation

The equations are linearly dependent:Take curl of the Faraday Law to obtain the Gauss Magnetic Law.Take curl of the Ampere Law to obtain the continuity equation. Notice that J imp, ρimp

must satisfy the compatibility condition:

∇ · J imp = −iωρimp .

Boundary Conditions (BC):

n× E = n× E imp on Γ1

n×H = n×H imp =: J impS︸︷︷︸

impressed surface current

on Γ2


Different variational formulations

n× E = n× E imp on Γ1

n×H = n×H imp on Γ2

∇× E + iωµH = 0 /φ (1)∇×H − (iωε+ σ)E = J imp /ψ (2)

To relax or not to relax ?

(1) (2) name energy setting

1 no no trivial (strong) E,H ∈ H(curl), φ, ψ ∈ L2

2 no yes standard E,ψ ∈ H(curl), H, φ ∈ L2

3 yes no standard H,φ ∈ H(curl), E, ψ ∈ L2

4 yes yes ultraweak E,H ∈ L2, φ, ψ ∈ H(curl)

The inf-sup constants for different variational formulations are equal or O(1)-equivalent(Closed Range Theorem at work).

Only standard variational formulations are eligible for the Bubnov-Galerkin method

(symmetric functional setting)


Standard variational formulation

Relax the Ampere equation:

I multiply with factor −iω,

∇× (−iωH) + (−ω2ε+ iωσ)E = −iωJ imp ,

I multiply with a test function ψ and integrate by parts the curl term,

(−iωH,∇× ψ) + 〈n× (−iωH), ψ〉+ ((−ω2ε+ iωσ)E,ψ) = −iω(J imp, ψ) ,

I build the second boundary condition into the formulation and eliminate the rest ofthe boundary term by not testing on Γ1, i.e. assuming n× ψ = 0 on Γ1,

(−iωH,∇×ψ)+((−ω2ε+iωσ)E,ψ) = −iω(J imp, ψ)+iω〈J impS , ψ〉Γ2 n×ψ = 0 on Γ1 .

Use the strong form of the Faraday equation to eliminate H:E ∈ H(curl,Ω), n× E = n× E imp on Γ1 ,

( 1µ∇× E,∇× ψ) + ((−ω2ε+ iωσ)E,ψ) = −iω(J imp, ψ) + iω〈J imp

S , ψ〉Γ2

ψ ∈ H(curl,Ω) : n× ψ = 0 on Γ1 .


Ultraweak variational formulation

Relax both equations:

E,H ∈ L2(Ω)

(E,∇× φ) + (iωµH, φ) = −〈n× Einc, φ〉φ ∈ H(curl,Ω), n× φ = 0 on Γ2

(H,∇× ψ)− ((iωε+ σ)E,ψ) = (J imp, ψ)− 〈n×Hinc, ψ〉ψ ∈ H(curl,Ω), n× ψ = 0 on Γ1

You may test on the whole boundary:

E ∈ L2(Ω), E ∈ H−1/2t (curl,Γ), n× E = n× Einc on Γ1

H ∈ L2(Ω), H ∈ H−1/2t (curl,Γ), n× H = n×Hinc on Γ2

(E,∇× φ) + (iωµH, φ) + 〈n× E, φ〉 = 0 φ ∈ H(curl,Ω)

(H,∇× ψ)− ((iωε+ σ)E,ψ) + 〈n× H, ψ〉 = (J imp, ψ) ψ ∈ H(curl,Ω)

where H−1/2t (curl,Γ) := trΓH(curl,Ω).


Outline






I Preconditioning.




The paradigm of breaking test functions

In each of the discussed formulations, we can eliminate BCs on test functions andreplace the test spaces with corresponding broken test spaces,

H1(Ω) −→ H1(Ωh)H(curl,Ω) −→ H(curl,Ωh)H(div,Ω) −→ H(div,Ωh) ,

i.e. we can “break test functions” at the expense of introducing additionalunknowns (Lagrange multipliers) that are traces of dual energy spaces to meshskeleton Γh := ∪K∈Th∂K,

H1(Ωh) introduces n · v ∈ H−1/2(Γh) := trΓhH(div,Ω)

H(curl,Ωh) introduces n× H ∈ H−1/2(div,Γh) := tr⊥ΓhH(curl,Ω)

H(div,Ωh) introduces u ∈ H1/2(Γh) := trΓhH1(Ω)


Formulations with broken test spaces for acoustics

Classical formulation:p ∈ H1(Ω), u · n ∈ H−1/2(Γh)(∇p,∇hq) + iω〈p, q〉Γ−〈u · n, q〉Γh

= (iωf + g, q), q ∈ H1(Ωh)

whereH−1/2(Γh) := trΓh

H0(div,Ω)

In other words, the additional unknown vanishes on domain boundary Γ.Ultraweak formulation:

p ∈ L2(Ω), u ∈ (L2(Ω))N

p ∈ H1/2(Γh), u · n ∈ H−1/2(Γh), p = u · n on Γ−(p, iωq + divhv)− (u, iωv + ∇hq)+〈p, v · n〉Γh

+ 〈u · n, q〉Γh= (f, q) + (g, v)

q ∈ H1(Ωh), v ∈ H(div,Ωh)

Above, ∇h, divh denote element-wise defined operators..


Lemma 1 (Brezzi’s argument)

We can “break” test functions in any well-posed variational problem,u ∈ Ub(u, v) = l(v) v ∈ V (Ω)

(1)

that goes into: u ∈ U, t ∈ (V (Ωh)′

b(u, v) + 〈t, v〉︸︷︷︸=:b((u,t),v)

= l(v) v ∈ V (Ωh) (2)

If (1) is well posed than so is (2).Sketch of the proof: We already control ‖u‖:

supv∈V (Ωh)

|b((u, t), v)|‖v‖ ≥ sup

v∈V (Ω)

|b(u, v)|‖v‖ ≥ γ‖u‖U

We control t in the dual norm:

〈t, v〉 = l(v)− b(u, v)

‖t‖(V (Ωh))′ = supv|〈t,v〉|‖v‖ ≤ ‖l‖+ ‖b‖‖u‖U ≤

(1 + ‖b‖

γ

)‖l‖


Lemma 2

The dual norm of t can be interpreted as a minimum-energy (quotient) norm. Assume:VC - Hilbert (test) space with graph norm

‖v‖2VC:= ‖Cv‖2 + ‖v‖2

t ∈ V ′. Then

‖t‖V ′C

:= supv∈VC

|〈t, v〉|‖v‖VC

= ‖t‖VC∗

where t is the minimum energy extension of t in VC∗ -norm.Sketch of the proof: Computation of the dual norm reduces to the inversion of theRiesz operator which leads to a Neumann problem:

(Cvt, Cδv) + (vt, δv) = 〈t, δv〉 ⇔

C∗ Cvt︸︷︷︸

t

+vt = 0 in Ω /C

Cvt = t on Γ

Computation of the minimum-energy extension norm reduces to the solution of aDirichlet problem;

C C∗t︸︷︷︸=−t

+t = 0 in Ω /C∗

t = t on Γ

andCvt = t, C∗t = −t, ‖t‖2VC∗ = ‖C∗t‖2 + ‖t‖2 = ‖vt‖2Vc


Theorem (C. Carstensen, L.D., J. Gopalakrishnan, 2015) ∗

Assume that the original variational problem is well posed. Then the variationalproblem with broken test spaces is well posed as well with a mesh-independentstability constant γ of order of inf-sup constant for the original problem, and thefollowing stability result holds:

supv∈V (Ωh)

|b((u, t), v)|‖v‖V (Ωh)

≥ γ(‖u‖2U + ‖t‖2C∗)1/2

where the Lagrange multiplier t is measured in the “minimum energy extension”(quotient) norm implied by the test norm.

Comment on the role of norms in U and V .

∗C. Carstensen, L.D. and J. Gopalakrishnan, “Breaking Spaces and Forms for the DPGmethod and Applications Including Maxwell Equations”, Comput. Math.Appl., 72(3): 494-522,2016.


Breaking test functions in Maxwell problems

Primal DPG formulation:

E ∈ H(curl,Ω), n× E = n× E imp on Γ1 ,

H ∈ H−1/2t (curl,Γh), n× H = n×Hinc on Γ2

( 1µ∇× E,∇h × ψ) + ((−ω2ε+ iωσ)E,ψ) + iω〈H, ψ〉 = −iω(J imp, ψ)

ψ ∈ H(curl,Ωh)

Ultraweak DPG formulation:

E ∈ L2(Ω), E ∈ H−1/2t (curl,Γh), n× E = n× Einc on Γ1

H ∈ L2(Ω), H ∈ H−1/2t (curl,Γh), n× H = n×Hinc on Γ2

(E,∇h × φ) + (iωµH, φ) + 〈n× E, φ〉 = 0 φ ∈ H(curl,Ωh)

(H,∇h × ψ)− ((iωε+ σ)E,ψ) + 〈n× H, ψ〉 = (J imp, ψ) ψ ∈ H(curl,Ωh)


Outline






I Preconditioning.




Approximation of optimal test functions (Practical DPGMethod)

Idea: Replace infinite dimensional test space V with a finite-dimensional enrichedspace V ⊂ V, dim V >> dim Uh.Natural choice: use element of higher order r := p+ ∆p.The enriched test space V =: V r may be determined a-priori or a-posteriori. In allreported experiments, typically, ∆p = 1, 2, 3.Comments: Other approximations of optimal test functions are possible, seeNiemi et al for use of subelement Shishkin meshes ∗ . A completely differentapproach has been proposed by Cohen et al. † where an additional (inner)adaptive loop is introduced to approximate error representation function ψ. Theresulting approximate optimal test space does not guarantee the discrete stability.Discuss the difference between the a-priori and a-posteriori approximation of ψ.

∗A. Niemi, N. Collier and V. Calo, Stable Discontinuous Petrov-Galerkin Methods for Stationary Transport Problems: Quasi-optimal Test Space

Norm, Comput. Math. Appl., 66(10): 2096-2113, 2013†

A. Cohen, W. Dahmen, and G. Welper, Adaptivity and variational stabilization for convection-diffusion equations, ESAIM Math. Model. Numer.Anal., 46(5):1247–1273, 2012.


Petrov-Galerkin method with optimal test functions

Ideal DPG method: uh ∈ Uh ⊂ Ub(uh, vh) = l(vh) ∀vh ∈ T (Uh)

where the ideal trial-to-test operator T : Uh → V is defined by

(Tδuh, δv)V = b(δuh, δv) uh ∈ Uh, δv ∈ V

Practical DPG method:uh ∈ Uh ⊂ Ub(uh, vh) = l(vh) ∀vh ∈ T r(Uh)

where the practical trial-to-test operator T r : Uh → V r ⊂ V is defined by

(T rδuh, δv)V = b(δuh, δv) uh ∈ Uh, δv ∈ V r


Minimum residual method

Ideal DPG method:

uh = arg minwh∈Uh‖l −Buh‖V ′ = arg minwh∈Uh

supv∈V

|l(v)− b(uh, v)|‖v‖V

Practical DPG method:

uh = arg minwh∈Uh‖l −Buh‖(V r)′ = arg minwh∈Uh

supv∈V r

|l(v)− b(uh, v)|‖v‖V


Mixed method

Ideal DPG method: ψ ∈ V, uh ∈ Uh(ψ, φ)V − b(uh, φ) = −l(φ) φ ∈ Vb(wh, ψ) = 0 wh ∈ Uh

where ‖ψ‖ = ‖l −Buh‖V ′ is the residual.Practical DPG method:

ψ ∈ V r, uh ∈ Uh(ψ, φ)V − b(uh, φ) = −l(φ) φ ∈ V rb(wh, ψ) = 0 wh ∈ Uh

where ‖ψ‖ = ‖l −Buh‖(V r)′ is the computable approximate residual.


Accounting for the error in computing test functions and ψ

Introduce a Fortin operator Π : V → V r,

b(uh,Πv − v) = 0 uh ∈ Uh ‖Πv‖V ≤ C‖v‖V

Theorem

‖u− uh‖U ≤MC

γinf

wh∈Uh

‖u− wh‖U

Theorem‡ There exists a family of commuting, element-wise defined Fortinoperators for standard energy spaces and the enriched spaces for ∆p = N .

H1(K)/IRgrad−→ H(curl,K)

curl−→ H(div,K)div−→ L2(K)

↓ Πgradp+3 ↓ Πcurl

p+3 ↓ Πdivp+3 ↓ Πp+2

Pp+3(K)/IRgrad−→ Np+3(K)

curl−→ Rp+3(K)div−→ Pp+2(K)

‡J.G. and W. Qiu, “An Analysis of the Practical DPG Method”, Math. Comp.,83(286):537-552, 2014, C. Carstensen, L.D. and J. G., “Breaking Spaces and Forms for the DPGmethod and Applications Including Maxwell Equations”, Comput. Math. Appl., 72(3): 494-522,2016.


Accounting for the error in computing test functions and ψ

The Fortin operator can also be used to assess the error in approximation of errorrepresentation function.Theorem §

Let η = ‖ψ‖V . Then

γ2‖u− uh‖2 ≤ η2 + (‖Π‖η + osc(l))2

andη ≤M‖u− uh‖U

For additional study of Fortin operators, see ICES Report 2015-22.

§C. Carstensen, L.D., J. Gopalakrishnan, A Posteriori Error Control for DPG Methods, SIAM J. Numer. Anal., 52(3): 1335-1353, 2014.


Outline






I Preconditioning.




Main point: ψ is condensed out elementwise

Group unknown (watch for the overloaded symbol):

uh := ( uh︸︷︷︸field

, th︸︷︷︸flux

)

Mixed system: G −B1 −B2

BT1 0 0

BT2 0 0

ψu

t

=

−l00

where B1,B2 correspond to (∇uh,∇hvh) and −〈th, vh〉, resp.Eliminate ψ to get the DPG system:(

BT1G−1B1 BT

1G−1B2

BT2G−1B1 BT

2G−1B2

) (u

t

)=

(BT

1G−1l

BT2G−1l

)After backsubstitution, ‖ψ‖V provides the element contribution to the globalresidual which drives adaptivity.


Are all variational formulations equal ?


1D acoustics, ω = 60, 50 elements, p = 2.

Spectrum of stiffness matrix after static condensation of interior dof, for classicalGalerkin, least squares (strong formulation) primal and ultraweak formulations.Comment on CG convergence


Corresponding solutions

Both least squares and ultraweak DPG methods are robust, i.e. the stabilityconstants are independent of ω but they converge in different norms: least squaresin graph norm, ultraweak in L2-norm.

Solutions obtained classical Galerkin, least squares (strong formulation) primaland ultraweak formulations. In one space dimension the ultraweak DPG method ispollution free!


In 1D the ultraweak DPG is pollution free

With the graph test norm,

‖(v, q)‖)2V := ‖A(v, q)‖2 + ‖(v, q)‖2

the corresponding UW DPG method is pollution free,

(‖(u− uh, p− ph)‖2 + ‖(u− uh, p− ph)‖2∗)1/2

≤ 1γ

( inf(wh,rh)

‖(u− wh, p− rh)‖2︸︷︷︸pollution free

+ inf(wh,rh)

‖(u− wh, p− rh)‖2∗︸︷︷︸=0

)1/2

Comment on the choice of the enriched space.


3D examples

All reported 3D examples were coded within hp3d - a three-dimensional hp code¶

supporting:

I hybrid meshes with elements of all shapes: hexas, tets, prisms and pyramids,

I first Nedelec family of H1-,H(curl)-, H(div)-, and L2-conforming elements,

I 1-irregular meshes and anisotropic hp-refinements.

The computations used a recently developed‖ suite of orientation embedded shapefunctions for elements of all shapes and the whole exact sequence that can bedownloaded from:

https://github.com/libESEAS/ESEAS

¶Developed with Paolo Gatto and Kyungjoo Kim.‖F. Fuentes, B. Keith, L. D. and S. Nagaraj, Orientation Embedded High Order Shape

Functions for the Exact Sequence Elements of All Shapes, em Comput. Math. Appl., 70:353-458, 2015.


3D Maxwell cavity problem, h-convergence

Unit domain, PEC BCs, µ = ε = 1, σ = 0, ω = 1, p = 1, 2, 3, ∆p = 2,manufactured solution:

E1 = sinπx sinπy sinπz, E2 = E3 = 0 .

Primal formulation, hexas, H(curl)-error and residual.





Primal formulation, tets, H(curl)-error and residual.





Primal formulation, prisms, H(curl)-error and residual.





Ultraweak formulation, hexas, L2-error and residual.





Ultraweak formulation, tets, L2-error and residual.





Ultraweak formulation, prisms, L2-error and residual.


DPG does not suffer form any preasymptotic behavior

and it enables adaptivity starting with very coarse meshes


Example: 2D acoustics: Gaussian beam


Mesh 1 and real part of pressure
























































































Residual and L2 error convergence history

Error (dotted line) decreases monotonically while residual (solid line) decreasesmonotically only in the p-refinement stage.


Fichera corner

Divide it into eight smaller cubes and remove one:


Fichera corner microwave

Attach a waveguide:

aaaa

ε = µ = 1, σ = 0ω = 5(1.6 wavelengths in the cube)

Cut the waveguide and use the lowest propagating mode for BC along the cut.ETH Zurich, August 2016 DPG Method 104 / 180


Standard variational formulation

(Primal DPG method)

Standard test norm:

‖ψ‖2V := ‖ψ‖2H(curl,Ω) = ‖ψ‖2 + ‖∇× ψ‖2


Initial mesh and real part of E1


Mesh and real part of E1 after 1st refinement


Mesh and real part of E1 after 2nd refinement


Mesh and real part of E1 after 3rd refinement


Mesh and real part of E1 after 4th refinement














Residual history

Residual decreases monotonically.



Ultraweak variational formulation

(Original DPG method)

Adjoint graph norm:

‖(φ, ψ)‖2V := ‖φ‖2 + ‖ψ‖2 + ‖∇× φ+ (iωε− σ)ψ‖2 + ‖∇× ψ − iωµφ‖2


Initial mesh and real part of E1


Mesh and real part of E1 after 1st refinement


Mesh and real part of E1 after 2nd refinement


Mesh and real part of E1 after 3rd refinement












Residual history

Residual decreases monotonically.


Comparison of primal and UW results

Fichera microwave problem. Comparison of results obtained with primal (left) andultraweak (right) formulations, meshes 3-9.


















Ficheramicrowave problem. Comparison of results obtained with primal (left) and

ultraweak (right) formulations, meshes 3-9.


A non-trivial question for Maxwell

Primal formulation:

I Does the satisfaction of the weak form of the Ampere equation imply thesatisfaction of the continuity equation?

I More precisely, does the minimization of the residual corresponding to theAmpere eqn imply the control of the residual corresponding to the continuityequation ? (Not the case for an explicit a-posteriori error estimation for thestandard Galerkin method.)

I The answer to the last two questions seems to be positive. Indeed,

supq

|b((E, H),∇hq)|‖q‖H1(Ωh)︸︷︷︸

controlled

≤ supq

|b((E, H),∇hq)|‖∇hq‖

≤ supψ

|b((E, H), ψ)|‖ψ‖H(curl,Ωh)︸︷︷︸minimized

since ∇hH1(Ωh) ⊂ H(curl,Ωh).



Primal formulation:




supq

|b((E, H),∇hq)|‖q‖H1(Ωh)︸︷︷︸

controlled

≤ supq

|b((E, H),∇hq)|‖∇hq‖

≤ supψ





Primal formulation:




supq

|b((E, H),∇hq)|‖q‖H1(Ωh)︸︷︷︸

controlled

≤ supq

|b((E, H),∇hq)|‖∇hq‖

≤ supψ





Primal formulation:




supq

|b((E, H),∇hq)|‖q‖H1(Ωh)︸︷︷︸

controlled

≤ supq

|b((E, H),∇hq)|‖∇hq‖

≤ supψ




More examples

DPG works when standard Galerkin may not


Full Wave Form Inversion

From Ph.D. Dissertation∗∗ of Jamie Bramwell:

(a) CG µ, low frequency (b) DPG µ, high frequency

(c) CG λ, low frequency (d) DPG λ, high frequency

Solution of Elastic Marmousi problem: Left: standard Galerkin, Right: DPG∗∗

Jamie Bramwell, CSEM (co-supervised with O. Ghattas), A Discontinuous Petrov-Galerkin Method for Seismic Tomography Problems, Ph.D.Thesis, University of Texas at Austin, April 2013.


Metamaterials: Model Problem 1 ††

u = 0 on Γ

−div(a(x)∇u) = f in Ω

where Ω = (−2, 1)× (0, 1), and

a(x) =

1.000 x1 < 0−1.001 x1 > 0

f(x) =

1 x1 < 00 x1 > 0

Mesh and corresponding solution

††Work with Patrick CiarletETH Zurich, August 2016 DPG Method 133 / 180

Metamaterials: Model Problem 1Convergence of DPG vs. standard Galerkin method

Left: residuals for DPG method. Right: Norm of the difference between fine andcoarse mesh solutions


Metamaterials: Model Problem 2 - Scattering

iωσ−1u +∇p = 0

iωηp +divu = 0

with impedance BC:

p− u · n = ginc := pinc − uinc · n

Coefficients σ, η are negative for the scatterer.Optimal Mesh and real part of u1 after 10 refinements:


Metamaterials: Model Problem 2Convergence of DPG vs. standard Galerkin method

Left: residual for DPG method. Right: Norm of the difference between fine andcoarse mesh solutions


2D acoustics (electromagnetics) cloaking problem ‡‡

You can solve efficiently cloaking problems if you select the right test norm.

Exact solution (pressure or magnetic field)

‡‡L.D and J. Li, “Numerical Simulations of Cloaking Problems using a DPG Method”,Comp. Mech., 51(5): 661-672, 2013.




An hp mesh (4 bilinear elements per wavelength)





Numerical solution (pressure or magnetic field)



Outline






I High frequency problems. Preconditioning.




High frequency time-harmonic wave propagation problems

I Linear Acoustics iωu+∇p = 0, in Ωiωp+ divu = 0p− u · n = g on ∂Ω

where u is the velocity and p is the pressure.

I DPG Ultra-weak Formulation (Relax both equations)

u ∈ (L2(Ω))d, p ∈ L2(Ω)u ∈ H−1/2(Γh), p ∈ H1/2(Γh)p− u = g, on ∂Ω

(iωu, v)− (p, divhv) + 〈p, v · n〉 = 0, v ∈ H(div,Ωh)(iωp, q)− (u,∇hq) + 〈u, q〉 = 0, q ∈ H1(Ωh)

where Γh is the mesh skeletonand H−1/2(Γh) := tr H(div,Ω) on Γh, H1/2(Γh) := tr H1(Ω) on Γh


High frequency time-harmonic wave propagation problemsRemarks on Ultra-weak formulation

I Denote Aω(u, p) = (iωu+∇p, iωp+ divu). Then Aω = −A∗ω.

I Ideally if ‖v‖V = ‖A∗v‖ then the method delivers L2 projection in U .

‖u‖E = supv∈V

|b(u, v)|‖v‖V

= supv∈V

|(u,A∗v)|‖A∗v‖ = ‖u‖

In practice, in order to resolve the optimal test functions, we use a weightedadjoint graph norm with the continuation parameter ω = max(ω, 6/h), whereh is the element size.

‖v‖V = ‖A∗ωv‖+ α‖v‖where α is O(1).

I Field unknowns u and p are condensed out of the final system and we solveonly for the traces and fluxes, reducing the size of the final systemsignificantly.


Resonating Cavity (300 wavelengths)Refinement 0

Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure



Mesh Pressure


Resonating Cavity (300 wavelengths)

Residual convergence

105 106 107

Degrees of Freedom

10−3

10−2

10−1

Res

idu

al

hp-adaptivity


Integration of solvers with adaptivity

I A multi-frontal solver is faster than any iterative solver for this kind ofmeshes (1D structure).

I However restarting the problem 400 times makes the cost prohibitive. Thiscalls for the use of an iterative solver where a partially converged solutioncould drive adaptivity.

I Being a minimum residual method, DPG always delivers a Hermitian positivedefinite stiffness matrix. This makes the Conjugate Gradient ideal but thereis still the need for a good preconditioner.

I Ultimate Goal: Integrate the solver with adaptivity by building apreconditioner for Conjugate Gradient that utilizes information from previousmeshes.


Two grid-like Preconditioner for Conjugate GradientDescription

I Step 1: Direct Solver. Use a direct solver to solve the problem at the nth

mesh and store the Cholesky decomposition of the matrix. This will be ourcoarse mesh.





I Step 2: Macro grid.

nth (Coarse) Grid (n+ k)th (Fine) Grid






nth (Coarse) Grid (n+ k)th (Macro) Grid







On the (n+ k)th, 1 ≤ k ≤ m (fine) mesh we statically condense out all the newdegrees of freedom which are not on the skeleton of the nth(coarse) mesh. Wename this the macro mesh.



I Step 3: Additive Schwarz Smoother









Given a macro grid we define a patch to be the support of a coarse grid vertexbasis function. A block is constructed by the interactions of the macro degrees offreedom within a patch.



I Step 4: Coarse grid correction









Construct prolongation (Imc ) and restriction (Icm = (Imc )∗) operators between thecoarse and the macro mesh. Given that the two meshes have the same geometrictopology and also the fact that all the degrees of freedom live on the skeleton ofthe meshes, the construction of such operators can be done edge-wise. (Face-wisein 3D.)



I Step 5: PCG with Additive Schwarz and coarse-grid correction.

We implement a two-grid cycle between the coarseand the macro grid.

Let S be the smoothing operator, C = Imc A−1c Icm

the coarse grid correction , I the identity operatorand let P = I − AmC , where Am denotes themacro grid stiffness matrix.

Then the resulting preconditioner is Hermitian andpositive definite and it’s given explicitly by

M = SP + P ∗S + C − SPAmS

Two grid cycle

Coarse grid solve

(Addititive Schwarz) (Addititive Schwarz)

(Back substitution)

Post-smoothPre-smooth

Restriction(Icm) Prolongation(Imc )


Two grid-like Preconditioner for Conjugate Gradient

I Results (40 wavelengths)

Additive Schwarz vs Two-grid Preconditioner

10000 20000 30000 40000 50000 60000 70000Degrees of freedom

0

100

200

300

400

Nu

mb

erof

Iter

atio

ns

Two Grid PCG

Additive Schwarz PCG

Direct Solve


Comments on the new solver

I The coarse grid correction seems to be necessary in order for the iterations toremain bounded.

I However, the cost of a single iteration increases within every refinement(static condensation, smoothing, coarse grid solve).

I The use of a relaxation parameter for smoothing is very essential. Hope forsome analysis.

I It is sufficient to use a partially converged solution to drive adaptivity.


Current work

I Convergence Analysis

I 3D acoustics and Maxwell (OpenMP/MPI implementations may benecessary.)


Outline










Motivation: Nonlinear Optics

I Study of EM-wave propagation in optical fibers

I Maxwell equations: ∇× E = −µ0∂H∂t

∇×H = ε0∂E∂t + ∂P

∂t ,

I Polarization vector P is composed of a linear and nonlinear part:

P = PL + PNL.

PL: linear Lorentz polarization, PNL: nonlinear Kerr and Raman polarization.

P = ε0(χ1 · E + χ2 : E⊗ E + χ3

...E⊗ E⊗ E)


Non Linear Schrodinger Equation in Optics

I The principal model of Maxwell equations in fiber optics adopts the slowlyvarying envelope approximation:

E(r, t) =1

2x (E(r, t) e−iω0t + c.c. )

I Additional assumptions and approximations imply an approximate solution in(r, θ, z):

E(r, θ, z) =1

2x (F (r, θ)u(z, t) ei(β0z−ω0t) + c.c. )

I Complexified amplitude function u(z, t) satisfies a nonlinear Schrodinger(NLS) type equation:

i∂u

∂z− β2

2

∂2u

∂t2+ γ|u|2u = 0

I Space and time are reversed


The Linear Schrodinger Equation in n Dimensions

I Ω0 ⊂ Rn is a bounded, Lipschitz domain and time t ∈ [0, T ] with T <∞.Ω := Ω0 × [0, T ]. Define ΓT = ∂Ω0 × [0, T ] ∪ Ω0 × t = T.

I Consider the LSE in n space dimensions:i∂u

∂t−∆u = f in Ω,

u(t, ·) = 0 on ∂Ω0,

u = 0 at t = 0.


The Linear Schrodinger Equation in n Dimensions

I Define Lu := −∆u and Schrodinger operator Au := iut + Lu.

I We have an orthonormal eigenbasis ek(x) and eigenvalues0 < ω2

1 ≤ ω22 ≤ ω2

k →∞ that satisfy:

Lek(x) = ω2kek(x).

and

u(t, x) =

∞∑k=1

uk(t)ek(x),

I Au = f implies, with fk(t) = (f, ek(x))L2(Ω):

iuk(t) + ω2kuk(t) = fk(t),

I This leads to an explicit formula for the solution:

u(t, x) =

∞∑k=1

(−i∫ t

0

e−iω2k(t−s)fk(s)ds)ek(x).


Ill-posedness of 1st Order Formulation

I The bound ‖u‖L2(Ω) ≤ C√T‖f‖L2(Ω) = C

√T‖Au‖L2(Ω) holds for some

C = C(Ω).

I However, we cannot control derivatives: ‖∇u‖L2(Ω), ‖ut‖L2(Ω) can beunbounded.

I Consequently, the naive 1st order formulationi∂u

∂t− div τ = f,

∇u− τ = g,

u(t, ·) = 0 on ∂Ω0,

u = 0 at t = 0.

is ill-posed in the L2 setting!

I Unpleasant reality: We cannot develop a DPG (or any) method for the abovefirst order system formulation!


Only 2 Variational Formulations in Space-Time

I We therefore have only 2 well-posed 2nd order formulations. The strong andultra-weak:

I Strong formulation:u ∈ w ∈ L2(Ω) : Aw ∈ L2(Ω) with B.C.’s

(Au, v)L2(Ω) = (f, v)L2(Ω)

I Mathematical challenges:• Non-standard graph energy space W = u ∈ L2(Ω) : Au ∈ L2(Ω)• Unlike H1, H(curl), H(div), we do not have notion of trace for W .

I Integration by parts gives:

(u,A∗v)L2(Ω) + 〈Du, v〉 = (f, v)L2(Ω)

I Need to develop a boundary map D : W → (W ∗)′

to account for relaxingthe strong formulation

I Need to develop specialized conforming element and best approximation errorestimates for the discrete solution


The Boundary Operator D

I The boundary operator D : W → (W ∗)′ defined as:

〈Du, v〉 = (Au, v)L2(Ω) − (u,A∗v)L2(Ω)

I D is continuous with graph norms on W,W ∗.

I D generalizes notion of trace: action of D can be thought as composition oftrace (when it exists) with duality pairing on boundary

I Using D, we interpret domain of operator A as∗:

V := dom(A) = ⊥(DV∗),

whereV∗ := DΓT

∩Wis the space of smooth functions in W that vanish on the boundary ΓT .

I Standard Friedrichs theory not applicable: A+A∗ not L2 bounded for linearSchrodinger operator

∗Ern, Guermond, Caplain, “An intrisic criterion for the bijectivity of Hilbert operators relatedto Friedrichs’ systems”, Communications in Partial Differential Equations, 32(2) 317-341, Feb2007


Well-posedness of the Strong and UW Formulations

I The linear Schrodinger operator A : V → L2(Ω) is a continuous bijection.Thus, the strong formulation

u ∈ V,(Au, v)L2(Ω) = (f, v)L2(Ω), v ∈ L2(Ω)

corresponding to A is inf-sup stable, i.e., A is bounded below.

I The UW formulation is obtained by treating Du as an additional unknownfrom the quotient u ∈ (W ∗)′/D(V )

u ∈ L2(Ω), u ∈(W ∗)′/D(V ),

(u,A∗v)L2(Ω) + 〈u, v〉(W∗)′,W∗ = (f, v)L2(Ω), v ∈W ∗

is also inf-sup stable.

I Here, u comes from the quotient space (W ∗)′/D(V ) and must be measuredin the quotient norm.


UW DPG Method

I We now have the broken UW variational formulation:u ∈ L2(Ω), u ∈(W ∗h )′/Dh(V ),

(u,A∗hv)L2(Ω) + 〈u, v〉h = (f, v)L2(Ω), v ∈W ∗

where A∗h is the operator A applied elementwise, W ∗h is the broken graphspace and Dh is the elementwise auxiliary pairing operator:

W ∗h :=∏K∈Ωh

W ∗(K)

〈Dhu, v〉h :=∑K∈Ωh

〈DuK , vK〉(W∗(K))′,W∗(K)

I The test norm on W ∗ is given as:

‖v‖2W∗ = (vt, vt)L2(Ω) + (∆v,∆v)L2(Ω) + i(vt,∆v)L2(Ω) − i(∆v, vt)L2(Ω)

I Notice the fourth order nature of the (∆v,∆v)L2(Ω) term


Conforming Discretization: 1D Space Case for Optics

I Stability of the ideal DPG method:

(‖u− uh‖2L2(Ω) + ‖u− uh‖Q)1/2 ≤ C( infwh,wh

(‖u− wh‖2L2(Ω) + ‖u− wh‖Q)1/2),

where ‖ · ‖Q is the quotient norm in (W ∗)′/D(V ).

I We need an A-operator conforming FE space Vh and an interpolation operatorΠ : V → Vh with a 1D space-time element whose degrees of freedom respectconformity.

The conforming element with p = 2

I Sufficient conditions for conformity:

ut, uxx ∈ L2(Ω) =⇒ Au ∈ L2(Ω).

I Vh := Pp+1 ⊗ Pp+2


Interpolation Error Estimates

I Applying Cauchy-Schwarz inequality gives:

‖D(u−Πu)‖ ≤ (‖A(u−Πu)‖2L2(Ω) + ‖u−Πu‖2L2(Ω))12 .

I We estimate ‖u−Πu‖L2(K), ‖(u−Πu)t‖L2(K) and ‖(u−Πu)xx‖L2(K).

I Overall,‖u−Πu‖V ≤ C(Ω)hp+1|u|Hp+2 .

I Here, p+ 1 is polynomial order in time and p+ 2 in space

I Lowest order element is order 2 in time and 3 in space


Rates of Convergence


Solution Plots

Plots of solutions: exact complex Gaussian (left), and manufactured Gaussian beamsolution with ω = 20 (right)


Solving 1D NLS

I Linearize the NLS and apply Newton iterations to the linearized problem toconverge to the solution of the NLS

I Define

A(u) = i∂u∂t − ∂2u∂x2 + γ|u|2u,

L(u) = i∂u∂t − ∂2u∂x2 .

I The linearized scheme is:

u0 : initial guessSolve for ∆un :L(∆un) + γ(2|un|2∆un + u2

n(∆un)∗) = −A(un)un+1 = un + ∆un

I Convergence criterion: ‖∆un‖ < ε


Summary

I Similar to other wave propagation problems (first order acoustics andMaxwell equations†), the LSE in space-time domain admits only twovariational formulations: strong and ultraweak.

I However, contrary to acoustics and Maxwell systems, the LSE does not admita well-posed, 1st order L2 stable variational formulation.

I Inversion of Gram matrix involving second derivative leads to conditioningissues

I Is there a better way to proceed?

†C.Wieners, “The Skeleton Reduction for Finite Element Substructuring Methods”,ENUMATH Proceedings 2015


Outline










εDPG method

Ultraweak variational formulation for the first order system of linear acousticsequations with impedance BC on Γ : un − p = g,

(p, iωq + divhv) + (u, iωv +∇hq) + 〈un − p, q〉Γ0h

+ 〈p, vn + q〉Γh = 〈g, q〉Γ

Abstract notation:(u,A∗hv) + 〈w, v〉 = (f, v) + 〈u0, v〉

where (watch for overloaded symbols):

u = (p, u)v = (q, v)

A∗hv = (ωq + divhv, ωv +∇hq)w = (un − p, p)

εDPG method minimizes residual in dual norm to the test norm:

‖v‖2V := ‖A∗hv‖2 + ε2‖v‖2


Things improve with ε→ 0: Dissipation

I Simulate a plane wavepropagating at θ = π/8

I Standard DPG (ε = 1) andleast-squares exhibit heavydissipation.

I Dissipation disappears asε→ 0 in εDPG method.





































Things improve with ε→ 0: Dispersion

Exact Wavevector:

~k = ω

[cos(θ)sin(θ)

].

Discrete Wavevector:

~kh = ωh

[cos(θ)sin(θ)

].

At every θ, we solve a nonlin-ear system for ωh, derived us-ing dispersion analysis.



Exact Wavevector:

~k = ω

[cos(θ)sin(θ)

].


~kh = ωh

[cos(θ)sin(θ)

].




Exact Wavevector:

~k = ω

[cos(θ)sin(θ)

].


~kh = ωh

[cos(θ)sin(θ)

].




Exact Wavevector:

~k = ω

[cos(θ)sin(θ)

].


~kh = ωh

[cos(θ)sin(θ)

].




Exact Wavevector:

~k = ω

[cos(θ)sin(θ)

].


~kh = ωh

[cos(θ)sin(θ)

].



Things improve with ε→ 0: Theorem

Theorem

Suppose the wave operator A with boundary conditions is bounded below in L2():

‖u‖L2 ≤ C0‖Au‖L2 ∀u ∈ D(A).

Then, there is a C1 depending only on C0 such that the DPG solution uh obeys

‖u− uh‖X ≤ (1 + C1 ε) infwh∈Xh

‖u− wh‖X .

I The theorem∗ guarantees that errors in fluxes and traces improves in anε-independent norm as ε→ 0.

∗J. Gopalakrishnan, I. Muga and N. Olivares, “Dispersive and Dissipative Errors in theDPG Method with Scaled Norms for Helmholtz Equation”, CAMWA, 2014.


Things improve with ε→ 0: Theorem

Theorem

Suppose the wave operator A with boundary conditions is bounded below in L2():

‖u‖L2 ≤ C0‖Au‖L2 ∀u ∈ D(A).

Then, there is a C1 depending only on C0 such that the DPG solution uh obeys

‖u− uh‖X ≤ (1 + C1 ε) infwh∈Xh

‖u− wh‖X .

I The theorem∗ guarantees that errors in fluxes and traces improves in anε-independent norm as ε→ 0.

∗J. Gopalakrishnan, I. Muga and N. Olivares, “Dispersive and Dissipative Errors in theDPG Method with Scaled Norms for Helmholtz Equation”, CAMWA, 2014.


εDPG method is a very good least squares method

In the case ω = 1:

maxθ

∣∣ k − kLSh

∣∣ = 0.228

maxθ

∣∣k − kFEMh

∣∣ = 0.080

maxθ

∣∣k − kDPGh

∣∣ = 0.046


Relation with Barret-Morton Optimal Test Functions†

ε-scaled adjoint operator graph norm:

‖v‖2V,ε := ‖A∗v‖2 + ε‖v‖2

(ε = ε2)

†J.W. Barret and K. W. Morton, Comp. Meth. Appl. Mech and Engng., 46, 97 (1984).




‖v‖2V,ε := ‖A∗v‖2 + ε‖v‖2

(ε = ε2)If A∗ (with adjoint BC ) is bounded below then

‖A∗v‖ ≥ δ‖v‖ =⇒ ‖A∗v‖2 ≤ ‖A∗v‖2 + ε‖v‖2 ≤ (1 +ε

δ)‖A∗v‖2

‖v‖V,0 = ‖A∗v‖ and ‖v‖V,ε are equivalent and approach each other for small ε.





‖v‖2V,ε := ‖A∗v‖2 + ε‖v‖2


‖A∗v‖ ≥ δ‖v‖ =⇒ ‖A∗v‖2 ≤ ‖A∗v‖2 + ε‖v‖2 ≤ (1 +ε

δ)‖A∗v‖2


Q: What optimal test functions do we get for ε = 0 ?





‖v‖2V,ε := ‖A∗v‖2 + ε‖v‖2


‖A∗v‖ ≥ δ‖v‖ =⇒ ‖A∗v‖2 ≤ ‖A∗v‖2 + ε‖v‖2 ≤ (1 +ε

δ)‖A∗v‖2


Q: What optimal test functions do we get for ε = 0 ?

(A:) (A∗v,A∗δv) = (wp, A∗δv) =⇒ A∗v = wp

We arrive at Barret-Morton optimal test functions that deliver L2-projection:

0 = (u− up, A∗v) = (u− up, wp) ∀wp



Weakly conforming least squares, mortar (FETI) methods

The point: For ε > 0, the scaled graph norm ‖v‖V,ε is localizable. The DPG methodcan be viewed as a localization technique for computing an approximation to globallyoptimal test functions in a weakly conforming enriched subspace:

V rp := v ∈ V r : 〈wp, v〉 = 0 ∀wp ⊂ V r

I discrete Friedrichs stability constant is mesh-dependent,δ = δ(h, p, r) =⇒ ε = ε(h, p, r)

I For ε→ 0, DPG delivers a (weakly-conforming) least squares approximation ofoptimal global test functions. Indeed

v ∈ V rp(A∗hv,A

∗hδv) = (u,A∗hδv) ∀δv ∈ V rp

is equivalent to:12‖A∗hv − up‖2 → min

v∈V rp

I Work in progress: Q: Are the weakly conforming least squares any good ? Q: Howdoes δ(h, p, r) depend upon h, p, r ? Q: Error analysis ?




V rp := v ∈ V r : 〈wp, v〉 = 0 ∀wp ⊂ V r



v ∈ V rp(A∗hv,A



v∈V rp





V rp := v ∈ V r : 〈wp, v〉 = 0 ∀wp ⊂ V r



v ∈ V rp(A∗hv,A



v∈V rp





V rp := v ∈ V r : 〈wp, v〉 = 0 ∀wp ⊂ V r



v ∈ V rp(A∗hv,A



v∈V rp



Illustration: weakly conforming least squares for acoustics

r = 4, p = 2

Solution (real part of pressure) and the corresponding error(bounds: −0.32E + 00− 0.32E + 00)



r = 5, q = 2

Solution (real part of pressure) and the corresponding error(bounds: -0.42E-02 - 0.43E-02)



r = 6, q = 2




r = 7, q = 2




r = 8, q = 2

Solution (real part of pressure) and the corresponding error(bounds: -0.34E+00 - 0.31E+00)


Support: AFOSR, NSF, ONR.

Thank You !


Documents

Discontinuous Petrov Galerkin (DPG)