Robust Petrov-Galerkin Method for Convection Problems

Robust Discontinuous Petrov-Galerkin Method
for Convection-Dominated Diﬀusion Problems

Mohammad Zakerzadeh

German Research School for Simulation Science

11 Jan. 2012

Introduction

Optimal trial/test spaces
Conceptual review and motivation
Abstract theory development
Optimal test spaces and minimizing of the residual
Optimal spaces in a nutshell

Discontinuous Petrov-Galerkin with the ultra-weak formulation
Basic settings and conventions
Canonical energy norm pairings

Convection-diﬀusion problem
Variational formulation
Optimal test norm for convection-diﬀusion problem
Construction of a test norm and adjoint problem

Numerical experiments

References

Introduction
References
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Singular perturbation problems and robustness
• Standard Bubnov-Galerkin methods tend to perform poorly for the
class of PDEs known as “singular perturbation problems”.

• These problems are often characterized by a parameter that may
be either very small or very large in the context of physical
problems (in contrast to regular perturbation problems).

• An additional complication of singular perturbation problems is
that, in the limiting case of the parameter, the PDE itself will
change types (e.g. from elliptic to hyperbolic).

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(contd.)
• In 1D, the convection-diffusion equation is

βu − u = f in Ω = [0, 1]

u(0) = u0 , u(1) = u1 .
• In the limit of an inviscid medium as → 0, the equation changes
type, from elliptic to hyperbolic, and from second order to first
order.
• For satisfying those Dirichlet boundary conditions, the solution
develops sharp boundary layers of width near the outflow.

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(contd.)
• Why poor performance?
From Cea’s Lemma:

u − uh H 1 (0,1) ≤ C( ) inf u − wh H 1 (0,1)
wh

where C( ) grows as → 0.
• This dependence on is referred as a loss of robustness
Finite element error is bounded more and more loosely by the best
approximation error.
• As a consequence, the finite element solution can diverge
significantly from the best finite element approximation of the
solution
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Introduction
References
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Inﬁnite dimensional setting, BNB theorem
Consider
Seek u ∈ U such that
b(u, v) = l(v) = f, v , ∀v ∈ V

• continuity of b(., .) constant M , |b(u, v)| ≤ M u U v V
b(u, v)
• b(., .) has a inf − sup constant γ, ∃γ > 0 : supv∈V ≥γ
u U v V
• and the injectivity of the adjoint operator,
b(u, v) = 0, ∀u ∈ U ⇒ v = 0
Banach-Nˇcas-Babuˇka’s theorem =⇒ it has a unique solution.
e s

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Finite dimensional setting, Babuˇka’s theorem
s
Now let Uh ⊂ U , Vh ⊂ V be two ﬁnite dimensional trial and test
spaces and consider

Seek uh ∈ Uh such that
b(uh , vh ) = l(vh ) = f, vh , ∀vh ∈ Vh

If dimUh = dimVh , and the following discrete inf − sup condition

b(uh , vh )
∃γh > 0 : inf sup ≥ γh
uh ∈Uh vh ∈Vh uh U vh V

holds, then the ﬁnite dimensional problem is well-posed by application
of Babuˇka’s Theorem.
s

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Babuˇka’s error estimate
s

Theorem (1)
( Babuˇka’s error estimate)
s
Suppose that both continuous and discrete problem are well-posed,
then
Mh
u − uh U ≤ (1 + ) inf u − wh U
γh wh ∈Uh

For Hilbert spaces:
Mh
u − uh U ≤ u − uh U
γh

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Best estimation
Immediately from Theorem 1,
Corollary (1)
If Mh = γh , then

u − uh U = inf u − wh U
wh ∈Uh

In particular, Mh = γh = 1 satisﬁes Corollary 1.

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Linear operator
Let’s look from operator point of view:
• Deﬁne the linear operator as

Bu, v V ×V := b(u, v), ∀v ∈ V

where B is an operator from U to V with the norm B L(U,V ) .
• Due to stability, for any closed subspace Uh ⊂ U ∀uh ∈ Uh ,
−1
B L(U,V ) l − Buh V ≤ u − uh U ≤ B −1 L(V ,U ) l − Buh V

• Approximation error u − uh can be estimated in the norm of U ,
only when the condition number

κU,V (B) := B L(U,V ) B −1 L(V ,U )

is moderate.
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Linear operator (contd.)

inf − sup condition and continuity of the bilinear form
⇓
boundedness of B L(U,V ) and B −1 L(V ,U ) :

1
B L(U,V ) ≤M B −1 L(V ,U ) ≤
γ
This shows that B is a norm-isomorphism.

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Main idea
Idea: If we can enforce that M = γ = 1 in inﬁnite-dimensional
setting and this property is inherited (trivially) by the ﬁnite
dimensional subspaces, then variational formulation is

well-posed and well-conditioned

with unity condition number and gives best estimate of the solution.
=⇒ ideal Petrov-Galerkin

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Introduction
References
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Main idea
• The inf − sup condition is typically defined by taking first the
supremum over the test space V and then the infimum over the
trial space U .
• As long as the well-posedness the distinction between the test and
trial spaces are irrelevant.

Lemma (1)
The infinite dimensional problem is well-posed if and only if

b(u, v) b(u, v)
∃γ > 0 : inf sup = inf sup ≥γ
u∈U v∈V u U v V v∈V u∈U u U v V

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Main idea (contd.)

Lemma (2)
The following are equivalent
i) M = γ = 1
b(u, v)
ii) ∀u ∈ U we have u U = supv∈V
v V
b(u, v)
iii) ∀v ∈ V we have v V = supu∈U
u U

If we start from trial or test space and prescribe a norm, this norm
induces its dual in the other space and leads to our ideal case
M = γ = 1 in desired norm.

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Main idea (contd.)

Theorem (2)
Suppose the continuity condition holds with unity constant (M = 1),

b(u, v) ≤ u U v V

Then there holds M = γ = 1 if either of the following conditions
holds:
i) For each u ∈ U {0}, there exists vu ∈ V {0} such that:

b(u, vu ) = u U vu V

ii) For each v ∈ V {0}, there exists uv ∈ U {0} such that:

b(uv , v) = uv U v V
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Main idea (contd.)
• In general, the continuity and the inf − sup conditions are not
related to each other. However, Theorem shows if
i) the continuity constant is unity
ii) the equality is attainable
then the inf − sup constant is unity as well.

• If both conditions hold, then the 3-tuple (U, V, b(., .)) is known as a
dual pair. That is, the bilinear form b(., .) puts U and V in duality.

• We call the norms in U and V spaces optimal norms If both
continuity and inf − sup constants are unity in these norms.

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Main idea (contd.), Discrete analogue

Lemma (3)
Let the assumption of the Theorem 2 holds respectively for i) and ii)
below:
i) Let Uh ⊂ U be a subspace and construct

Vh = span{vuh : uh ∈ Uh , b(uh , vuh ) = uh U vuh V }.

ii) Let Vh ⊂ V be a subspace and construct

Uh = span{uvh : vh ∈ Vh , b(uvh , vh ) = vh V uvh U }.

If the pair of test space Vh and trial space Uh are constructed by
either i) or ii), then there holds Mh = γh = 1 and the discrete
problem is well-posed if dim Uh =dim Vh .
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Main idea (contd.)
• Theorem 2 and Lemma 3 do not explicitly specify either the
optimal test function or the optimal trial function.

• A general-purpose approach for choosing optimal pair of functions
is through the Riesz representation theorem

• If a basis in the trial space Uh is speciﬁed then we can determine
the corresponding test space (primal approach)and vice versa (dual
approach), so that the ﬁnite-dimensional problem is well-posed
with Mh = γh = 1.

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Optimal space, Primal approach

Theorem (3)
We have a map B : U → V , B(u) = Bu as Bu, v V ×V = b(u, v).
Denote vBu as the Riesz representation of Bu in V . Suppose B(., .)
is continuous with unity constant and assumption i) of Theorem 2
holds. Take Uh ⊂ U and deﬁne

Vh = span{vBuh : uh ∈ Uh }

Then the following hold,
i) Mh = γh = 1.
ii) Let Uh = span{φi }n , where φi ∈ U, i = 1, . . . , n. Then {vBφi }
i=1
is a basis of Vh .

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Optimal space, Dual approach

Theorem (4)
We have adjoint map B : V → U , B (v) = B v as
B v, u U ×U = b(u, v). Denote uB v as the Riesz representation of
B v in U . Suppose B(., .) is continuous with unity constant and
assumption ii) of Theorem 2.5 holds. Take Vh ⊂ V and deﬁne

Uh = span{uB vh : vh ∈ Vh }

Then the following hold,
i) Mh = γh = 1.
ii) Let Vh = span{φi }n , where φi ∈ V, i = 1, . . . , n. Then {uB φi }
i=1
is a basis of Uh .

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Optimal space
• In the proof of theorem 3 we have shown

2
b(uh , vBuh ) = Buh , vBuh V ×V = vBuh V = uh U vBuh V

• It shows that equality in Theorem 2 is ataiable by choosing vu
from Reisz representation.

• There is no distinguish between vu and vBu as well as uv and uB v
as long as we work exclusively with the Riesz representations.

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Introduction
References
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Optimal test space
• For Uh ⊂ U we can deﬁne optimal test space as:

Vh,opt = R−1 BUh

Where R : U → V is the Riesz map.

Lemma (4)
Considering discrete problem and choosing test space as optimal test
spaces, Vh = R−1 BUh , it holds that uh = arg minuh ∈Uh l − B uh V .
¯ ¯

• Solving of Petrov-Galerkin in optimal test space can be called
optimal Petrov-Galerkin in the sense of minimizing the residual in
V .
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Optimal test space (contd.)
• For any uh ∈ Uh , one has l − Buh V = B(u − uh ) V .
Equipping U with energy norm, . U,E := B. V

=⇒ u − uh U,E = B(u − uh ) V = l − Buh V

• Using optimal test spaces property

u − uh U,E = inf u − wh U,E
w∈Uh

uh is the best approximation in . U,E (quasi-optimal w.r.t. . U ).

• Corollary 1 =⇒ test spaces constructed by Riesz =⇒
i) Well-conditioned system, Mh = γh = 1,
ii) Residual minimization property in this energy norm
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Energy norm pairings
• In the optimal view, norms on trial and test space induces each
other,
b(φ, v) b(w, v)
φ U = sup where v V,U = sup
v∈V v V,U w∈U w U

We call such a pair of norms as an energy norm pairing.

• Stronger energy norm in U generates a weaker norm in V and vice
versa.
u U,1 ≤c u U,2

b(w, v) b(w, v)
v V,U,2 = sup ≤ c sup =c v V,U,1
w∈U w U,2 w∈U w U,1

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Finding optimal test functions
• Uh ⊂ U is spanned by finite number of basis functions, {φi }n .
i=1
We make a basis for optimal test space Vh,opt by finding the
corresponding optimal test function for each φi as below:

vφi = R−1 Bφi in V ⇐⇒ Rvφi = Bφ in V

(vφ , v )V = Rvφ , v
ˆ ˆ V ×V = Bφ, v
ˆ V ×V = (R−1 Bφ, v )V = b(φ, v )
ˆ ˆ
• With definition T = R−1 B, optimal test functions can be
determined by solving the auxiliary variational problem

(T φ, v )V = (vφ , v )V = b(φ, v )
ˆ ˆ ˆ ∀ˆ ∈ V
v

• Symmetric, Coercive

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Advent of DG
• In standard H 1 and H(div)-conforming ﬁnite element methods,
test functions are continuous over the entire domain, requires a
global operation over the entire mesh, rendering the method
impractical.
• A breakthrough −→ discontinuous Galerkin (DG)
Basis functions are discontinuous. Considering variational
formulations in a “broken space”, the variational problems that
determine Vh become local problems that can be solved in an
element-by-element fashion.
• Even this local problem can not be solved exactly; e.g. ﬁnite
degree of polynomials in DG test functions in each element.
• In practice we can just have an estimate of the optimal test space,
called a nearly optimal test space.
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δ-proximal
• Some works ([Dahmen et al.]) on estimating the eﬀect of this
approximation on the condition number of the operator and
optimality of the solution, uh .
δ δ
• New concept of δ-proximal for Uh , Vh ⊂ V with dimVh = dimUh
and this property
δ
∀0 = vh ∈ Vh , ∃˜h ∈ Vh
v such that vh − vh
˜ V ≤ δ vh V

Then it has been proved that:
−1 1
Bh L(U,V ) ≤ 1, Bh L(V ,U ) ≤
1−δ
and
2−δ
u − uh U ≤inf u − wh U
1 − δ wh ∈Uh
for δ < 1 the problem is reasonably near ideal.
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p-enrichment
• Optimal test functions are approximated using the standard
˜
Bubnov-Galerkin method on an “enriched” subspace Vh such that
˜h ) > dim(Uh ) element-wise.
dim(V

˜
• In special case it is in the form Vh ≈ P p+∆p (K). where p is
K
the polynomial order of the trial space on a given element K.

• More details can be found in [Demkovicz et al. ’12].

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Introduction
References
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• Set a norm on trial or test space as we wish. This norm induces a
norm in the other space and provide us with unity inf − sup and
continuity constant in infinite dimensional case.
• Here we also have a rule for constructing of the test space as
optimal one in finite dimensional settings.
• Trivial inheritance of unity inf − sup and continuity constant from
infinite dimensional case.
• Optimality of the solution in our desired norm in U, usually L2
norm. This also can be inferred from Babuˇka’s theorem with
s
Mh = γ h = 1

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Introduction
References
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• Partitioning the domain Ω into N non-overlapping elements
Kj , j = 1, . . . , N such that Ωh = N Kj and Ωh = Ω. Here h is
j=1
¯ ¯
defined as h = maxj∈{1,...,N } diam(Kj ).
N
• Denote the mesh “skeleton” by Γh = j=1 ∂Kj ;
the set of all
faces/edges e, each which comes with a normal vector ne . The
internal skeleton is then defined as Γ◦ = Γh ∂Ω.
h
• If a face/ edge e ∈ Γh is the intersection of ∂Ki and ∂Kj , i = j,
we define the following jumps:

[[v]] = sgn(n− )v − + sgn(n+ )v + , [[τ .n]] = n− .τ − + n+ .τ +

For e belonging to the domain boundary, ∂Ω, we define

[[v]] = v, [[τ .n]] = ne .τ
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Basic settings and conventions (contd.)
• By introducing as trace variable and ignoring boundary conditions
for now, the ultra-weak formulation for Bu = f on Ωh reads
∗
b(u, ) := , [[v]] Γh − (u, Bh v)Ωh = (f, v)Ωh
∗
where Bh is the formal adjoint.
• Regularity requirement on solution variable u is relaxed. The
trade-oﬀ is that u does not admit a trace on Γh even though it did
originally.
• The setting is,
∗
u ∈ L2 (Ωh ) ≡ L2 (Ωh ), v ∈ V = D(Bh ), ∈ tr(D(B))
∗ ∗
D(Bh ) is the broken graph space of Bh , and tr(D(B)) the trace
space of the graph space of operator B.
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Introduction
References
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Canonical trial/test norm
• The canonical norm for the group variable (u, ) is set as:

2 2 2
(u, ) U = u L2 (Ω) + L2 (Ω)

∗
• Since v ∈ D(Bh ) the canonical norm for v is the broken graph
norm:
2 ∗ 2 2
v V = Bh v L2 (Ω) + v L2 (Ω)
• We chose these canonical norm as those that more desirable in
practice and also for calculation.

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Canonical trial/test norm (contd.)
• Using Canonical norm on U ,

b((u, ), v) ≤ (u, ) U v V,U

∗ 2 , [[v]] Γh 2
v V,U = Bh v L2 (Ω) +( sup )
∈tr(D(B))

The norm v V,U is called optimal test norm.
• Using Canonical norm on V ,

b((u, ), v) ≤ (u, ) U,V v V

2 , [[v]] Γh 2
(u, ) U,V = u L2 (Ω) + (supv∈D(Bh ))
∗ )
v V
The norm v V is called quasi optimal test norm.
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Canonical trial/test norm (contd.)
• The optimal test norm is non-localizable due to the presence of the
jump term [[v]].
• A localizable norm, v V (Γh ) can be written as

v V (Ωh ) = v V (K) .
K∈Ωh

• Evaluation of jump terms requires contributions from all the
elements in the mesh, making the optimal test norm impractical.
• Quasi optimal test norm is localizable and hence practical.
However it generates a complicated norm on trial space.
• Construction another equivalent norm in test space.

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Introduction
References
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• For some Lipschitz domain Ω ⊂ Rn , A ∈ L∞ (Ω)n×n ,
b ∈ L∞ (Ω)n , consider the boundary value problem

−divA u + b. u = f on Ω
u=0 on ∂Ω

• with some assumption on b such that

(Bu)(v) := Ω A u. v + b. uv = Ω f v,
B : H0 (Ω) → H −1 (Ω) is boundedly invertible.
1

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Convection-diﬀusion problem (contd.)
• In the case of convection domination one can easily shows

1
B L(U,V ) ≤ b ∞ B −1 L(V ,U ) ≤

Problem is well-posed for all >0

• Noticing that

b ∞
κU,V ≤ B L(U,V ) B −1 L(V ,U ) ≤

It becomes ill-conditioned as → 0 or when the convection part
b ∞ dominates the diﬀusion .

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Introduction
References
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Mixed formulation
• Introducing σ = A u the problem in mixed form reads as:

σ − A u = 0
 on Ω
−divσ + b. u = f on Ω

u=0 on ∂Ω


By deﬁning the group test function as (τ , v) this system can be
changed to variational form.

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Mild formulation
• Neither of the equations is integrated by parts.

U := H(div, Ω) × H0 (Ω), V = L2 (Ω)n × L2 (Ω),

1

given l ∈ L (Ω), f ind (σ, u) ∈ U such that f or all (τ , v) ∈ V,

 2

 b(σ, u, τ , v, ) : = (σ − A u).τ + v(−divσ + b. u)

Ω



= l(v)


• The solution of this mild variational formulation with optimal test
space solves the common ﬁrst order least squares problem
2 2
arg(σh ,uh )∈Uh min A uh −σ h L2 (Ω)n + l+divσ h −b. uh L2 (Ω)n

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Mild-weak formulation
• Second equation is integrated by parts introducing a new
independent variable on the skeleton, called “ﬂux”.
• The second equation is integrated by parts with reading
(ub − σ)|∂Ωh .n as an additional independent variable θ

U := L2 (Ω)n × H0 (Ω) × H −1/2 (∂Ωh ), V = L2 (Ω)n × H0 (Ωh ),
1 1


1

given l ∈ H0 (Ωh ) , f ind (σ, u, θ) ∈ U such that f or all (τ , v) ∈ V,





b(σ, u, θ, τ , v, ) : = (σ − A u).τ + (σ − ub). vh − divh buv

Ω





 + [[v]]θ
∂Ωh




= l(v)

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Ultra-weak formulation
• Both equations are integrated by parts. It also has two new
independent variable as “trace” and “ﬂux”.
• Restrict ourselves to b with divb = 0
 1/2
U := L2 (Ω)n × L2 (Ω) × H00 (∂Ωh ) × H −1/2 (∂Ωh ),


V = H(div, Ω ) × H 1 (Ω ), given l ∈ H 1 (Ω ) ,



 h h h
f ind (σ, u, , θ) ∈ U such that f or all (τ , v) ∈ V,




b(σ, u, , θ, τ , v, ) : = (A−1 σ.τ + u divh τ + (σ − ub). hv
Ω





= [[v]]θ − [[τ .n]]





 ∂Ωh

= l(v)


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Ultra-weak formulation (contd.)
• and θ replace the trace u|∂Ωh and ﬂux (ub − σ)|∂Ωh .n,

• In compact form and by assuming A = I and divb = 0 we have

−1
b(u, σ, , θ) = (u, .τ − b. v)Ωh + (σ, τ+ v)Ωh
− [[τ .n]], Γh + θ, [[v]] Γh

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Some space and norm deﬁnitions
• We set
1 1
H0 (Ωh ) := {v ∈ L2 (Ω) : v|K ∈ H0 (K)(K ∈ Ωh )}
H −1/2 (∂Ωh ) := {q|∂Ωh .n : q ∈ H(div; Ω)}
1/2 1
H00 (∂Ωh ) := {u|∂Ωh : u ∈ H0 (Ω)}
equipped with the “broken” norm
2 2
v H 1 (Ωh ) := v|K H 1 (K)
K∈Ωh

and quotient norms
θ H −1/2 (∂Ωh ) := inf{ q H(div;Ω) : θ = q|∂Ωh .n}
1
1/2
H00 (∂Ωh )
:= inf{ u H 1 (Ω) : u ∈ H0 (Ω), = u|∂Ωh }
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Well-posedness
• In infinite dimensional, using optimal norms defined in Lemma 1
means that M = γ = 1. What remains to be proved is the adjoint
injectivity condition.
• For further details refer to [Lazarov et al. 94] for mild, [Stevensen
et al. ’12] for mild-weak and [Demkovicz et al. ’10] for ultra-weak
formulation of convection-diffusions problems.
• For optimal test space, we have trivial well-posedness of the finite
dimensional problem.

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Introduction
References
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Optimal test norm
• Optimal test norm of the convection diﬀusion problem as

(v, τ ) V,U = .τ − b. v 2
L2 + −1 τ + v 2
L2
, [[τ .n]] Γh 2
+ (sup ∈tr(D(B)) )

q n , [[v]] Γh 2
+ (supq∈H(div,Ω) )
qn
• Unfortunately, this norm is non-localizable, quasi-optimal test norm
generates a trial norm that might not be easy to work with.
• Look for other norms that might not be ideal but a robust one in
the range of .

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Optimal test norm (contd.)
• The chosen norm should not generate boundary layers in the
solution of the optimal test functions.
• The chosen test norm should be equivalent to our optimal test
norm
τ , v V,U τ , v V,i(=1,2)
where τ , v V,i(=1,2) are constructed of a chosen test norm like
2 2 2 2 2 2
τ,v V := v L2 + v L2 + b. v L2 + 1/ τ L2 + .τ L2

by choosing some -robust coeﬃcients C as
2 2 2 2
(τ , v) V,i = Cv v L2 + C v v L2 + Cb. v b. v L2
2 2
+ Cτ / τ L2 +C .τ .τ L2

though not optimal but at least near optimal and also robust.
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Introduction
References
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Construction of a test norm, Adjoint problem
• Require a priori that the test norm has separable τ and v
components. Problem then decouples and it is easier to conclude
whether or not there are boundary layers in the solutions.
• The choose the test norm. This is implied by the mathematics of
the adjoint problem.
Consider u, by choosing (τ , v) ∈ H 1 (Ω) × H(div, Ω) such that
1
.τ − b. v = u τ+ v=0
2
u L2 = b ((u, σ, , θ) , (τ , v)) ≤ (u, σ, , θ) U,V (τ , v) V

Now if (τ , v) V u L2 then
2
u L2 (u, σ, , θ) U,E

• Show the equivalence of the energy norm to explicit norms on U .
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Introduction
References
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• Numerical test in one-dimension taken from [Stevensen et al. ’12].
The model problem is one-dimensional equation

u +u =f on Ω
u=0 on ∂Ω

This problem has an analytical solution as
1 1
u(x) = x2 + x + ( + )(ex/ − 1)/(1 − e1/ )
2 2
which has a “layer” at the outﬂow boundary x = 1.
• In this one-dimensional setting, the optimal test functions can be
determined analytically.

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L2 (0, 1)-error in uh vs. 1/h in the Galerkin, and in the Petrov-Galerkin approximations (mild/mild-weak, and
ultra-weak) for the one-dimensional convection-diﬀusion equation with = 10−4 .

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Exact solution u and the Galerkin (left), mild/mild-weak, and ultra-weak Petrov-Galerkin approximations uh for
h = 16 and = 10−4
1

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References

W. Dahmen, C. Huang, C. Schwab, G. Welper, Adaptive
Petrov-Galerkin Methods for First Order Transport Equations. , SIAM
J. Numer. Anal., 55(5): 2420-2445, 2012.
J. Chan, N. Heuer, T. Bui-Thanh, L. Demkowicz, Robust DPG
method for Convection-Dominated Diffusion Problems II: a Natural in
Flow Condition. ICES Report 12-21, University of Texas at Austin,
2012.
D. Broersen, R. Stevensen, A Petrov-Galerkin Discretization wuth
Optimal Test Space of a Mild-Weak Formulation of
Convection-Diffusion Equations in Mixed-Form. November 2012.
L. Demkowicz, N. Heuer, Robust DPG Method for
Convection-Dominated Diffusion Problems. ICES Report 11-33,
University of Texas at Austin, 2011.
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References

T. Bui-Thanh, L. Demkowicz, O. Ghattas, Constructively Well-Posed
Approximation Methods with Unity INF-SUP and Continuity. ICES
Report 11-10, University of Texas at Austin, 2011.
L. Demkowicz and J. Gopalakrishnan, Analysis of the DPG Method
for the Poisson Equation. ICES Report 11-33, University of Texas at
Austin, 2010.
L. Demkowicz , Babuˇka ↔ Brezzi?. Tech. Rep. 06-08, Institute for
s
Computational Engineering and Sciences, the University of Texas at
Austin (Appril 2006). 2006.

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Robust Petrov-Galerkin Method for Convection Problems

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