Finite Element Methods for the Advection-Diffusion Equation

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```					Intro                CG Method               DG Method             C-DG Method   The Future

Finite Element Methods for the

John Chapman

Mathematical Sciences, Durham University

8th May 2008

John Chapman                                                                       Durham
Intro                CG Method        DG Method   C-DG Method   The Future

Outline

Introduction

The Continuous Galerkin Method

The Discontinuous Galerkin Method

The Continuous-Discontinuous Galerkin Method

Work for the Future

John Chapman                                                      Durham
Intro                CG Method           DG Method         C-DG Method   The Future

We will be considering methods of solution for the model
advection-diﬀusion (A-D) equation which is given by

−ε∆u + b. u = f    in Ω ⊂ Rn
u = g on ∂Ω

with ε > 0 the diﬀusion coeﬃcient, b ∈ [L2 (Ω)]n the velocity
vector ﬁeld and f, g ∈ L2 (Ω). We take Ω to be bounded and
polygonal.

John Chapman                                                               Durham
Intro                CG Method        DG Method       C-DG Method        The Future

Physical Interpretations and Applications

Many relevant physical applications.
These examples are not exhaustive.
Two diverse problems to demonstrate the width of application.
Examples drawn from “Numerical Solution of
Convection-Diﬀusion Problems”, Morton (1996).

John Chapman                                                               Durham
Intro                CG Method            DG Method          C-DG Method   The Future

Pollutant dispersal in a river

Consider a river surface as a two dimensional domain which
contains a depth averaged pollutant concentration c. Then the
steady state representation of its dispersal may be given by

v. c =    .(D c) + S

where S is the source of the pollutant, v a horizontal velocity ﬁeld
and D a diﬀusivity coeﬃcient. This problem is representative of
advection diﬀusion problem if D is independent of c.

John Chapman                                                                   Durham
Intro                CG Method               DG Method    C-DG Method       The Future

Semi-conductor equations

Two parts of the basic model for the steady state distribution of
electrons in a semi-conductor are
qR =    .Jn
Jn = qµn (Ut n − n ψ)

Clearly a substitution of the second equation into the ﬁrst results
in a advection-diﬀusion problem for n.

John Chapman                                                                  Durham
Intro                  CG Method                DG Method        C-DG Method   The Future

Analytic Solutions and Behaviour

Analytic Solution of A-D Eqns.
In simple cases it is possible to solve our model A-D equations
analytically using asymptotic expansions or the method of
characteristics. We won’t go into detail here, but use some results
to further our understanding of the behaviour of the equations.
Let us consider the simple equation

−εu + u = 1 in (0, 1),             u(0) = u(1) = 0

which has the solution
exp(− 1−x ) − exp(− 1 )
ε            ε
u(x) = x −
1 − exp(− 1 )
ε

John Chapman                                                                     Durham
Intro                  CG Method      DG Method         C-DG Method            The Future

Analytic Solutions and Behaviour

Boundary Layers
Plotting this equation for several values of ε we see the onset of a
boundary layer.

Figure: Advection Diﬀusion Boundary Layer for ε = 1.0, 0.1, 0.01

John Chapman                                                                     Durham
Intro                 CG Method       DG Method         C-DG Method         The Future

General Finite Element Methods

A Short Introduction to Galerkin Finite
Element Methods
The emphasis here is on short, with details being expanded in the
next section.
Find the weak formulation of the problem.
Replace the inﬁnite dimensional space with a ﬁnite
dimensional subspace, resulting in a problem we can solve.
Take successively larger subspaces and show that the solution
converges to the solution of the original problem.
This is the Galerkin method.
If the subspace is of piecewise polynomials on a triangulation
of the region, we call it a Finite Element Method.

John Chapman                                                                    Durham
Intro                CG Method                DG Method         C-DG Method   The Future

CG FEMs for the Advection Diﬀusion Equation

Formulation of the Problem

Returning to our model problem

−ε∆u + b. u = f         in Ω ⊂ Rn
u = 0 on ∂Ω

with g = 0 we follow the steps of the previous slide. The weak
formulation is
1
Bε (u, v) = ε( u, v) + (b. u, v) = (f, v) ∀ v ∈ H0 (Ω)

with ( , ) being the usual L2 inner product.

John Chapman                                                                    Durham
Intro                CG Method                 DG Method                C-DG Method   The Future

CG FEMs for the Advection Diﬀusion Equation

Existence and Uniqueness of Solution

Provided we can show

|Bε (u, v)| ≤ α u        1
H0     v      1
H0   continuity,

2
Bε (u, u) ≥ β u      H01    coercivity

and have f ∈ V , existence and uniqueness follow from the
Lax-Milgram theorem.

John Chapman                                                                            Durham
Intro                CG Method                DG Method          C-DG Method   The Future

CG FEMs for the Advection Diﬀusion Equation

Finite Element Approximation

Let T be a shape regular triangulation of Ω. We consider a space

Vh := {v ∈ H 1 (Ω) : v|T ∈ P k (T ), v|∂Ω = 0, ∀T ∈ T },

where P k (T ) is the space of polynomials of degree at most k
supported in T . Then the continuous Galerkin ﬁnite element
approximation to the A-D equation is: Find uh ∈ Vh such that

Bε (uh , v) = (f, v) ∀v ∈ Vh .

and existence and uniqueness again follows from Lax-Milgram.

John Chapman                                                                     Durham
Intro                CG Method                  DG Method            C-DG Method            The Future

CG FEMs for the Advection Diﬀusion Equation

Stability of CG
We want to demonstrate that this ﬁnite element approximation has
not got good error characteristics as ε → 0.
e
We derive from C´a’s lemma

(ε2 + b   2 )1
2                       ∞ 2                        2
u − uh         1
H0 (Ω)   ≤                     (1 + c) u − vh        1
H0 (Ω)
ε
which indicates a problem as
1
(ε2 + b        2 )1                          b 2             2
∞ 2
(1 + c) = (1 + c) 1 + 2∞
ε                                       ε
→ ∞ as ε → 0.

John Chapman                                                                                  Durham
Intro                CG Method                DG Method   C-DG Method       The Future

CG FEMs for the Advection Diﬀusion Equation

Behaviour of Numerical Solutions

Figure: Instability in CG Method (Ern and Guermond, p.166)

John Chapman                                                                  Durham
Intro                CG Method        DG Method        C-DG Method          The Future

The Discontinuous Galerkin Method

Clearly the application of a continuous Galerkin method is
limited.
One alternative is the discontinuous Galerkin method.
We replace discontinuous test space with a discontinuous one.
There are other solutions too, but we do not cover them here.

John Chapman                                                                  Durham
Intro                 CG Method                   DG Method   C-DG Method   The Future

History and Development of the Discontinuous Method

Initial Work

Introduced by Reed and Hill (1973) for solving the Neutron
Transport Equation.
Analysis followed, notably by LeSaint and Raviart (1974) who
proved a rate of convergence of (∆x)k .
This rate has been improved upon, with Peterson (1991)
proving an optimal order in the general case.
Better results have been proven for the case of special meshes
(e.g. Richter (1988)).

John Chapman                                                                  Durham
Intro                 CG Method                   DG Method   C-DG Method    The Future

History and Development of the Discontinuous Method

Extension to the A-D Equations

Richter (1992) presented a extension to the linear A-D
equations, including an optimal order of convergence in the
Bassi and Rebay (1997) formulated a method of DG space
discretisation for the compressible Navier Stokes equations
(related to this problem).
Cockburn and Shu (1998) introduced the Local Discontinuous
Galerkin (LDG) method, rewriting the original system as a
larger ﬁrst order system which is then solved using DG
methods.

John Chapman                                                                   Durham
Intro                 CG Method                   DG Method   C-DG Method   The Future

History and Development of the Discontinuous Method

u
Houston, Schwab and S¨li (2000) considered an extension to
the method that does not rely on streamline stabilization.
Buﬀa, Hughes and Sangalli (2006) presented a new approach
to the stability of the DG method for the A-D equation, which
we will look at in more detail later.

John Chapman                                                                  Durham
Intro                CG Method                      DG Method   C-DG Method   The Future

The DG Method for the Advection Diﬀusion Equation

Formulation of the Problem

The test space is now discontinuous

Vh := {v ∈ L2 (Ω) : v|T ∈ P k (T ) ∀T ∈ T }

and we introduce the notation for averages and jumps across edges
1
{v} = (v + + v − ) [v ] = v + n+ + v − n−
2
1
{v} = (v+ + v− ) [v] = v+ n+ + v− n−
2

John Chapman                                                                    Durham
Intro                CG Method                      DG Method        C-DG Method   The Future

The DG Method for the Advection Diﬀusion Equation

Elements and Normals

T+
n-
T-

n+

Figure: Normals and ± Regions

John Chapman                                                                         Durham
Intro                CG Method                      DG Method   C-DG Method   The Future

The DG Method for the Advection Diﬀusion Equation

More Formulation of the Problem

Call the boundary of the region ∂Ω = Γ and take the inﬂow and
outﬂow boundaries Γ− , Γ+ with

Γ− = {x ∈ Γ : b(x).n(x) ≤ 0}
Γ+ = {x ∈ Γ : b(x).n(x) > 0} .

For a triangulation T the skeleton Eh can be decomposed into Eh ,o

the internal edges and Γ for the external edges (abusing notation).
For each T ∈ T we can similarly denote the inﬂow and outﬂow
edges by ΓT .

John Chapman                                                                    Durham
Intro                CG Method                      DG Method        C-DG Method           The Future

The DG Method for the Advection Diﬀusion Equation

Representation of Region and Element

Γ-                 Γ+

Γ-          T          +
T
Γ
T

b

Figure: Boundary of Region Ω and Element T

John Chapman                                                                                 Durham
Intro                CG Method                      DG Method   C-DG Method   The Future

The DG Method for the Advection Diﬀusion Equation

There are diﬀerent presentations of the DG method corresponding
to diﬀerent penalizations of the use of discontinuous spaces. Here
we the the symmetric interior penalty method (SIPG), but other
approaches are just as valid. See Arnold et al. (2001) for a
thorough approach to diﬀerent methods (applied to the elliptic
equation).

John Chapman                                                                    Durham
Intro                CG Method                         DG Method                  C-DG Method            The Future

The DG Method for the Advection Diﬀusion Equation

The DG Method
Deﬁne the bilinear forms

Bd (u, v) =                        u. v −                     { u.n}[v ] + { v.n}[u]
T ∈T        T                      o
e∈Eh     e

−                 ( u.n)v + ( v.n)u +                           σh−1 [u].[v ]
⊥
e∈Γ       e                                      e∈Eh   e

Ba (u, v) =                      (b. u)v −                        (b.n)[u]v +
T                                e
T ∈T                             e∈Γ−
T

−                     (b.n)u+ v +
e
e∈Γ−
Bε = εBd + Ba

John Chapman                                                                                                  Durham
Intro                CG Method                      DG Method    C-DG Method   The Future

The DG Method for the Advection Diﬀusion Equation

The DG method

The discontinuous Galerkin method is: Find uh ∈ Vh such that

Bε (uh , v) = (f, v) ∀v ∈ Vh .

John Chapman                                                                     Durham
Intro                CG Method                      DG Method                  C-DG Method   The Future

The DG Method for the Advection Diﬀusion Equation

A Note on Norms

The norms used are
2
v   d   = Bd (v, v)
1
v   2
a   =       ( .b)v 2 +              |b.n| [v ]2
Ω 2                     Eh
2               2           2
v   DG      =ε v    d   + v     a
2              2                             2
v       SDG     = v    DG   +            τT b. v     T
T ∈T

John Chapman                                                                                   Durham
Intro                CG Method                      DG Method   C-DG Method           The Future

The DG Method for the Advection Diﬀusion Equation

Stability Using the Method of Buffa, Hughes
and Sangalli

Theorem
There exists σ such that for all σ ≥ σ ,
ˆ                       ˆ
Bε (v, w)
inf sup                    ≥ζ>0
v∈Vh w∈Vh        v SDG w SDG

where ζ is independent of h,ε,b and the domain.

Proof.
Show w SDG       v SDG and Bε (v, w) ≥ β v                 2      using the
SDG
method of Buﬀa, Hughes and Sangalli (2006).

John Chapman                                                                            Durham
Intro                CG Method                      DG Method   C-DG Method          The Future

The DG Method for the Advection Diﬀusion Equation

Why inf sup?

˜           ∗
Consider the map B : Vh → Vh .
˜
What can one say about B −1 ?
˜
In fact, the inf sup constant gives a bound on B −1            Vh →Vh ,
∗

provided it is independent of ε.
This implies stability.

John Chapman                                                                             Durham
Intro                 CG Method          DG Method           C-DG Method            The Future

Limitations of the Methods

Advantages and Limitations of the DG
Method

Pros
Can be made arbitrarily accurate by increasing the order of
approximating polynomial element by element.
Highly parallelizable.
Suitable for complex geometries.
Easy to apply mesh reﬁnement due to lack of continuity.
Cons
Increase of degrees of freedom over CG methods.
More costly to compute (in general).

John Chapman                                                                          Durham
Intro                CG Method        DG Method   C-DG Method       The Future

CDG Method

Characteristics of a New Method

Global stability.
Ease of implementation.
Low cost.
We consider the work started by Max Jensen, Andrea Cangiani and
Emmanuil Georgoulis in 2006.

John Chapman                                                          Durham
Intro                CG Method        DG Method        C-DG Method        The Future

CDG Method

Description of the Problem

We look for a method that implements
CG away from boundary layers in a region ΩC (low cost).
DG around the boundary layer in a region ΩD (high stability).
New behaviour at the junction of the CG and DG regions.
We look at each of these points separately.

John Chapman                                                                Durham
Intro                CG Method                                                  DG Method        C-DG Method   The Future

CDG Method

Ω

ΩD
Assumptions for unperturbed region valid

ΩC

Figure: Partition into CD and DG Regions

John Chapman                                                                                                     Durham
Intro                CG Method             DG Method         C-DG Method   The Future

CDG Method

Finding a CG region

Consider again the original A-D equation, with ε small. Here we
want the solution u to be close to a solution uC of an unperturbed
problem, such as

b. uC =f    in ΩC ⊂ Rn ,
uC =0 on ∂ΩC− .

We can formalize this relationship and prove that for a solution
u = uC + uD then u → uC in H 1 (ΩC ) as ε → 0. The assumptions
made to show this result are reasonable for smooth domains away
from boundary layers.

John Chapman                                                                 Durham
Intro                CG Method          DG Method           C-DG Method       The Future

CDG Method

Stability on the CG region

Theorem
Let uε,h be the DG approximation to u = uC + uD on Vh . Assume
uC ∈ H 2 (ΩC ). Assume k ≥ 1. Assume shape-regularity. Assume
− .b ≥ γ > 0 on ΩC . Then
√                1
|u − uε,h |H 1 (ΩC ) ≤ C(f )(k 2 ε + k 4 + k 2 h 2 + h)

Proof.
Show the result on uC and uD , then use linearity of the method to
1
extend to u. This holds provided h ∈ (1, 2 ] and ε ∈ [0, 1].

John Chapman                                                                    Durham
Intro                CG Method             DG Method        C-DG Method     The Future

CDG Method

Finding a DG region

On the region ΩD we can use the results of Buﬀa, Hughes and
Sangalli, as deﬁned previously. We take a space of test functions

Vh :={v ∈ L2 (Ω) : v|T ∈ P k (T ),
v|∂Ω∪∂ΩC = 0, v|ΩC ∈ H 1 (ΩC ) ∀T ∈ T }.

Notice that this space means that on ΩC the usual continuous
Galerkin method will be implemented. We have now partitioned
our region as shown on a previous slide.

John Chapman                                                                  Durham
Intro                CG Method         DG Method            C-DG Method        The Future

CDG Method

Region Interaction

It’s not clear if extra proof is needed to resolve the region
boundary.
We have shown that the DG method works on the CG region
and on the DG region.
I present the current, not ﬁnal, thinking.
Take the streamline diﬀusion parameter τ to be switched oﬀ
in the boundary region.
Consider the solution across critical edges.

John Chapman                                                                     Durham
Intro                CG Method                     DG Method                   C-DG Method   The Future

CDG Method

No man's land

ΩC                                                        ΩD

τ=0                      τ≠0

Critical edge
Assumptions for unperturbed region valid

Figure: Element Boundary

John Chapman                                                                                   Durham
Intro                CG Method        DG Method        C-DG Method         The Future

CDG Method

Numerical Experiments

Results of numerical experiments performed by Cangiani,
Georgoulis and Jensen (2006).
The results are for a 10 × 10 grid with Ω = (0, 1)2 , ε = 10−2 ,
b = (1, 1)T , f = 1.

John Chapman                                                                 Durham
Intro                   CG Method                          DG Method                              C-DG Method                                        The Future

CDG Method

2                                                                   2

1.5                                                                 1.5

1                                                                   1

0.5                                                             1   0.5                                                              1
0.8                                                                  0.8
0.6                                                                  0.6
0                                            0.4                    0                                             0.4
0    0.2                                                            0    0.2
0.4                        0.2                                     0.4                          0.2
0.6    0.8                                                            0.6   0.8
1   0                                                                1   0

Figure: Comparison of DG (left) and CDG (right) Methods with f=1,
ε = 10−2 on a 10x10 grid

John Chapman                                                                                                                                           Durham
Intro                CG Method             DG Method   C-DG Method        The Future

The Future

Stability Across the Whole Region

Conjecture
There exists σ such that for all σ ≥ σ ,
ˆ                       ˆ
Bε (v, w)
inf sup               ≥ζ>0
v∈Vh w∈Vh   v SDG w SDG

where ζ is independent of h,ε,b and the domain.

Idea for Proof
Follow the method as for DG and adapt it to handle the critical
edges. The diﬃcult question is how to turn τ oﬀ.

John Chapman                                                                Durham
Intro                CG Method         DG Method         C-DG Method   The Future

The Future

Complete Stability Analysis

Complete proof above.
Incorporate the second join into the stability proof.
Perform some numerical experiments.

John Chapman                                                             Durham
Intro                CG Method        DG Method        C-DG Method     The Future

The Future

Conclusions

equation.
The development of stable DG methods.
Search for a suitable combined method with favourable
characteristics of both CG and DG.

John Chapman                                                                Durham