stability analysis of scalar advection diffussion equation

7/30/2019 Stability Analysis of Scalar Advection Diffussion Equation

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Stability Analysis of Scalar Advection-Diffusion Equation

M. Behr

http://www.cats.rwth-aachen.de/library/research/technotes

Abstract

This note recounts detailed stability and accuracy analysis of a scalar

advection-diffusion equation.

1. Introduction

Although stability and accuracy proofs are possible for most of the finite el-ement formulations, they are often avoided because of their complexity. Twodesirable propertiesstability and consistencyresult in convergent methods.Consistency, and to a lesser extent, stability, are typically easily determined for agiven weak form, and are slightly more difficult for discrete (Galerkin or Petrov-Galerkin) forms. In contrast, the convergence behavior is much harder to analyze;yet this analysis is crucial for determining both the order of convergence and theoptimal design of stabilization parameters.In this note, we carry out the convergence analysis for a (simple!) scalar advection-diffusion equation to its bitter end.

2. Strong Form

Consider the scalar advection-diffusion equation:

Lu = f on Rd, (1)

u = 0 on = , (2)

whereLu

.= a u u. (3)

The following non-essential assumptions are made in order to simplify an alreadycomplex proof:

the velocity field a satisfies a = 0,

is a positive constant.

We will also define the element Peclet number which quantifies the relativestrength of advective or diffusive terms:

=h|a|

2. (4)
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Stability of Scalar Advection-Diffusion Equation M. Behr

3. Galerkin Form

The Galerkin form is given as: Find uh

Vh

H10 such that:

B(wh, uh) = L(wh) wh Vh, (5)

where

B(wh, uh).

=

wha uh +wh uh

d, (6)

L(wh).

=

whf d, (7)

and B(, ) and L() are bilinear and linear forms, respectively, and H1

0

denotes theSobolev space of functions with square-integrable first derivatives and satisfyingthe homogenous boundary condition on .

3.1. Analysis Outline

In obtaining the convergence estimates, it is necessary to first determine the con-sistency and stability of the discrete form. These two properties can be thenapplied to obtain bounds on the error of the discrete solution in terms of inter-polation errors which are strictly related to element size only. At this final step,the proper form for the stabilization parameter, if any, should become apparent.

3.2. Consistency

The discrete form is consistent, if it is satisfied identically by the exact solutionu. In other words:

B(wh, u) = L(wh) wh Vh, (8)

orB(wh, e) = 0 wh Vh, (9)

where e = uhu is the error. The Galerkin form (5) is consistent by construction.

3.3. Stability

The discrete form is stable, if small deviations from the given inputsboundary conditions, domain shaperesults only in small deviations in thesolution. A more precise statement of stability is: For each uh, there exists wh

such that:B(wh, uh) > 0 if uh > 0, (10)

That also means that any non-negligible component of the solution uh will pro-duce non-negligible change in the value of the bilinear form for at least one of thepossible test functions wh. The inequality in (10) must be strict.

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Lets look at the stability of the Galerkin form. For each uh we are lookingfor a corresponding wh that would make the bilinear form non-zero. The best

candidate is often uh itself (or uh). But the Galerkin bilinear form gives:

B(uh, uh) =

uha uh +uh uh

d

=

a(uh)

2

2d +

(uh)2

2a n d

+ 1

2uh2 uh Vh. (11)

The first two terms on the right hand side are zero due to the assumptions we

made (otherwise this reasoning can still proceed but with some extra effort).The problem with the Galerkin form is now clearly seen: as 0, its stabilityvanishes. For advection-dominated flows, the bilinear form can be arbitrarilysmall for non-zero uh.Note that this instability is not proven directly here; we are just unable to find atest function that would give us a lower bound on the form value. In some senseproving stability is easier than proving instability.

4. Artificial Diffusion

A crude way of gaining stability for advection-dominated flows has been theartificial diffusion (AD) method. The bilinear form becomes:

BAD(wh, uh)

.=

wha uh +wh uh

d

+

wh uhd, (12)

where = O(h) is a positive parameter. Due to the presence of that parameter,the form is stable in the sense of the preceding section, even as 0, but it isnot consistent. The convergence proof as shown below could not proceed.

5. Galerkin Least-Squares Form

Now let us see how the picture changes if a stabilized Galerkin Least-Squares(GLS) formulation is used. The bilinear form becomes:

BGLS(wh, uh)

.=

wha uh +wh uh

d

+e

e

a wh wh

Lwh

Luhd (13)

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and it clearly remains consistent. To determine stability, we look at:

BGLS(uh

, uh

) = uha uh +uh uh d+

e

e

LuhLuhd

= 1

2uh2 + 1

2 Luh2h |||uh|||2 uh Vh. (14)

The expression BGLS(uh, uh) is thus seen to define a norm, which we will denote

as |||uh|||2. This is equivalent with stability. Why do we only include elementinteriors in the integration in the stabilization term? If the integral was over theentire domain, we would have to require the first derivatives of the interpolationfunctions to be continuous across the element boundaries, so that second deriva-

tives remain square-integrable even when the element boundaries are included.

6. Convergence

Armed with consistency and stability, we may proceed with the convergenceanalysis. We will split the solution error into two components:

e = uh uh ehVh

+ uh u H1

0

, (15)

where eh is the discrete error, is the interpolation error, and uh is the interpolant,

i.e. a function in the trial function space which coincides at the nodes with theexact solution, as shown in Figure 1. In one sense, interpolant is as close as we

eh

uh

uuh

Figure 1. Components of the discrete error.

can expect to get to the real solution (although not necessarily in the sense ofthe norms we will use). Then interpolation error is an error that cannot beavoided; it is dependent only on the mesh size, and not on the discretizationmethod we employ. The discrete error is tied to the discretization method itself,and we hope to obtain bounds for this error which are similar to the existingbounds on the interpolation error. The two bounds will combine to give us thefollowing bound on the total error:

|||e||| =

O(hk) diffusive limit

O(hk+1

2 ) advective limit(16)

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This bound will only be possible if is appropriately defined. Since we are alwaysdealing with the GLS form from now on, the GLS subscript will be dropped from

the bilinear form.

|||eh|||2 = B(eh, eh)

= B(eh, e) B(eh, )

(first term is zero by consistency)

=

eha d +

eh d

3

+

ee

LehLd

4

(integrating by parts and adding and subtracting a term)

=

a ehd +

a nehd

+e

e

eh

d

e

e

eh

d

+ 3 + 4 (boundary term is zero due to boundary conditions)

=

e

e

Lehd 1

e

e

eh

d

2

+ 3 + 4 =

1 + 2 + 3 + 4 . (17)

All four terms are mixed in the sense that they contain products of expressionsin eh and . We will try to make them bounded by a sum of expressions in eh

alone and in alone. Then the eh terms will be absorbed by the left hand side|||eh|||2, and the bound will depend on the interpolation error alone! For terms 1

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and 2, we will use this identity:

12 a2

1

2 b2

= a24

+ b2

ab > 0, (18)

and for terms 3 and 4, another identity:a2

b2

=a2

4+ b2 ab > 0. (19)

to obtain bounds on ab. Consequently,

1 1

4

1

2 Leh2h + 1

2 2 (20)

+

2 1

4

1

2

eh

2h +

12 2 (21)

+

3 1

4

1

2eh2 + 1

22 (22)

+

4 1

4

1

2 Leh2h + 1

2 L2h. (23)

absorbed by |||eh|||2

The only eh term that does not have a matching term inside |||eh|||2 is the firstof the two terms in (21); for an appropriate definition of (forthcoming), it canbe replaced in the inequality using the inverse estimate:

1

2 eh 2h 12eh2. (24)The first set of terms on the right-hand side of the inequalities add up to 1

2|||eh|||2,

which can be subtracted from the left-hand side. The remaining right-hand sidedepends only on the interpolation error:

1

2|||eh|||2 2

1

2 2 +

1

22 + 1

2 L2h

||||||2

, (25)

or|||eh|||2 4

1

2 2 + 2||||||2. (26)

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Finally, the total error can be bounded by:

|||e|||2

2 |||eh|||2 + ||||||2 8 12 2 + 6||||||2. (27)From interpolation theory it is known that interpolation errors using polynomialsof order k obey:

||m = O(hk+1m), (28)

as long as elements are quasi-uniform, or not too distorted. Therefore, the normson the right-hand side of (27), are of the order:

||||||2 =O(h2k) + 2O(h2(k1)) diffusive limit, (29)

||||||2 = |a|2O(h2k) advective limit, (30)

1

2 2 =1

O(h2(k+1)). (31)

We can see that should be a polynomial in h to improve the last, weak, boundin (29), but of order low enough not to destroy the bound in ( 31). The optimalconvergence (16) is obtained with this design:

=O(h

|a|), advective limit, (32)

= O(

h2

), diffusive limit. (33)

Serendipitously (finite element people like this word), with this choice, the in-equality (24) also holds. Note that (16) does not indicate that if we use linearinterpolation functions, the solution will converge e.g. linearly in the diffusivelimit. The norm used involves derivatives, and it is the first derivatives thatwill converge linearly. The solution itself will then converge quadratically in thediffusive limit.

7. Inverse Estimate

For 1D linear element, the inverse estimate can be stated as:

1

C0||q||20

K

h2K ||q||20

K

, (34)

1

C0||q||20

K

h2K ||q||20

K

. (35)

Denoting by q the average value of q in the element (at midpoint), and by q therange of q within that element, we can write:

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||q||20K =

hK0

q q2

+ q

hK

2 d = q2 + q212hK, (36)

||q||20K

=

hK0

q

hK

2d =

q2

hK, (37)

1

C0||q||20

K

=1

C0

q2 +

q2

12

hK

1

C0

q2

12hK

q2

hKh2K = h

2K ||q||

20

K

. (38)

The second-to-last inequality is satisfied if C0 =112

.

History

May 11, 2001 Written based on the Tom Hughes notes.November 21, 2001 Added inverse estimate section.June 6, 2004 Corrected Eq. (31).

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stability analysis of scalar advection diffussion equation

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