OXFORD AUCKLAND ROSTON JOHANNESBURG MELBOURNE NEW DELHI

Fifth edition

UTTERWORTHEINEMANN

R.L. TaylorProfessor in the Graduate School

Department of Civil and Environmental EngineeringUniversity of California at Berkeley

Berkeley, California

The Finite ElementMethod

Volume 3: Fluid Dynamics

O.c. Zienkiewicz, (BE, FRS, FREngUNESCOProfessor of Numerical Methods in Engineering

International Centre for Numerical Methods in Engineering, BarcelonaEmeritus Professor of Civil Engineering and Director of the Institute for

Numerical Methods in Engineering, University of Wales, Swansea

Butterworth-HeinemannLillacre House, Jordan Hill, Oxford OX2 8DP

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First published in 1967 by McGraw-HiliFifth edition published by Butterworth-Heinemann 2000

((',O.C Zienkiewicz and R.L. Taylor 2000

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f-Ott f\EU nIlE lIL'T M f'WWH. hUlTIUI....OKllHIf.ll'£IVJC,'1"lU, , ...r fOf! R11S 1 0 PUI'-l ,\,'\11 G ,n POll ,,\ lfll:f

il'~j

1:'·",.~

.~"~

(B. 1)

(B.2)k(O) ~~{~l-gdx -and¢(L) = ¢

• J.T. Oden, personal communication, 1999.

Discontinuous Galerkin methodsin the solution of the

convection-diffusion equation*

d¢ d ( d¢) .u-- - ~ k(x)- = Idx dx dx .

where II is the convection velocity, k = k(x) the diffusion (conduction) coefllcient(always bounded and positive), and.r = .r(x) the source term. We add boundaryconditions to Eq. (B.1); for example,

In Volume I of this book we have already mentioned the words 'discontinuousGalerkin' in the context of transient calculations. In such problems the discontinuitywas introduced in the interpolation of the function in the time domain and somecomputational gain was achieved.

In a similar way in Chapter 13 of Volume I, we have discussed methods which havea similar discontinuity by considering appropriate approximations in separateclement domains linked by the introduction of Lagrangian multipliers or otherprocedures on the interface to ensure continuity. Such hybrid methods are indeedthe precursors of the discontinuous Galerkin method as applied recently to lluidmechanics.

In the context of fluid mechanics the advantages of applying the discontinuousGalerkin method are:

• the achievement of complete flux conservation for each element or cell in \vhich theapproximation is made:

• the possibility of using higher-order interpolations and thus achieving highaccuracy for suitable problems;

• the method appears to suppress oscillations which occur with convective tennssimply by avoiding a prescription of Dirichlet boundary conditions at the flowexit; this is a feature which we observed to be important in Chapter 2.

To introduce the procedure we consider a model of the steady-state conl'ection-diffusion problem in one dimension of the form

294 Appendix B

(B.8)

(B.7)

(8.6)

(B.3)

(B.5)

(BA)

It = -[v](x,,)

/ dll)\ k dx (x,.) = °and

III2.:P[¢] + µ(k d¢ dx) }(xe)e=1

[cf>](x,.) = 0

111 f" (d¢ dv d(uv) ) III {\ dV) \ d¢) }2.:, k---.--~--:-¢ dx+2.: k---. [¢]- k- [v] (xc)e = I \,- I d.\ dx dx e = I dx dx

( dl') ( ddJ)+ kef;': ¢(L) -I) k d~ (L) +V¢-lI(O)

III j" d= [;. -',:_1 Jl!dx + k d~ (L)q; + v(O)g - IIL'(O)¢

where .\ and µ an: the multipliers. A simple calculation shows that the multiplierscan be identified as average fluxes and interface jumps:

3. The Dirichlet boundary conditions (an inflow condition) enter the weak form onthe left-hand side, an uncommon property, but one that permits discontinuousweight functions at relevant boundaries.

4. The signs of the second term on the left side (Le{(kv'[¢]) - (k(,?')[v]}) can bechanged without affecting the equivalence of Eq. (B.3) and Eqs (B.I) and (B.2),but the particular choice of signs indicated turns out to be crucial to the stabilityof the discontinuous Galerkin method (DGM).

5. We can consider the conditions of continuity of the solution and of the fluxes atinterelement boundaries, conditions (8.6), as constraints on the true solution.Had we used Lagrange multipliers to enforce these constraints then, instead ofthe second sum on the left-hand side of Eq. (B.3), we would have terms like

it being understood that Xe_t = Iim,_.o(xe ± E), Vi = dvjdx etc.The particular structure of the weak statement in Eq. (B.3) is significant. We make

the following observations concerning it:

I. Ih/J = ¢(x) is the exact solution ofEqs (B.I) and (B.2), then it is also the (one and only)solution of Eg. (B.3); i,e. Egs (B.I) and (B.2) imply the problem given by Eg. (B.3).

2. The solution of Egs (B.l) and (B.2) satisfies Eg. (B.3) because if> is continuous andthe l1uxes k d¢jdx arc continuous:

and [.] denote jumps

for arbitrary weighl functions v. Here (.) denotes (flux) averages

(kv')(x<,) = k1)' (x~~-± .~~,'(xc-=:)2

As usual the domain n = (0, L) is partitioned into a collection of N elements(intervals) He = (Xe- 1, Xe), e = 1,2, ... , m. In the present case, we consider the specialweak form of Egs (B. I) and (B.2) defined on this mesh by

'II.

ilil:. I, /,

".,111 1,. ,I' ~

Appendix B 295

(B.9)

(B.IO)j= 1,2, .... Pe' e= 1,2, ... ,111

This is the DGM approximation of Eq. (B.3). Some properties of Eq. (B.IO) arenoteworthy:

I. The shape [unctions Nk need not be the usual nodal based functions; there are nonodes in thisformulation. We can take N'k to be any monomial we please (represent-ing, for example, complete polynomials up to degree Pe for each element n,. andeven orthogonal polynomials). The unknowns are the coefficients ilk which arenot necessarily the values of ¢ at any point.

2. We can usc different polynomial dcgrees in each element n..; thus Eq. (B.IO)provides a natural setting for lip-version finite element approximations.

3. Suppose u = O. Then the operator in Eq. (B. I) is symmetric. Even so. theformulation in Eq. (B.IO) leads to an unsymmetric stiffness matrix owing to thepresellt.:e of the jump terms and averages 011 the element interfaces. HO\vever. itcan be shown that the resulting matrix is always positive definik. the choice ofsigns in the boundary and interface terms bcing critical for preserving thisproperty.

4. In general, the formulation in Eq. (B.IO) involves more degrees of freedom thanthe conventional continuous (conforming) Galerkin approximation of Eqs (8.1)and (8.2) owing to the fact that the usual dependencies produced in enforcingcontinuity across element interfaces are now not present. However, the verylocalized nature of the discontinuous approximations wntrihutes to the surprisingrobustness of the DGM.

p,o ; ""'" e .,e( )q>~<p= Lak1vk x

k=O

where the ilk are undetermined constants and N'k =.l are monomials (shapefunt.:tions) of degree k each associated only with ne. Introdut.:ing Eq. (B.9) into(B.3) and using, for example. complete polynomials Ne of degree Pe for weight func-tions in each element, we arrive at the discrete system

Introducing Eq. (B.8) into Eq. (B.7) gives the second term on the left hand side ofEq. (B.3). Incidently, had we construt.:ted independent approximations of), and µ,a setting for the construction of a hybrid finite clement approximation of Eq. (B. I)and Eq. (8.2) would be obtained (see Chaptcr 13, Volume I).

We are now ready to construct the approximation of Eqs (B. I) and (8.2) by theDGM. Rctuming to Eq. (B.3), we introduce over each dement ne a polynomialapproximation of (p;

296 Appendix B

(B. I I )

1.00.6

Exactp=2-----p= 3 ---------p=4 ..p=5----

0.2ox

fd¢ IX,

,I dx + k - -\ = 0• !!< dx x,_,

-0.6 -0.2-2.5

-1.0

-2.0

-1.0 .-

-0.5 .-

-1.5

~ 0

Fig. 81. Continuous Galerkin approximation.

0.5 .-

1.0

2.5,2.0 I"1.5

5. While the piecewise polynomial basis {N i,...,N;, ' ... :NI' : ... , N;,,} containscomplete polynomials from degree zero up to p = P minePe, numerical experi-ments indicate that stability demands p ~ 2, in general.

6. The DGM is elementwise conservative while the standard finite element approxi-mation is conservative only in element patches. In particular, for any element ne,

we always have

This property holds for arbitrarily high-order approximations Pe'

The DGM is robust and essentially free of the global spurious oscillations ofcontinuous Galerkin approximations when applied to convection-diffusionproblems.

We now consider the solution to a convection-diffusion problem with a turningpoint in the middle of the domain. The Hemker problem is given as follows:

k~~<i> + x~di!. = _kJr2 cos (7rx) - 7rxsin(r.x) on [0, I]dX2 x

with ¢(-I) = -2. 1>( 1) = O. Exact solution for above shows a discontinuity of

cP(x) = cos(7rx) + erf(xIVfk)/erf( l/~)

Figures B.1 and B.2 show thc solutions to the above problem (k = 10-10 andII = 1/10) obtained with the continuous and discontinuous Galerkin method,respectively. Extension to two and three dimensions is discussed in references givenin Chapter 2.

Fig. 82. Discontinuous Galerkin approximation.

Appendix B 297

1.00.6

Exactp=2-----p= 3 u_u_._.p=4 ' .p=5----

0.2ox

-0.2j

-0.6

2.5

1.5

2.0

1.0

0.5

-1.5

-2.0

-2.5-1.0

;:, 0