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Appendix BDiscontinuous Galerk in m ethodsin th e solution of th econvection-diff usion equation

In Volume 1 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 1, we have discussed methods which havea similar discontinuity by considering appropriate approximations in separateelement 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 fluidmechanics.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 which 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 termssimply 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 convection-dzflision problem in one dimension of the form3-+g)=fx dx O d X d L

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

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

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294 AppendixAs usual the domain R 0,L) is partitioned into a collection of N elements(intervals) 0 (x+ 1, x, , e 1 , 2 , . . . m n the p resen t case, we consid er the specialweak form of Eqs ( B.l) an d (B.2) defined on this mesh by

for arb itrary weight functions w Here . ) den otes (flux) averages

it being understood that x,+ l imE-O(x, , v dv/dx etc .Th e particu lar structu re of the weak statem ent in Eq. (B.3) is significant. W e mak ethe following observations conc erning it:1. If 4 (x) s the exact solution of Eqs (B. 1) an d (B.2), then it is also the (one an d on ly)2 . Th e so lution of E qs (B. 1) an d (B.2) satisfies E q. (B.3) because q I is continuous andsolution of E q. (B.3); i.e. Eqs (B. 1) an d (B.2) imply the pro blem given by E q. (B.3).the fluxes k dq5ldx are con tinuou s:

[4](xe) 0 and ( k g ) ( x e ) 03. Th e Dirichlet bou ndary cond itions (an inflow condition) enter the weak form o nthe 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 (Ce{( kv [4] ) k )[v])) can bechanged without affecting the equivalence of Eq. (B.3) and Eqs (B.l) and (B.2),

but the particular choice of signs indicated turn s o ut to be crucial to the stabilityof the discontinuous Galerkin method (D GM ).5. We ca n consider the conditions of continuity of the solution and of the fluxes a tinterelement boundaries, conditions (B.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

where X an d are the multipliers. A simple calcu lation sh ow s th at the multiplierscan be identified as average fluxes and interface jumps:

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Appendix 295Introducing Eq. (B.8) into Eq. (B.7) gives the second term on the left hand side ofEq. (B.3). Incidently, had we constructed independent approximations of X and pa setting for the construction of a hybrid finite element approximation of Eq. (B. 1and Eq. (B.2) would be obtained (see Chapter 13, Volume I .

We are now ready to construct the approximation of Eqs (B.1) and (B.2) by theDGM. Returning to Eq. (B.3), we introduce over each element 0 a polynomialapproximation of 4;

k O

where the a; are undetermined constants and Nt xk are monomials (shapefunctions) of degree k each associated only with 0 . Introducing Eq. (B.9) into(B.3) and using, for example, complete polynomials N, of degree p e for weight func-tions in each element, we arrive at the discrete system

{ k s ) [ q ] } a Z+ { (k%)Nt(L) N; (kT) (L) + NT(O)u(O)Nt-(O) i.j = l , 2 . . . , p , e = 1 , 2 ... B.lO)

This is the DGM approximation of Eq. (B.3). Some properties of Eq. (B.lO) arenoteworthy:1 . The shape functions Ni need not be the usual nodal based functions; there are nonodes in this o rm ulatio n. We can take Nl to be any monomial we please (represent-

ing, for example, complete polynomials up to degree p , for each element 0 andeven orthogonal polynomials). The unknowns are the coefficients uz which arenot necessarily the values of t any point.

2. We can use different polynomial degrees in each element 0 ; thus Eq. (B.lO)provides a natural setting for Izp-version finite element approximations.3 . Suppose u 0. Then the operator in Eq. (B.l) is symmetric. Even so, the

formulation in Eq. (B.lO) leads to an unsymmetric stiffness matrix owing to thepresence of the jump terms and averages on the element interfaces. However, itcan be shown that the resulting matrix is always positive definite, the choice ofsigns in the boundary and interface terms being critical for preserving thisproperty.

4. In general, the formulation in Eq. (B. 10) involves more degrees of freedom thanthe conventional continuous (conforming) Galerkin approximation of Eqs (B. 1)and (B.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 contributes to the surprisingrobustness of the DGM.

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296 Appendix5. While the piecewise polynomial basis { N ; , . ,NF,,. . . N ; , . . . ,N;,,} containscomplete polynomials from degree zero up to p =pmin,p,, 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 Re,

we always havejQ.fdx + k g 0 (B.ll)

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 turning

point in the middle of the domain. The Hemker problem is given as follows:d2d dd 2k - +x- kx cos(.lrx) xsin(7rx) on [0, I ]dx2 dx

with (-1) -2, 4(1) 0. Exact solution for above shows a discontinuity of4 x) cos(7i.x)+ e r f ( x / G ) / e r f ( l / G )

Figures B.l and B.2 show the solutions to the above problem k lo-'' andh 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. B1. Continuous Galerkin approximation.

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Appendix 297

Fig. B2. Discontinuous Galerkin approximation.