Appendix B Discontinuous Galerkin methods in the solution of the convection-diff usion equation* In Volume 1 of this book we have already mentioned the words ‘discontinuous Galerkin’ in the context of transient calculations. In such problems the discontinuity was introduced in the interpolation of the function in the time domain and some computational gain was achieved. In a similar way in Chapter 13 of Volume 1, we have discussed methods which have a similar discontinuity by considering appropriate approximations in separate element domains linked by the introduction of Lagrangian multipliers or other procedures on the interface to ensure continuity. Such hybrid methods are indeed the precursors of the discontinuous Galerkin method as applied recently to fluid mechanics. In the context of fluid mechanics the advantages of applying the discontinuous Galerkin method are: 0

0

0

the achievement of complete flux conservation for each element or cell in which the approximation is made; the possibility of using higher-order interpolations and thus achieving high accuracy for suitable problems; the method appears to suppress oscillations which occur with convective terms simply by avoiding a prescription of Dirichlet boundary conditions at the flow exit; 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 convectiondzflision problem in one dimension of the form

3-+g)=f dx

dx

OdXdL

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 boundary conditions to Eq. (B. 1); for example,

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

294 Appendix B

As usual the domain R = (0,L) is partitioned into a collection of N elements (intervals) 0, = (x+ 1, x,), e = 1,2, . . . ,m.In the present case, we consider the special weak form of Eqs (B.l) and (B.2) defined on this mesh by

for arbitrary weight functions w. Here (.) denotes (flux) averages

it being understood that x,+ = limE-O(x, f E ) , v’ = dv/dx etc. The particular structure of the weak statement in Eq. (B.3) is significant. We make the following observations concerning it: 1. If 4 = #(x) is the exact solution of Eqs (B. 1) and (B.2), then it is also the (one and only) solution of Eq. (B.3); i.e. Eqs (B. 1) and (B.2) imply the problem given by Eq. (B.3). 2 . The solution of Eqs (B. 1) and (B.2) satisfies Eq. (B.3) because q!I is continuous and the fluxes k dq5ldx are continuous:

[4](xe)= 0

and

(kg)(xe)

=0

3. The Dirichlet boundary conditions (an inflow condition) enter the weak form on the left-hand side, an uncommon property, but one that permits discontinuous weight functions at relevant boundaries. 4. The signs of the second term on the left side (Ce{(kv’[4]) - (k$’)[v])) can be changed without affecting the equivalence of Eq. (B.3) and Eqs (B.l) and (B.2), but the particular choice of signs indicated turns out to be crucial to the stability of the discontinuous Galerkin method (DGM). 5. We can consider the conditions of continuity of the solution and of the fluxes at interelement boundaries, conditions (B.6), as constraints on the true solution. Had we used Lagrange multipliers to enforce these constraints then, instead of the second sum on the left-hand side of Eq. (B.3), we would have terms like

where X and p are the multipliers. A simple calculation shows that the multipliers can be identified as average fluxes and interface jumps:

Appendix B 295 Introducing Eq. (B.8) into Eq. (B.7) gives the second term on the left hand side of Eq. (B.3). Incidently, had we constructed independent approximations of X and p, a setting for the construction of a hybrid finite element approximation of Eq. (B. 1) and 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 the DGM. Returning to Eq. (B.3), we introduce over each element 0, a polynomial approximation of 4;

k=O

where the a; are undetermined constants and Nt = xk are monomials (shape functions) 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 functions in each element, we arrive at the discrete system

-

{ ( k s ) [ q ] } a Z+ { (k%)Nt(L)

-

j = l , 2 , . . . , p p , e = 1 , 2 ,..., rn

N;(kT)(L)

1

+ NT(O)u(O)Nt-(O) ui.

(B.lO)

This is the DGM approximation of Eq. (B.3). Some properties of Eq. (B.lO) are noteworthy: 1. The shape functions Ni need not be the usual nodal based functions; there are no nodes in thisformulation. We can take N l to be any monomial we please (representing, for example, complete polynomials up to degree p , for each element 0, and even orthogonal polynomials). The unknowns are the coefficients uz which are not necessarily the values of at 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 the presence of the jump terms and averages on the element interfaces. However, it can be shown that the resulting matrix is always positive definite, the choice of signs in the boundary and interface terms being critical for preserving this property. 4. In general, the formulation in Eq. (B. 10) involves more degrees of freedom than the conventional continuous (conforming) Galerkin approximation of Eqs (B. 1) and (B.2) owing to the fact that the usual dependencies produced in enforcing continuity across element interfaces are now not present. However, the very localized nature of the discontinuous approximations contributes to the surprising robustness of the DGM.

4

296 Appendix B

5. While the piecewise polynomial basis { N ; , . . . ,NF,, . . . ,N ; , . . . , N;,,} contains complete polynomials from degree zero up to p =pmin,p,, numerical experiments indicate that stability demands p > 2, in general. 6. The DGM is elementwise conservative while the standard finite element approximation is conservative only in element patches. In particular, for any element Re, we always have jQ.fdx

+k

g

-,:I

(B.ll)

=0

This property holds for arbitrarily high-order approximations pe. The DGM is robust and essentially free of the global spurious oscillations of continuous Galerkin approximations when applied to convection-diffusion problems. 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:

+

k -d2d x-dd = -kx 2 cos(.lrx) - rxsin(7rx) dx2 dx

on

[0, I ]

with $(-1) = -2, 4(1) = 0. Exact solution for above shows a discontinuity of

4(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-'' and h = 1/10) obtained with the continuous and discontinuous Galerkin method, respectively. Extension to two and three dimensions is discussed in references given in Chapter 2.

Fig. B1. Continuous Galerkin approximation.

Appendix B

Fig. B2. Discontinuous Galerkin approximation.

297