Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

Appendix B
Discontinuous Galerkin methods
in the solution of the
convection±diffusion 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 ¯uid
mechanics.
In the context of ¯uid mechanics the advantages of applying the discontinuous
Galerkin method are:
. the achievement of complete ¯ux 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 ¯ow
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 convection±
di€usion problem in one dimension of the form
u d d
ÿ
dx dx
k…x† d
dx
ˆ f 0 4 x 4 L …B:1†
where u is the convection velocity, k ˆ k…x† the di€usion (conduction) coe cient
(always bounded and positive), and f ˆ f …x† the source term. We add boundary
conditions to Eq. (B.1); for example,
…L† ˆ and
d …0†
k…0† ˆ g
dx
…B:2†
J.T. Oden, personal communication, 1999.