By Ivan, V. Meshchersky, N.M. Sinelnikova

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That is, %o(0) *sme set °f a^ functions u in "H1 (SI) such that u is the limit in Hl (fi) of a sequence {us}'^-1 whose us are in (70° (fi). 5. 2 Extension to vector-valued functions In thefiniteelement analysis of flow problems consideration will be given not only to scalar functions (such as temperature or pressure) but also to vector- valued functions (such as fluid velocity). For vector- valued functions with m components, that is u, v : 0 —> Rm, the procedure is in fact essentially the same as for scalar functions.

1 Piecewise linear approximation in 1D As a model problem and for illustration purposes we consider a ID scalar equation. Moreover, in order to determine the element matrices the convection and diffusion 38 STEADY TRANSPORT PROBLEMS coefficients, a and v, are assumed constants. However, the source term s is variable with x. The treatment of the boundary conditions is simplified by imposing homogeneous Dirichlet conditions on each side. Note, however, that other boundary conditions could easily be implemented.

1977; Heinrich and Zienkiewicz, 1979; Griffiths and Mitchell, 1979). These were based on modified weighting functions such that the element upstream of a node is weighted more heavily than the element downstream of a node. 8. In this way, the finite element equivalent 52 STEADY TRANSPORT PROBLEMS Galerkin Fig. 8 Upwind-type weighting function, element e element e + 1 f j- 1 J J+1 Fig. 9 Node numbering for two consecutive linear elements. of upwind differencing was invented in the form of a Petrov—Galerkin method based upon an upwind distortion of the weighting function.