an:00166692
Zbl 0772.65072
Lube, G.
An asymptotically fitted finite element method for convection dominated convection-diffusion-reaction problems
EN
Z. Angew. Math. Mech. 72, No. 3, 189-200 (1992).
00009591
1992
j
65N30 65N50 65N15 35J65
error estimates; finite element; streamline-diffusion; convection- dominated convection-diffusion-reaction problems; boundary layers; numerical examples; mesh refinement; exponentially fited method
The author presents convergence of an asymptotically fitted variant SDFEM-A of the finite element method of streamline-diffusion type (SDFEM) for singularly perturbed elliptic boundary value problems modelling convection-dominated convection-diffusion-reaction problems. The method is based on the observation that for \(\varepsilon \leq C \cdot h\) and \(\varepsilon \leq C \cdot h^{3/2}\), respectively, any unrefined mesh cannot resolve the downstream and characteristic boundary layers, respectively [cf. \textit{C. Johnson}; \textit{A. H. Schatz, L. B. Wahlbin}: Math. Comput. 49, 25-38 (1987; Zbl 0629.65111)].
The idea consists of replacing the sharp layers by smooth layers. As explained by \textit{O. Axelsson} [I.M.A. J. Numer. Anal. 1, 329-345 (1981; Zbl 0508.76069)], this method can he viewed, in some sense as a limit case (\(\varepsilon \ll h\)) of using exponentially weighted functions.
Without perturbing the simple finite element shape structure and desirable linearization properties of SDFEM, the SDFEM-A allows for global error estimates in \(L_ 2\)-norm and sometimes in a weighted \(W'_ 2\)-norm which are uniformly valid with respect to \(\varepsilon\). Such global results are not valid for SDFEM. As a result it is concluded that boundary layers are better approximated by the SDFEM-A.
Theoretical superiority of SDFEM-A is demonstrated through four numerical examples with the help of tables and graphs. Computed results establish that local oscillations of SDFEM-solutions in boundary layers are suppressed by SDFEM-A. Consequently SDFEM-A is in some situations an alternative to mesh refinement methods or exponentially fitted methods to resolve the layers.
The question of optimal local \(L_ \infty\)-estimates for SDFEM is still open. The paper by \textit{C. Johnson} et al. (loc. cit.) concerning a modified streamline diffusion method SDFEM-C is a step forward to solve the problem.
The streamline diffusion schemes are now a common method for solving as well transport-dominated problems as more complicated convection- dominated flow problems, in particular incompressible and compressible Euler or Navier-Stokes equations.
H.K.Verma (Ludhiana)
Zbl 0629.65111; Zbl 0508.76069