Finite element solutions of the energy equation at high peclet number

Abstract

Galerkin finite element solutions of the energy equation, like their central difference counterparts, sometimes display non-physical spatial oscillations at high Peclet number. This work compares the behaviour of closed-form solutions of the steady-state one-dimensional energy equation produced by quadratic finite elements, linear finite elements, central differencing and upwind differencing. Examples with different boundary conditions and source distributions are examined to determine the dependence of oscillation amplitudes on these factors. Finally, a two-dimensional numerical experiment is used to show how the qualitative results of the analysis can be extrapolated to more realistic flows. The paper concludes that Galerkin finite element methods can lead to oscillatory behaviour, but the solutions are generally more robust in this respect than the corresponding central difference solutions. For both constant and discontinuous sources in one dimension, boundary conditions can be chosen to eliminate any oscillation. This is in contrast with the central difference method where the solution is always oscillatory if the source is discontinuous. In this connection, the most suitable downstream boundary conditions are (natural) temperature-gradient conditions, which cannot impose spuriously high temperature variations at outlet. In some real flows, where such boundary conditions are not appropriate, large steamwise temperature gradients occur naturally. In these cases it is likely that local mesh refinement would have to be used if oscillations are to be avoided.