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Keywords:

  • Variational formulation;
  • computational fluid dynamics;
  • balance equations.

Abstract

Dynamical processes modeled with a suitable set of differential equations can be approximately computed with finite element method. However, especially in fluid dynamics, numerical instabilities may occur. In order to circumvent such problems many techniques have been realized over the last four decades. Often the lack of stability is linked to numerical insufficiency, therefore the solution is sought by tuning functional spaces, cf. [5] or by adding stabilizing terms, cf. [18, 19, 41]. Stabilizing terms are needed and helpful, however, they involve coefficients to be found depending on the underlying problem. The purpose of this work is to propose another approach based on first principles producing similar terms so that the numerical approximation converges without using any other parameter but the material constants, which are measurable. The approach eliminates stability problems, first, by setting the primary variables in a fundamental way and, second, by using the balance equations in their global form in a discrete manner instead of a continuous one. Some applications at the end emphasize the usefulness of the lengthy calculations, which we develop in a straightforward and consistent way.