Numerical analysis of viscoelastic flows

In order to build up and implement efficient numerical schemes for partial differential equations, it is necessary to have informations on the mathematical properties of the system of equations—this has been done in the previous sections—as well as on the stability and the convergence properties of the schemes this is the purpose of numerical analysis. 

In the context of viscoelastic fluid flows, numerical analysis has been performed for differential models only, and for the following types of approximations finite element methods for steady flows, finite differences in time and finite element methods in space for unsteady flows. Finite element methods are the most popular ones in numerical simulations, but some other methods like finite differences, finite volume approximations, or spectral methods are also used. 

Moreover although the results we present here are only valid for regular and/or slow flows, they give some confidence in the use of these numerical schemes in more realistic situations, where for example the domain of the flow is not regular. 

A typical result of numerical analysis is an estimate of the error U — Uh between the solution U of the continuous problem (. e., the solution of the initial boundary value problem) and the solution Uh of the discrete problem (also called approximate problem). In what follows the error estimates are obtained with the assumption that U is sufficiently regular. In many realistic situations the geometry of the flow has singularities (corners for example), the solution U is not regular, and these results do not apply. (As a matter of fact existence of a solution has not been shown yet in those singular situations.) 

In order to simplify the presentation we make the following assumptions  

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