Numerical analysis and simulation of a \(\theta\)-scheme type method approximating nonstationary viscoelastic fluid flow. (Analyse et simulation numériques d’une méthode de type \(\theta\)-schéma approximant l’écoulement de fluides viscoélastiques instationnaires.) (French. English summary) Zbl 1254.76012
Summary: In the framework of numerical analysis and simulation of transient viscoelastic fluid flows, we present an approximation scheme based on an implicit Euler \(\theta\)-scheme for time and a finite element method for space.
For non-zero \(\theta\), the resulting algorithm is strongly coupled, which brings some difficulties related to the existence of fixed points and error estimates. However, it has the advantage that it can be interpreted as a variant of the alternating directions method. This latter has been presented by P. Saramito [RAIRO, Modélisation Math. Anal. Numér. 28, No. 1, 1–34 (1994; Zbl 0820.76051)] as a stable scheme, allowing relatively high Weissenberg numbers, especially in the numerical simulation over complex geometry.
We show by a fixed point method that the approximate problem has a solution and we give an error bound. We also give some relationships between the continuous solution of the problem, the Weissenberg number \((\lambda)\) and the approximation scheme \((\theta)\).
For non-zero \(\theta\), the resulting algorithm is strongly coupled, which brings some difficulties related to the existence of fixed points and error estimates. However, it has the advantage that it can be interpreted as a variant of the alternating directions method. This latter has been presented by P. Saramito [RAIRO, Modélisation Math. Anal. Numér. 28, No. 1, 1–34 (1994; Zbl 0820.76051)] as a stable scheme, allowing relatively high Weissenberg numbers, especially in the numerical simulation over complex geometry.
We show by a fixed point method that the approximate problem has a solution and we give an error bound. We also give some relationships between the continuous solution of the problem, the Weissenberg number \((\lambda)\) and the approximation scheme \((\theta)\).
MSC:
| 76A15 | Liquid crystals |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
