Sergio Caucao, Gabriel N. Gatica, Ricardo Oyarzúa:
A posteriori error analysis of an augmented fully-mixed formulation for the non-isothermal Oldroyd-Stokes problem
In this paper we consider an augmented fully-mixed variational formulation that has been recently proposed for the non-isothermal Oldroyd-Stokes problem, and develop an a posteriori error analysis for the 2D and 3D versions of the associated mixed finite element scheme. More precisely, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the bilinear forms of the continuous formulation, suitable assumptions on the domain and the data, stable Helmholtz decompositions, and the local approximation properties of the Clement and Raviart--Thomas operators. On the other hand, inverse inequalities, the localisation technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the a posteriori error estimators and illustrating the performance of the associated adaptive algorithms are reported.
This preprint gave rise to the following definitive publication(s):
Sergio CAUCAO, Gabriel N. GATICA, Ricardo OYARZúA: A posteriori error analysis of an augmented fully-mixed formulation for the non-isothermal Oldroyd-Stokes problem. Numerical Methods for Partial Differential Equations, vol. 35, 1, pp. 295-324, (2019).