A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem. Issue 1 (8th September 2018)

• Record Type: Journal Article
• Authors: Caucao, Sergio; Gatica, Gabriel N.; Oyarzúa, Ricardo
• Is Part Of: Numerical methods for partial differential equations. Volume 35:Issue 1(2019)
• Page Start: 295
• Page End: 324
• Publication Date: 2018-09-08
• Subjects: a posteriori error analysis -- augmented fully mixed formulation -- fixed‐point theory -- mixed finite element methods -- nonisothermal -- Oldroyd–Stokes problem -- stress‐velocity formulation; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22301
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : In this article, we consider an augmented fully mixed variational formulation that has been recently proposed for the nonisothermal Oldroyd–Stokes problem, and develop an a posteriori error analysis for the 2‐D and 3‐D 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 nonconvex) 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 Clément and Raviart–Thomas operators. On the other hand, inverse inequalities, the localization 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.