Finite element analysis of a coupled thermally dependent viscosity flow problem

Abstract

Abstract. In this work, a stationary Stokes flow with thermal effects is studied both mathematically and numerically. First, existence, uniqueness and regularity of the weak solution of the problem are established. Next, fi nite element approximation to the problem, based on a fi xed point algorithm, is proposed. Then, an error estimate between continuous solution and discrete one is obtained. Finally, some numerical tests are presented to confi rm the theoretical results.Mathematical subject classification: 35J, 35Q, 65N30, 76.

Finite element analysis of thermally coupled nonlinear Darcy flows

“…Similarly to our former studies [9,17,18], we first establish existence, uniqueness, and regularity of the weak solution of problem (1.2). We apply a fixed point algorithm to the nonlinear coupled problem and then propose a finite element approximation to the iterative solution.…”

“…The existence study for the system of equations can be found in, e.g. [3,[5][6][7], and [2,8,9] for non-Newtonian flows with viscous heating. As our knowledge, there is no mathematical analysis (such as uniqueness and regularity) and numerical analysis for problem (1.2) up to now.…”

“…In [17,18], we gave a complete mathematical and numerical studies such as existence, uniqueness, regularity, finite element approximations based on an iterative algorithm and convergence analysis. We extended our study to the Stokes flows with viscous heating in [9]. In this article, we give mathematical and numerical analyses for problem (1.2).…”