A posteriori finite element output bounds with adaptive mesh refinement: application to a heat transfer problem in a three-dimensional rectangular duct

Abstract

Numerical simulations based on an a posteriori finite element bound method with adaptive mesh refinement are presented for the three-dimensional convection–diffusion equation. The bound method provides relevant, quantitative, inexpensive, and rigorous lower and upper bounds to the output on a very fine discretization (“truth” discretization) at a cost close to the coarse mesh calculation (“working” discretization). To achieve a desired bound gap (i.e., difference between upper and lower bounds) at the lowest cost, an adaptive mesh refinement technique is used to refine the mesh only where needed. An optimal stabilization parameter is also applied to improve the sharpness of the bound gap. In this paper, the output of a heat transfer problem in a rectangular duct with a given velocity field is investigated. The average temperature at one section of the duct is bounded for a given inlet temperature and heat flux. For this problem, the adaptive mesh refinement strategy provides the same bound gap with only half the number of elements required by an uniform mesh refinement strategy. The hybrid flux calculation on the coarse mesh introduced for the domain decomposition approach is compared with the hybrid flux calculation on the fine mesh to analyze the contribution of the hybrid flux to the bound gap.