A Goal-Oriented Error Estimator for a Class of Homogenization Problems


We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.

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TC was supported by the German Research Council (DFG) through project “Multiscale modeling and numerical simulations of Lithium ion battery electrodes using real microstructures” (CA 633/2-1).

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Carraro, T., Goll, C. A Goal-Oriented Error Estimator for a Class of Homogenization Problems. J Sci Comput 71, 1169–1196 (2017). https://doi.org/10.1007/s10915-016-0338-y

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  • A posteriori error estimation
  • Dual based adaptivity
  • Finite element method
  • Homogenization problems
  • Stokes/Darcy coupling
  • Fuel cell application