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**A discontinuous Galerkin finite element method for heat conduction problems with local high gradient and thermal contact resistance.**
*(English)*
Zbl 1257.80004

Summary: A discontinuous Galerkin (DG) finite element method for the heat conduction problems with local high gradient and thermal contact resistance is presented. The DG formulation is constructed by employing the stabilization term and the Bassi-Rebay numerical flux term. The stabilization term is defined by a penalization of the temperature jump at the interface. By eliminating the penalization term of the temperature jump in the region of local high gradient and imperfect contact interfaces, the present DG method is applied to solve problems involving local high gradient and thermal contact resistance where the numerical flux is obtained from the definition of the thermal contact resistance. This treatment leads to a novel approach to capture the peak value of the heat flux in local high gradient field and the temperature jump at the imperfect contact interface. Moreover, an iterative procedure and a relaxation technique are also adopted herein to simulate the nonlinear thermoelastic coupling between thermal contact resistance, temperature and stress field, which can easily avoid numerical instability and gets reasonable results. Several numerical examples are given to demonstrate the accuracy and the reliability of the present DG finite element method. From our investigations, we find that the DG method is an attractive and competitive approach for solving heat conduction problems with local high gradient and thermal contact resistance, and also can easily avoid numerically instability in dealing with nonlinear coupling problems caused by thermal contact resistance.

### MSC:

80M10 | Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer |

80A20 | Heat and mass transfer, heat flow (MSC2010) |