Local discontinuous galerkin formulations for heat conduction problems involving high gradients and imperfect contact surfaces

A Local Discontinuous Galerkin (LDG) method is described here which provides a unified mathematical setting and framework for solving various kinds of heat conduction problems to include thermal contact conductance/resistance, sharp/high gradient problems and the like. For these applications, the LDG method does not require much modifications to the basic formulation or the need to employ ad hoc approaches as with the Continuous Galerkin (CG) finite element methods. In this paper, we describe the LDG formulation for elliptic heat conduction problems which is then extended to parabolic problems. The advantages of the LDG method over the CG method are shown using two classes of problems - problems involving sharp/high gradients, and imperfect contact between surfaces. So far, interface/gap elements have been primarily used to model the imperfect contact between two surfaces to solve thermal contact resistance problems. The LDG method eliminates the use of interface/gap elements and provides a high degree of accuracy. It is further shown in the problems involving sharp/high gradients, that the LDG method is less expensive (requires less number of degrees of freedom) as compared to the CG method to capture the peak value of the gradient. Several illustrative 1-D/2-D applications highlight the effectiveness of the present the LDG formulation.