Numerical treatment of microscale heat transfer processes arising in thin films of metals

The microscale heat transport equation (MHTE) is an important model in the microtechnology. The MHTE differs from the classical model of heat diffusion since it includes temperatures derivatives of second- and third-order with respect to time, and space and time, respectively. This paper studies the application of the localized radial basis function partition of unity (LRBF-PU) method for finding the MHTE solution. The proposed algorithm discretizes the unknown solution in two phases. First, the discretization of time dimension is obtained through the finite difference with second order accuracy. In addition, the unconditional stability and the convergence of the temporal semi-discretization are analysed with the help of the discrete energy method in an appropriate Sobolev space. Second, the discretization of the spatial dimension is accomplished with the help of the LRBF-PU. One of the main disadvantages of global collocation techniques is the high computational cost caused by the associated dense algebraic system. Using the proposed strategy, one can divide the original domain into a number of sub-domains through a kernel approximation over every sub-domain. The LRBF-PU method avoids the ill-conditioning driven by global collocation RBF-based methods and decreases the related computational cost. Numerical examples highlight the robustness and accuracy of the LRBF-PU.