Wednesday, 03 September, 2014
SPEAKER: Prof. Max Jensen, Sussex
TITLE: Wellposedness and Finite Element Convergence for the Joule Heating Problem
ABSTRACT: The stationary Joule heating problem is a two way coupled system of non-linear partial differential equations modelling the heat and electrical potential in a body. The electrical current acts as a heat source in a resistive material while the temperature feeds back to the electrical potential through the electrical conductivity. Joule heating is important in many micro-electromechanical systems, where the effect is used to achieve very exact positioning at the micro scale. In applications boundary conditions of mixed type are typically used.
In this talk we present the existence proof for finite energy solutions of the Joule heating problem in three dimensions with mixed boundary conditions, using only very mild assumptions on the computational domain and the data. In particular, we show how previously established results can be extended to mixed boundary conditions. Furthermore, we prove strong convergence (of subsequences in case of non-unique exact solutions) of conforming finite element approximations.
Under the additional assumption of a so-called creased domain together with a sufficiently weak temperature dependency in the electrical conductivity we also prove optimal global regularity estimates together with local estimates guaranteeing smooth solutions away from the boundary given smooth data. We further discuss a priori and a posteriori error bounds for conforming finite element approximations on shape regular meshes.
The presented material is joint work with Axel Målqvist (Uppsala, Sweden).