A finite element analysis of thermal convection problems with the Joule heat

Error estimates are introduced in a finite element approximation to thermal convection problems with the Joule heat. From the Joule heat, a nonlinear source term arises in the convection-diffusion part of problems. The derivation of the estimates is based on the fact that a finite element approximation to the electric potential appearing in such a nonlinear term is uniformly bounded in the cubic summable norm with derivatives up to the first order. The error estimates are optimal, and do not require any stability conditions. Numerical results show that the numerical convergence rates agree well with the theoretical ones, and that phenomena in a simplified electric glass furnace are influenced by the electric potential distributions associated with the location of electrodes.