Mixed discontinuous Galerkin analysis of thermally nonlinear coupled problem

A stabilized mixed discontinuous Galerkin (SMDG) method based on Brezzi–Hughes–Marini–Masud [F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Masud, Mixed discontnuous Galerkin methods for Darcy flow, J. Sci. Comput. 22 (2005) 119–145.] is proposed to solve a thermally coupled nonlinear elliptic system modeling a large class of engineering problems. A fixed point algorithm is adopted to solve the nonlinear systems. Convergence analysis and error estimates are presented for equal order linear or bilinear discontinuous Lagrangian finite element interpolations for all fields. Numerical results are presented confirming the predicted convergence rates and illustrating the performance of the proposed formulation solving problems with globally stable and blowing up solutions.