Finite element approximation to nonlinear coupled thermal problem (Q1006009)

The authors study the nonlinear problem \[ \begin{aligned} &-\nabla (\mu(\;) \nabla u) =f,\text{ in }\Omega , \\ &-\Delta \theta =\mu (\theta) | \nabla u| ^2\text{ in }\Omega , \\ &u=0\;,\theta =0\text{ on }\partial \Omega , \end{aligned} \] which models problems with thermal effects. After establishing existence and regularity of the weak solution of the problem, they utilize the \(L^p\) theory for giving a convergence analysis of the standard finite element approximation in the general case where \(f\) is assumed only to be bounded. The authors also obtains optimal a priori error estimates in both \(W^{1,p}\)-norm and \(L^p\)-norm for the finite element approximations to the nonsingular solutions.