Quantitative justification of simplifications in some heat conduction problems (Q1903674)

The stationary heat equation \(\text{div}(k(x)\nabla T(x))= f(x)\) on a multidimensional domain \(\Omega\) is considered. For \(k(x)= k_0+ e(x)\) with \(e\) being a ``small'' perturbation, a posteriori \(L^2\)-error estimates between \(\nabla T\) and \(\nabla T_0\), where \(T_0\) is the solution to the problem \(\text{div}(k_0 \nabla T(x))= f(x)\), are derived. Using a convex duality theory, similar a posteriori bounds are derived for a nonlinear perturbation of the heat-conductivity coefficient having then the form \(k= k(x, |\nabla T|)\), for perturbations of (linear) boundary conditions, and for a linearization of a nonlinear boundary condition of Stefan-Boltzmann type (i.e. \(k\partial T/\partial n+ \gamma T^4= g\)) and of the natural-convection type (i.e. \(k\partial T/\partial n+ \gamma T^{5/4}= g\)).