Finite element approximation of a nonlinear steady-state heat conduction problem (Q2708325)

This paper deals with the nonlinear elliptic problem NEWLINE\[NEWLINE-\text{div}(A(.,u)\text{grad }u)= f\tag{1}NEWLINE\]NEWLINE in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\Omega\subset \mathbb{R}^d\) is bounded domain with a Lipschitz continuous boundary \(\partial\Omega\), \(d\in\{1,2,\dots\}\), \(f\in L^2(\Omega)\), and \(A= (a_{ij})^d_{i,j=1}\) is a uniformly positive definite matrix. To state a weak formulation of the problem (1) the author assumes that the entries of \(A= A(.,.)\) are bounded measurable functions. Now, comparison and maximum principles are proved. Also, a discrete analogue of the maximum principle for linear elements which is based on nonobtuse tetrahedral partitions is presented.