We deal with the numerical solution of heat conduction problems featuring steep gradients. In order to solve the associated partial differential equation a finite volume technique is used and unstructured grids are employed. A discrete maximum principle for triangulations of a Delaunay type is developed. To capture thin boundary layers incorporating steep gradients an anisotropic mesh adaptation technique is implemented. Computational tests are performed for an academic problem where the exact solution is known as well as for a real world problem of a computer simulation of the thermoregulation of premature infants.