In the paper, the non-continuous FEM with Trefftz base functions (FEMT) applied to direct and inverse problem of heat conduction equation has been presented. For the finite number of base functions in each finite element the temperature field becomes non-continuous on the border between elements. This non-continuity has been decreased with the penalty function added to optimised functional. The numerical entropy distribution and energy dissipation function have been analysed on the common boundaries of elements. Increasing the number of base functions in the finite element substantially decreases the inaccuracies of direct and inverse problem solution. (c) 2006 Elsevier Ltd. All rights reserved.