The inverse method has been developed to determine three dimensional heat flux and heat transfer coefficient distributions in space and time. The numerical tests conducted for simulated temperature sensor indications have shown that the dedicated heat conduction model has to be employed to achieve correct solutions for limited number of temperature sensors. The dedicated three dimensional finite element method based on nonlinear shape functions has been developed to effectively solve the heat conduction problem. The accuracy of 5 finite element models has been compared to analytical solution and to a reference finite element solution. The reduced nonlinear finite element model with 384 degrees of freedom has given in direct simulation of the temperature field errors at a level of 2 degrees C only. Heat transfer boundary condition over the cooled surface has been approximated by serendipity family elements with cubic shape functions. Heat transfer coefficients at surface element nodes have been extended in time of cooling with the parabolic spline functions. Inverse solutions based on the developed three dimensional heat condition and boundary condition models have been obtained without additional regularization. Solutions have been achieved for measured temperatures as well. Temperature of EN 1.4724 steel plate heated to 900 degrees C and then cooled has been measured by thermocouples located 2 mm below the cooled surface. The plate has been cooled by 1 and 2 water jets. Equations for heat transfer coefficient as functions of dimensionless plate surface temperature have been developed and verified in direct simulations of EN 1.4724 steel cooling. (C) 2014 Elsevier Ltd. All rights reserved.