On the effectiveness of convolutive type variational principles in the numerical solution of the heat conduction problem

We consider the transient heat conduction problem with boundary conditions regarding both the temperature field and its flux across the external surface of the solid, which is supposed to be thermally inhomogeneous but, for simplicity, isotropic. We formulate some variational principles by virtue of a bilinear form of the convolutive and bi-convolutive type with respect to the time variable. In particular, new formulations are obtained by dividing the time domain into two equal subintervals and doubling, as a consequence, the variables and the equations of the problem. Thanks to the properties of some terms arising from the decomposition of the time interval, we deduce some min-max and min-stat principles. The formulations are used for the numerical solution of the heat conduction problem through spatial and temporal discretizations considering a finite time range. Several numerical examples are included and numerical results are compared with the aim of investigating the effectiveness of the variational formulations in the numerical solution of the transient heat conduction problem.Finally, hints are given on how to extend the functionals presented here to the nonlinear case.