Modern manufacturing technologies allow heterogeneous materials with complex inner
structures (e.g., foams) to be easily produced. However, their utilization is not
straightforward, as the classical constitutive laws are not necessarily valid. According
to various experimental observations, the Guyer–Krumhansl equation is a promising
candidate for modeling such complex structures. However, practical applications need
a reliable and efficient algorithm capable of handling both complex geometries and
advanced heat equations. In the present paper, we derive new two-field variational
formulations which treat the temperature and the heat flux as independent field variables,
and we develop new, advanced hp -type mixed finite element methods, which can be reliably
applied. We investigate their convergence properties for various situations, challenging
in relation to stability and the treatment of fast propagation speeds. That algorithm
is also proved to be outstandingly efficient, providing solutions four magnitudes
faster than commercial algorithms.
