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.

现代制造技术可以轻松生产具有复杂内部结构的异质材料（例如泡沫）。然而，它们的使用并不简单，因为经典本构定律不一定有效。根据各种实验观察，Guyer-Krumhansl 方程是模拟此类复杂结构的有希望的候选方程。然而，实际应用需要一种可靠且高效的算法，能够处理复杂的几何形状和先进的热方程。在本文中，我们推导了新的两场变分公式，将温度和热通量视为独立场变量，并开发了新的、先进的hp型混合有限元方法，可以可靠地应用。我们研究了它们在各种情况下的收敛特性，在稳定性和快速传播速度的处理方面具有挑战性。该算法也被证明非常高效，提供的解决方案比商业算法快四个数量级。