We study the existence of solutions for Darcy’s problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The model studied involves thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case where thermal diffusion is independent of temperature, we propose an a posteriori error estimator and study its reliability and efficiency properties. We illustrate the theory with numerical examples.

中文翻译：

---

奇异强迫下达西问题与热方程耦合：分析和离散化


我们研究了奇异强迫下达西问题与热方程耦合的解的存在;热方程的右侧对应于 Dirac 测度。所研究的模型涉及热扩散和粘度，具体取决于温度。我们提出了一种有限元求解技术并分析了它的收敛特性。在热扩散与温度无关的情况下，我们提出了一个后验误差估计器，并研究了它的可靠性和效率特性。我们用数字例子来说明该理论。