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：

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达西问题与奇异强迫下的热方程相结合：分析和离散化

研究奇异强迫下达西问题与热方程耦合解的存在性；热方程的右侧对应于狄拉克测度。研究的模型涉及取决于温度的热扩散和粘度。我们提出了一种有限元求解技术并分析了其收敛特性。在热扩散与温度无关的情况下，我们提出了后验误差估计器并研究了其可靠性和效率特性。我们用数值例子来说明该理论。