In Lipschitz domains, we study a Darcy-Forchheimer problem coupled with a singular heat equation by a nonlinear forcing term depending on the temperature. By singular we mean that the heat source corresponds to a Dirac measure. We establish the existence of solutions for a model that allows a diffusion coefficient in the heat equation depending on the temperature. For such a model, we also propose a finite element discretization scheme and provide an a priori convergence analysis. In the case that the aforementioned diffusion coefficient is constant, we devise an a posteriori error estimator and investigate reliability and efficiency properties. We conclude by devising an adaptive loop based on the proposed error estimator and presenting numerical experiments.

翻译：在Lipschitz区域中，我们研究了一个由依赖于温度的非线性强迫项耦合的Darcy-Forchheimer问题与奇异热方程。所谓“奇异”是指热源对应于Dirac测度。我们建立了在允许热方程扩散系数依赖于温度的模型下解的存在性。针对此类模型，我们还提出了有限元离散格式并进行了先验收敛性分析。当上述扩散系数为常数时，我们设计了一个后验误差估计器并研究了其可靠性和效率特性。最后，基于所提出的误差估计器设计了自适应循环并给出了数值实验。