In Lipschitz two- and three-dimensional domains, we study the existence for the so-called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to H−1(ϖ, Ω), where ϖ is a weight in the Muckenhoupt class A2 that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and ϖ−1 ∈ A 1, show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability. We also explore efficiency estimates.

中文翻译：

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奇异强迫下的平稳 Boussinesq 问题

在 Lipschitz 二维和三维域中，我们研究了奇异强迫下热驱动对流的所谓 Boussinesq 模型的存在。单数是指允许热源属于H-1(ϖ, Ω)， 在哪里ϖ是 Muckenhoupt 类中的权重一种2在边界附近是规则的。我们提出了一个有限元方案，并且假设域是凸的并且ϖ-1 ∈ 一种 1，显示其收敛性。在热扩散和粘度为常数的情况下，我们提出了一个后验的误差估计器并显示其可靠性。我们还探讨了效率估计。