一种用于达西问题与热方程耦合的虚拟单元法
A virtual element method for Darcy's problem coupled with a heat equation
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中文总结 AI 辅助
研究二维空间中达西问题与非线性热方程耦合问题，采用虚拟单元法求解，证明了数值格式适定性，推导了最优误差估计，并通过数值测试补充理论结果。
中文摘要 AI 辅助
在二维空间中，我们开发了一种虚拟单元法来求解与非线性热方程耦合的达西问题。该耦合模型考虑了热扩散以及粘度随温度的变化。在标准离散化假设和数据的适当假设下，我们证明了所提出数值格式的适定性。在解的适当正则性假设和数据的适当假设下，我们还推导了最优误差估计。最后通过在不同网格族上进行的一系列数值测试对理论结果进行补充。
英文摘要
In two dimensions, we develop a virtual element method to solve the Darcy's problem coupled with a nonlinear heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions and appropriate assumptions on the data, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under suitable regularity assumptions for the solution and appropriate assumptions on the data. We conclude with a series of numerical tests performed on different families of meshes that complement the theoretical findings.
