The div-curl system arises in many fields including electromagnetism and fluid dynamics. We are particularly interested in the div-curl problem in 2D multiply-connected domains, as a simplified model of flow around airfoils. In such domains, well-posedness of the problem depends on the prescription of additional line integrals (circulation), apart from standard boundary conditions. We apply the DPG method to the ultraweak formulation of the problem and impose the circulation condition in a constrained minimization framework, which results in a mixed problem. We prove the discrete stability with Brezzi's theory and demonstrate convergence with numerical experiments in a toroidal domain. We also perform -adaptive refinements based on the DPG a posteriori error estimator, and obtain much faster convergence rate in the presence of singularity. The method is lastly applied to simulate the non-lifting flow around a NACA 0012 airfoil.

中文翻译：

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解决平面div-curl问题的DPG方法

div-curl 系统出现在许多领域，包括电磁学和流体动力学。我们对二维多重连通域中的 div-curl 问题特别感兴趣，作为翼型周围流动的简化模型。在这些领域中，除了标准边界条件之外，问题的适定性还取决于附加线积分（循环）的规定。我们将 DPG 方法应用于问题的超弱表述，并将循环条件施加在约束最小化框架中，这导致了混合问题。我们用 Brezzi 理论证明了离散稳定性，并用环形域中的数值实验证明了收敛性。我们还基于 DPG 后验误差估计器执行自适应细化，并在存在奇点的情况下获得更快的收敛速度。该方法最后应用于模拟 NACA 0012 翼型周围的非升力流。