论文标题
Div-Curl问题和$ \ Mathbf {H}^1 $ - regular流函数在3D Lipschitz域中
Div-Curl Problems and $\mathbf{H}^1$-regular Stream Functions in 3D Lipschitz Domains
论文作者
论文摘要
我们考虑恢复无脱水的速度字段$ {\ mathbf u} \ in \ mathbf {l}^2（ω） $ω\ subset \ mathbb {r}^3 $。为此，我们为给定的$ {\ mathbf f} \ in {\ Mathbf H}^{ - 1}（ω）$求解了“ Div-Curl问题”。该解决方案是根据向量电位（或流函数）$ {\ Mathbf a} \ in {\ Mathbf H}^1（ω）$表示的，这样$ {\ MathBf U} = \ Mathrm {curl} \，{\ MathBf u} =在讨论了解决方案和相关矢量电位的存在和唯一性之后，我们提出了一个适合流函数的构造。提出了基于这种结构的数值方法，实验证实所产生的近似值比另一种常见方法的规律性更高。
We consider the problem of recovering the divergence-free velocity field ${\mathbf U}\in\mathbf{L}^2(Ω)$ of a given vorticity ${\mathbf F}=\mathrm{curl}\,{\mathbf U}$ on a bounded Lipschitz domain $Ω\subset\mathbb{R}^3$. To that end, we solve the "div-curl problem" for a given ${\mathbf F}\in{\mathbf H}^{-1}(Ω)$. The solution is expressed in terms of a vector potential (or stream function) ${\mathbf A}\in{\mathbf H}^1(Ω)$ such that ${\mathbf U}=\mathrm{curl}\,{\mathbf A}$. After discussing existence and uniqueness of solutions and associated vector potentials, we propose a well-posed construction for the stream function. A numerical method based on this construction is presented, and experiments confirm that the resulting approximations display higher regularity than those of another common approach.
