论文标题
$ hp $ fem的指数收敛用于光谱分数扩散多边形
Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
论文作者
论文摘要
对于频谱分数扩散运算符，$ 2S \ in（0,2）$ in（0,2）$ in（0,2）的曲线，曲线曲线多边形域$ω$ $ω$，我们证明在分析数据的假设下，两类$ hp $ iveptized的指数收敛，而没有任何边界sobolev nord n nard n narm n narm n nard n nard n narm n narm n narm n nard n nard n narm of MathBB^s（HP）。第一个$ hp $ iptizatization基于编写解决方案作为$ 2+1 $维本地，线性椭圆边界价值问题的共同衍生物，将$ hp $ -fe的离散化应用于其中。扩展变量中的对角线化降低了光谱分数扩散算子倒数的数值近似，以降低$ω$中局部，脱钩的二阶反应扩散方程系统的数值近似。在二阶$ hp $ -fem的可靠指数收敛上利用结果，线性反应扩散边界价值问题$ω$，解决方案的指数收敛速率$ u \ in \ mathbb {h}^s（h}^s（ω）$ $ \ mathcal {l}^s u = f $的$ u \。此$ HP $ -FEM中的关键成分是边界拟合的网眼，并具有几何网状细胞的细化，朝$ \partialΩ$。 第二个离散化是基于$ \ Mathcal {l}^{ - s} $的Balakrishnan积分代表的指数收敛的正交近似，并结合了$ HP $ -FE -FE-FE-fe-fe离散化的本地，线性，线性，奇异的反应反应 - 反应反应 - $ω$ω$ω$的分离系统。两种方法的当前分析均扩展到多边形子集$ \ widetilde {\ Mathcal {m}} $分析性，紧凑$ 2 $ -2 $ -MANIFOLDS $ \ MATHCAL {M MATHCAL {M} $。对于非凸多边形域中模型问题和不兼容数据的数值实验证实了理论结果。 推导了kolmogoroff $ n $ widths solutions seths seths seths seths seths seths seths频谱分数扩散的趋势的小范围。
For the spectral fractional diffusion operator of order $2s\in (0,2)$ in bounded, curvilinear polygonal domains $Ω$ we prove exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without any boundary compatibility, in the natural fractional Sobolev norm $\mathbb{H}^s(Ω)$. The first $hp$ discretization is based on writing the solution as a co-normal derivative of a $2+1$-dimensional local, linear elliptic boundary value problem, to which an $hp$-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in $Ω$. Leveraging results on robust exponential convergence of $hp$-FEM for second order, linear reaction diffusion boundary value problems in $Ω$, exponential convergence rates for solutions $u\in \mathbb{H}^s(Ω)$ of $\mathcal{L}^s u = f$ follow. Key ingredient in this $hp$-FEM are boundary fitted meshes with geometric mesh refinement towards $\partialΩ$. The second discretization is based on exponentially convergent sinc quadrature approximations of the Balakrishnan integral representation of $\mathcal{L}^{-s}$, combined with $hp$-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in $Ω$. The present analysis for either approach extends to polygonal subsets $\widetilde{\mathcal{M}}$ of analytic, compact $2$-manifolds $\mathcal{M}$. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogoroff $n$-widths of solutions sets for spectral fractional diffusion in polygons are deduced.
