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Exponential convergence of hp FEM for spectral fractional diffusion in polygons

机译：hp FEM 在多边形中光谱分数扩散的指数收敛

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摘要

For the spectral fractional diffusion operator of order 2s, s is an element of (0, 1), in bounded, curvilinear polygonal domains omega subset of R2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm Hs(omega). 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 omega. Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in omega, exponential convergence rates for solutions u is an element of Hs(omega) of Lsu = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towards part; omega. The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of L-s combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in omega. The present analysis for either approach extends to (polygonal subsets M? of) analytic, compact 2-manifolds M, parametrized by a global, analytic chart chi with polygonal Euclidean parameter domain omega subset of R2. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.
机译：对于阶数为 2s 的谱分数阶扩散算子，s 是 （0， 1） 的元素，在 R2 的有界曲线多边形域 omega 子集中，我们证明了在解析数据（系数和源项，没有任何边界兼容性）的假设下，在自然分数阶 Sobolev 范数 Hs（omega） 中，两类 hp 离散化的指数收敛性。第一个 hp 离散化基于将解写为 2+1 维局部线性椭圆边界值问题的协法导数，并对其应用 hp-FE 离散化。扩展变量中的对角化将谱分数扩散算子逆的数值近似简化为 omega 中局部解耦二阶反应-扩散方程组的数值近似。利用二阶hp-FEM的稳健指数收敛性结果，omega中的线性反应扩散边界值问题，解u的指数收敛速率是Lsu = f跟随的Hs（omega）的一个元素。该hp-FEM的关键成分是边界拟合网格，其几何网格细化为&部分;欧米茄。第二种离散化基于L-s的Balakrishnan积分表示的指数收敛数值sinc正交近似，并结合omega中局部、线性、奇异扰动反应-扩散方程的解耦系统的hp-FE离散化。目前对这两种方法的分析都扩展到（多边形子集 M？）解析的紧凑 2 流形 M，由具有多边形欧几里得参数域 omega 子集 R2 的全局解析图 chi 进行参数化。在非凸多边形域和不相容数据下对模型问题的数值实验证实了理论结果。推导了曲线多边形中谱分数扩散和解析源项的解集的Kolmogorov n宽度的指数小边界。

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