论文标题
椭圆算子的分数幂的Padé-Parametric FEM近似
Padé-parametric FEM approximation for fractional powers of elliptic operators on manifolds
论文作者
论文摘要
本文重点介绍了椭圆运算符的分数幂的数值近似，$ 2 $ -D歧管。首先，使用参数有限元方法来离散原始问题。然后，我们通过有理函数的乘积近似离散椭圆运算符的分数幂，每个函数都是对角线的近对角性padé近似值。进行了严格的错误分析并提出了急剧错误界限，这表明该方案对于$α\ rightarrow 0^+$和$α\ rightarrow 1^ - $。所提出的算法的成本正在解决一些椭圆问题。由于该方法相对于求解的数量是指数收敛的，因此非常有效。进行了一些数值测试以确认我们的理论分析和算法的鲁棒性。
This paper focuses on numerical approximation for fractional powers of elliptic operators on $2$-d manifolds. Firstly, parametric finite element method is employed to discretize the original problem. We then approximate fractional powers of the discrete elliptic operator by the product of rational functions, each of which is a diagonal Padé approximant for corresponding power function. Rigorous error analysis is carried out and sharp error bounds are presented which show that the scheme is robust for $α\rightarrow 0^+$ and $α\rightarrow 1^-$. The cost of the proposed algorithm is solving some elliptic problems. Since the approach is exponentially convergent with respect to the number of solves, it is very efficient. Some numerical tests are given to confirm our theoretical analysis and the robustness of the algorithm.
