Numerical Approximation of Fractional Powers of Regularly Accretive Operators
Numerical Approximation of Fractional Powers of Regularly Accretive Operators
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正则增值算子分数幂的数值逼近
DOI：
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发表时间：
2015
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通讯作者：
J. Pasciak
中科院分区：
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作者：
A. Bonito;J. Pasciak
本文研究了增值算子分数次幂的数值逼近。也就是说，如果 $A$ 是与 $L^2(\Omega)$ 中包含的希尔伯特空间 $\mathbb V$ 上定义的累加倍半线性形式 $A(\cdot,\cdot)$ 相关的累加运算符，我们将 $A^{-\beta}$ 近似为 $\beta\in (0,1)$。分数幂是根据所谓的 Balakrishnan 积分公式定义的。给定有限元近似空间 $\mathbb V_h\subset \mathbb V$，$A^{-\beta}$ 近似为 $A_h^{-\beta}\pi_h$，其中 $A_h$ 是与 $A(\cdot,\cdot)$ 形式相关的运算符，限制为 $\mathbb V_h$ 且 $\pi_h$ 是 $L^2(\Omega)$ 到 $\mathbb V_h$ 的投影。我们首先提供 Sobolev 范数中 $(A^\beta-A_h^{\beta}\pi_h)f$ 的误差估计，索引在 [0,1] 中，以适合适当的 $f$。这些结果取决于涉及 $A(\cdot,\cdot)$ 形式的变分解的椭圆正则性质，并且对于小于完全椭圆正则的情况有效。我们还构建并分析了定义 $A_h^{\beta}\pi_h f$ 的 Balakrishnan 积分的指数收敛 sinc 求积近似。最后，给出了说明该方法的数值计算结果。
We study the numerical approximation of fractional powers of accretive operators in this paper. Namely, if $A$ is the accretive operator associated with an accretive sesquilinear form $A(\cdot,\cdot)$ defined on a Hilbert space $\mathbb V$ contained in $L^2(\Omega)$, we approximate $A^{-\beta}$ for $\beta\in (0,1)$. The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space $\mathbb V_h\subset \mathbb V$, $A^{-\beta}$ is approximated by $A_h^{-\beta}\pi_h$ where $A_h$ is the operator associated with the form $A(\cdot,\cdot)$ restricted to $\mathbb V_h$ and $\pi_h$ is the $L^2(\Omega)$-projection onto $\mathbb V_h$. We first provide error estimates for $(A^\beta-A_h^{\beta}\pi_h)f$ in Sobolev norms with index in [0,1] for appropriate $f$. These results depend on elliptic regularity properties of variational solutions involving the form $A(\cdot,\cdot)$ and are valid for the case of less than full elliptic regularity. We also construct and analyze an exponentially convergent sinc quadrature approximation to the Balakrishnan integral defining $A_h^{\beta}\pi_h f$. Finally, the results of numerical computations illustrating the proposed method are given.