This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the influence of the optical parameters. The correction spaces are build from low-order spherical harmonics approximations generalizing well-known diffusion approximations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that $\mathcal{H}$- or $\mathcal{H}^2$-matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples.
翻译:本文审议了由球体上不连续元素的厌食性散热传输方程式离散产生的线性系统的迭代解决方案。 为了在离散参数和我们在希尔伯特空域上开发的理查德森必备迭代参数方面实现有力的趋同行为, 我们证明由此形成的方案是趋同的。 先决条件是分为两步构建的。 第一步从矩阵分裂中借用想法, 确保网状独立。 第二步使用子空间校正技术来减少光学参数的影响。 校正空间是用低顺序球形调近似构建的, 将众所周知的扩散近似归纳为通用的。 我们详细讨论离散操作器的高效实施和应用。 特别是对于考虑不连续的球形元素, 散开操作器变得密度大, 我们显示, $\macal{H}$- 或 $\mathcal{H ⁇ 2$-matrix 压缩可以黑箱方式应用, 在应用相应的近似近似时获得几乎线性或线性复杂度。 提议的方法的有效性在数字示例中显示。