Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit
Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit
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扩散极限下辐射输运方程的迎风间断伽辽金近似的渐近分析
DOI：
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发表时间：
2010
影响因子：
2.9
通讯作者：
G. Kanschat
中科院分区：
文献类型：
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作者：
J. Guermond;G. Kanschat
我们重新审视 M. L. Adams [Nucl.科学。工程，137（2001），第 298-333 页]。利用泛函分析工具证明了标准迎风间断伽辽金逼近在扩散域中收敛到正确极限解的充分必要条件是逼近空间包含连续函数的线性空间，并且该空间的函数对每个网格单元的限制都包含线性多项式。此外，如果边界数据是各向同性的，则离散扩散极限在 Sobolev 空间 $H^1$ 中收敛到连续扩散极限。对于各向异性边界数据，会出现边界层，并且仅在破断 Sobolev 空间 $H^s$ 中且 $s<\frac{1}{2}$ 中收敛。
We revisit some results from M. L. Adams [Nucl. Sci. Engrg., 137 (2001), pp. 298-333]. Using functional analytic tools we prove that a necessary and sufficient condition for the standard upwind discontinuous Galerkin approximation to converge to the correct limit solution in the diffusive regime is that the approximation space contains a linear space of continuous functions, and the restrictions of the functions of this space to each mesh cell contain the linear polynomials. Furthermore, the discrete diffusion limit converges in the Sobolev space $H^1$ to the continuous one if the boundary data is isotropic. With anisotropic boundary data, a boundary layer occurs, and convergence holds in the broken Sobolev space $H^s$ with $s<\frac{1}{2}$ only.