关于费米细束方程的先验误差估计及计算研究
A priori error estimates and computational studies for a Fermi pencil-beam equation
AI总结：
本文研究了费米细束方程的先验误差估计，采用标准伽辽金和流体扩散有限元方法，通过二维数值实验验证了理论结果，并展示了自适应算法显著降低计算误差。
摘要：
我们推导了标准伽辽金和流体扩散有限元方法在费米细束方程中的先验误差估计，该方程源自空间变量x=(x,y,z)和速度变量ṽ=(μ, η, ξ)的三维福克-普朗克方程。福克-普朗克项出现在单位球面的拉普拉斯-贝尔特里算子中。费米方程中的扩散项是FP算子在单位球面极点(1,0,0)处的切平面和方向v₀=(1,η, ξ)上的投影。因此，费米方程在三维空间域x=(x,y,z)中仅依赖两个速度变量v=(η, ξ)。由于某些截面有闭式解析解，因此无需后验误差估计过程，在我们的自适应算法中局部网格细化采用先验方法。不同二维数值实验验证了理论结果。实施显示使用我们的自适应算法显著降低了计算误差。
英文摘要：
We derive a priori error estimates for the standard Galerkin and streamline diffusion finite element methods for the Fermi pencil-beam equation obtained from a fully three dimensional Fokker-Planck equation in space ${\mathbf x}=(x,y,z)$ and velocity $\tilde {\mathbf v}=(μ, η, ξ)$ variables. The Fokker-Planck term appears as a Laplace-Beltrami operator in the unit sphere. The diffusion term in the Fermi equation is obtained as a projection of the FP operator onto the tangent plane to the unit sphere at the pole $(1,0,0)$ and in the direction of $ {\mathbf v}_0=(1,η, ξ)$. Hence the Fermi equation, stated in three dimensional spatial domain ${\mathbf x}=(x,y,z)$, depends only on two velocity variables ${\mathbf v}=(η, ξ)$. Since, for a certain number of cross-sections, there is a closed form analytic solution available for the Fermi equation, hence an a posteriori error estimate procedure is unnecessary and in our adaptive algorithm for local mesh refinements we employ the a priori approach. Different numerical examples, in two space dimensions are justifying the theoretical results. Implementations show significant reduction of the computational error by using our adaptive algorithm.
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