Multigrid in a weighted space arising from axisymmetric electromagnetics
Multigrid in a weighted space arising from axisymmetric electromagnetics
复制标题
由轴对称电磁学产生的加权空间中的多重网格
作者：
D. Copeland;Jay Gopalakrishnan;M. Oh
考虑二维向量函数的空间，其分量和旋度关于由径向变量给出的退化权是平方可积的。当在轴对称下建模电磁问题并通过柱坐标进行降维时，自然会出现这种空间。我们证明了，如果原来的三维域是凸的，那么多重网格V-循环适用于在这个空间的内积收敛，提供一定的现代smooters被使用。对于收敛性分析，我们首先证明了几个中间结果，例如，加权范数中交换投影的逼近性质，以及加权空间中对偶混合方法的超收敛估计。多重网格收敛速度的均匀性相对于网格大小，然后建立理论和数值实验说明。
Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex, then the multigrid V-cycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a dual mixed method in weighted spaces. The uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments.