论文标题
在弯曲边界上的几何跨越幽灵平滑策略
A ghost-point smoothing strategy for geometric multigrid on curved boundaries
论文作者
论文摘要
我们提出一个边界局部傅立叶分析（BLFA），以优化多机框架中边界条件的放松参数。该方法是在嵌入均匀笛卡尔网格中嵌入的曲面域上求解椭圆方程的，尽管该方法的设计用于弯曲域中的一般PDE，无论在任何地方都可以实现多方面技术。边界是由级别集合函数隐式定义的，并采用了幽灵点技术来处理边界条件。文献中现有的策略在整个边界上采用恒定的放松参数。在本文中，松弛参数是根据幽灵点和边界之间的距离进行了优化的，其目的是沿切向方向平滑残差。理论结果通过1D，2D和3D中的几个数值测试证实，表明与内部方程上的平滑因子相关的收敛因子不会因边界效应而降低。
We present a Boundary Local Fourier Analysis (BLFA) to optimize the relaxation parameters of boundary conditions in a multigrid framework. The method is implemented to solve elliptic equations on curved domains embedded in a uniform Cartesian mesh, although it is designed to be extended for general PDEs in curved domains, wherever a multigrid technique can be implemented. The boundary is implicitly defined by a level-set function and a ghost-point technique is employed to treat the boundary conditions. Existing strategies in literature adopt a constant relaxation parameter on the whole boundary. In this paper, the relaxation parameters are optimized in terms of the distance between ghost points and boundary, with the goal of smoothing the residual along the tangential direction. Theoretical results are confirmed by several numerical tests in 1D, 2D and 3D, showing that the convergence factor associated with the smoothing on internal equations is not degraded by boundary effects.
