一种结合四阶紧致格式的三维泊松方程 extrapolation cascadic 多网格方法
An extrapolation cascadic multigrid method combined with a fourth order compact scheme for 3D poisson equation
AI总结：
本文提出了一种EXCMG方法，利用不等网格大小的紧致有限差分方法求解三维泊松方程，结合Richardson外推和三元四次拉格朗日插值提高精度，减少松弛次数，实现高效求解。
摘要：
本文开发了一种EXCMG方法，用于在矩形域上求解三维泊松方程，采用具有不同坐标方向不等网格大小的紧致有限差分（FD）方法。通过将Richardson外推与三元四次拉格朗日插值结合，可以产生极高的数值解近似，从而大大减少松弛次数。此外，基于中点外推公式的方法用于两层网格上的四阶FD解，以在整体细网格上以第六阶精度廉价高效地实现。通过求解由四阶紧致FD离散化得到的三对角线性系统，可以轻松获得数值解的梯度。数值结果表明，所提出的EXCMG方法比经典的V-循环和W-循环多网格方法更高效。此外，当精确解属于C^6时，仅需少量CG迭代即可在L^2范数和L^∞范数下实现完全四阶精度。最后，数值结果表明，当精确解具有较低正则性时，所提出的EXCMG方法仍然有效，扩大了该方法的应用范围。
英文摘要：
In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the $L^2$-norm and $L^{\infty}$-norm for the solution and its gradient when the exact solution belongs to $C^6$. Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.
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