论文标题
多式元素用于分级网格上有限体积元素离散的双面分数微分方程
Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes
论文作者
论文摘要
众所周知，保守的稳态两侧分数扩散问题的解会在边界附近表现出奇异性。因此，由于问题的保守性，我们对通用的非均匀网格采用了有限的体积要素离散方法。我们专注于由平滑函数映射的网格，该函数由奇异性附近的分级网格组合和溶液平滑的均匀网格组成。这样的选择产生了类似于toeplitz的离散矩阵，因此允许矩阵矢量产物的计算成本较低和详细的频谱分析。获得的光谱信息用于开发GMRE的无临时参数的多式预处理，该信息在数值上显示出可产生良好的收敛性会导致在存在的分级网格中，该分级网格由在奇异性附近积累点积累点的功率函数所映射。将所考虑的分级网格的近似顺序与文献中的某些复合网格之一进行比较，该复合网仍然导致类似于Toeplitz的线性系统，然后仍然非常适合我们的Multigrid方法。几个数值测试证实，电源分级网格的近似误差比复合误差较低，并且我们的求解器具有广泛的适用性。
It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consist in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix-vector product and a detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability.
