The Influence of Higher Order FEM Discretisations on Multigrid Convergence
The Influence of Higher Order FEM Discretisations on Multigrid Convergence
复制标题
高阶有限元离散化对多重网格收敛的影响
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
S. Turek
中科院分区：
文献类型：
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作者：
M. Köster;S. Turek
与线性方法相比，二次元甚至高阶有限元具有更好的近似性质，是求解偏微分方程数值解的有趣候选者。由基础（椭圆）偏微分方程离散化产生的方程组通常由迭代求解器求解，如预处理krylow空间方法，而多网格求解器仍然很少使用-这可能是由于与实现必要的数据结构以及平滑和网格间转移算子相关的高努力引起的。本文讨论了在多网格求解器中二次型有限元的数值分析。使用“正确的”网格转移算子与二次有限元近似相结合，可以制定改进的近似性质，增强多网格的（渐近）行为：如果m表示平滑步骤的数量，收敛速率表现为渐近的O(1/m2)，而不是线性FEM的O（1/m）。
Abstract Quadratic and even higher order finite elements are interesting candidates for the numerical solution of partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. The systems of equations that arise from the discretisation of the underlying (elliptic) PDEs are often solved by iterative solvers like preconditioned Krylow-space methods, while multigrid solvers are still rarely used – which might be caused by the high effort that is associated with the realisation of the necessary data structures as well as smoothing and intergrid transfer operators. In this note, we discuss the numerical analysis of quadratic conforming finite elements in a multigrid solver. Using the “correct” grid transfer operators in conjunction with a quadratic finite element approximation allows to formulate an improved approximation property which enhances the (asymptotic) behaviour of multigrid: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O(1/m2) in contrast to O(1/m) for linear FEM.