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Preconditioning high order $$H^2$$ H 2 conforming finite elements on triangles
Numerische Mathematik ( IF 2.1 ) Pub Date : 2021-06-11 , DOI: 10.1007/s00211-021-01206-7
Mark Ainsworth , Charles Parker

We develop a nonoverlapping Additive Schwarz Preconditioner for the p-version finite element with \(H^2\)-conforming elements on triangles based on Argyris elements. With a particular choice of basis functions corresponding to the \(C^2\) degrees of freedom (dofs), we give a preconditioner consisting of eliminating the interior dofs, global solves of the \(C^0\) and \(C^1\) vertex dofs, diagonal solves of the \(C^2\) vertex dofs, and block diagonal solves of the edge dofs. We show that the condition number of the preconditioner system grows at most like \({\mathscr {O}}(1+\log ^3 p)\) independent of the mesh size h. The analysis permits the use of inexact interior solves which lends itself to a more efficient implementation while maintaining the same \({\mathscr {O}}(1 + \log ^3 p)\) asymptotic growth of the preconditioned system.



中文翻译:

预处理高阶 $$H^2$$ H 2 符合三角形上的有限元

我们为基于 Argyris 元素的三角形上具有\(H^2\)符合元素的p 版本有限元开发了一种非重叠加法 Schwarz 预处理器。通过与\(C^2\)自由度 (dofs)对应的基函数的特定选择,我们给出了一个预处理器,包括消除内部自由度、\(C^0\)\(C ^1\)顶点自由度,\(C^2\)顶点自由度的对角线求解,以及边自由度的块对角线求解。我们表明,预处理器系统的条件数最多增长为\({\mathscr {O}}(1+\log ^3 p)\)独立于网格大小h. 该分析允许使用不精确的内部求解,这有助于更有效的实现,同时保持预处理系统的相同\({\mathscr {O}}(1 + \log ^3 p)\)渐近增长。

更新日期:2021-06-11
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