拟协调元空间的紧致性和拟协调元法的收敛性
On the Compactness of Quasi-Conforming Element Spaces and the Convergence of Quasi-Conforming Element Method
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摘要: 本文首先讨论拟协调元空间的紧致性,把Rellich紧致定理推广到拟协调元空间序列,进而把广义Poincare、Friedrichs和Poincare-Friedrichs等不等式推广到拟协调元空间.然后讨论拟协调元法的收敛性和误差估计.本文证明了如果拟协调元空间具有逼近性和强连续性、满足单元秩条件且通过检验IPT,则近似解是收敛的.做为例子,我们证明了6参、9参、12参、15参、18参及21参拟协调元的收敛精度在L2,2(Ω)范数下分别是O(hτ)、O(hτ)、O(hτ2)、O(hτ2)、O(hτ3)及O(hτ4)量级.Abstract: In this paper,the compactness of quasi-conforming clement spices and the convergence of quasi-conforming element method are discussed.The well-known rellich compactness theorem is generalized to the sequences of quasi-conforming element spaces with certain properties,and the generalized poincare inequality,the generalized Friedrichs inequality and the generalzed inequality of Poincare-Friedrichs are proved true for them.The error estimates are also given.It is shown that the quasi-conforming element method is convergent if the quasi-conforming element spaces have the approximability and the strong continuity,and satisfy the rank condition of element and pass test IPT.As practical examples,6-parameter,9-parameter,12-parameter,15-parameter,18-parameter and 21-parameter quasi-conforming elements are shown to be convergent,and tlieir.L2,2(Ω)-errors are O(hτ)、O(hτ)、O(hτ2)、O(hτ2)、O(hτ3)and O(hτ4)respectively.
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