Poisson方程谱元法的一个有限元预条件分析

Analysis for a Finite Element Preconditioned Spectral Element Method of the Poisson Equation

考虑一维Poisson方程的谱元法离散系统的预条件求解问题。分析基于整体Gauss-Lobatto-Legendre节点上的线性有限元刚性矩阵S_h作为谱元离散系统A_hU=F_h预条件的代数性质。证明了区域分解情形下(S_hU,U)_(l_2)与(A_jU,U)_(l_2)的等价性,即存在与h无关的两个正常数c_0,c_1,使得S_h^(-1)A_h的任一特征值λ_k满足c_0≤λ_k≤c_1。

In this paper, we analyze the spectrum of a preconditioned spectral element approximation to the Poisson problem. The analysis is carried out based on the algebraic properties of the stiffness matrix (Sh) of the linear finite element method associated to the global Gauss-Lobatto-Legendre nodes,which is used as the preconditioner of the spectral element system AhU = Fh. We show the equivalence between (ShU,U)l2 and (AhU,U)l2 in the case of domain decomposition. Precisely, we prove that there exist two positive constants c0, C1, such that λk∈[c0,c1]for all eigenvalues λk of Sh-1Ah.