基于张量乘积的快速谱元算法  

Tensor Product Based Fast Spectral Element Solvers

作　　者：容志建[1] 许传炬[1] 

机构地区：[1]厦门大学数学科学学院,福建厦门361005

出　　处：《数学研究》2008年第3期264-271,共8页Journal of Mathematical Study

基　　金：国家自然科学基金重点项目(10531080);973"高性能科学计算研究"项目(2005CB321703);教育部"新世纪优秀人才支持项目"(NCET-05-0562)

摘　　要：针对椭圆型方程的谱元离散系统构造了一种基于张量乘积的快速直接解法.分析显示,新算法的计算量仅相当于迭代方法迭代K_x+K_y次的计算量(这里K_x,K_y分别为x,y方向的区域剖分数),特别适合那些网格不多但多项式阶数较高的谱元离散.我们还将张量乘积方法推广到具有Neumann边界条件的奇异泊松问题的求解,给出了具体的实现方法.最后,利用张量乘积构造了变形区域上椭圆型方程的预条件子,数值结果显示预条件系统的条件数与多项式阶数无关.In this paper we first propose a fast direct spectral element solver for the Poisson equation based on the tensor product method（TPM）. Our analysis shows that the cost of the new solver is equivalent to that of an iterative method with Kx ＋ Ky iterations, where Kx, Ky are the numbers of macro-element in the x, y direction respectively. Therefore the proposed solver is advantagous than the classical iterative algorithms in the case few elements and high order polynomial are used in the approximation. We then extend the TPM to the singular Poisson problem with Neumann boundary condition. Finally, we use the new solver to construct two preconditioners for the Poisson problem defined on deformed domains, which usually cannot be solved by a TPM. The efficiency of the preconditioners are confirmed by some numerical tests. It is shown that the condition numbers of the preconditioned systems are independent of the polynomial degree.

关 键 词：椭圆型方程 谱元法 张量乘积法 直接解法 

分 类 号：O175.25[理学—数学]