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中文摘要：
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Kx＋Ky次的计算量（这里Kx，Ky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．
英文摘要：
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.
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