Unstructured Additive Schwarz-Conjugate Gradient Method for Elliptic Problems with Highly Discontinuous Coefficients
Unstructured Additive Schwarz-Conjugate Gradient Method for Elliptic Problems with Highly Discontinuous Coefficients
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DOI：
10.1137/s1064827596305593
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发表时间：
1999-05
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影响因子：
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通讯作者：
I. Graham;M. J. Hagger
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文献类型：
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作者：
I. Graham;M. J. Hagger
本文讨论了在非结构三角形或四面体网格上用线性有限元法离散的二维或三维空间中的分段常系数对称椭圆问题的迭代解法。首先假设在域中有d个固定区域，在这些区域中系数取常数正值a =（a1，a2），a =（a3，a4），a =（a5，a6），a =（a6，a7），a =（a7，a8），a =（a7，a8），a =（a8，a9）。. .，ad），然后考虑某些（正）系数序列{a（m）}，其中这些值中的一些接近$0$或$\infty$为$m \rightarrow \infty$。我们考虑的性能添加剂施瓦茨区域分解预处理器构造的局部解决自动生成的子域上的一些粗糙的网格上的全局解决。假设系数函数为常数的区域与子域或粗网格之间没有关系，我们证明了预条件共轭梯度法的收敛性（在能量和欧几里得范数中），迭代次数仅在最大跳跃${\cal J}^{（m）}的大小上以几何方式增长：= \max\{a_k^{（m）}/a_l^{（m）} \：\ k，l = 1，\ldots，d\} $，as $m\rightarrow \infty$.结果是通过对预条件矩阵的仔细分析得到的。它不能通过估计条件数的通常过程获得：给出一个简单的例子，其中原始和预处理刚度矩阵的条件数在${\cal J}^{（m）}$中线性退化。Chan，Smith和Zou最近的结果表明，通过使用这种预条件与共轭梯度法，迭代次数可以独立于网格直径有界，前提是子域的重叠与粗网格的大小相称。我们的研究结果表明，这种方法也是高度弹性的不连续系数，即使没有注意到系数的不连续性的求解器的建设。
This paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes. The effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a = (a1, . . . , ad)and then considering certain (positive) coefficient sequences {a(m)} in which some of these values approach $0$ or $\infty$ as $m \rightarrow \infty$. We consider the performance of additive Schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid. Assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid, we show that the preconditioned conjugate gradient method converges (in both the energy and the Euclidean norms) with a number of iterations which grows only logarithmically in the size of the maximum jump ${\cal J}^{(m)} := \max\{a_k^{(m)}/a_l^{(m)} \ : \ k, l = 1, \ldots , d\} $, as $m\rightarrow \infty$. The result is obtained by a careful analysis of the preconditioned matrix. It cannot be obtained by the usual procedure of estimating condition numbers: a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in ${\cal J}^{(m)}$. Recent results of Chan, Smith, and Zou have shown that by using this preconditioner together with the conjugate gradient method, the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh. Our results now show that this method is also highly resilient to discontinuous coefficients, even if no attention is paid to the coefficient discontinuity in the construction of the solver.