A new multiscale finite element method for high-contrast elliptic interface problems
A new multiscale finite element method for high-contrast elliptic interface problems
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DOI：
10.1090/s0025-5718-2010-02372-5
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发表时间：
2010-01
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通讯作者：
C. Chu;I. Graham;T. Hou
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作者：
C. Chu;I. Graham;T. Hou
介绍了一种新的多尺度有限元方法，该方法只需用粗的拟均匀网格就能精确地求解高对比度系数的椭圆界面问题，而不需要分解界面。一个典型的应用是模拟多孔介质中的流动，其中包含许多低(或高)渗透率的包裹体，嵌入高(分别低)渗透率的基质中。我们的方法是H^1协调的，在三角形网格的结点处具有自由度，并且需要解跨系数界面但使用标准线性逼近的单元上的基函数的次网格问题。其中一个关键点是引入了新的次网格问题的系数相关边界条件。在适当的假设下，我们证明了我们的方法在能量范数下具有O(H)的(最优)收敛速度，在L_2范数下具有O(h^2)的(最优)收敛速度，其中h是(粗略的)网格直径，并且这些估计中的隐藏常数与偏微分方程系数的“对比度”(即最大与最小值之比)无关。对于标准元素，能量范数中的最佳估计是O(h^(1/2−e))，它有一个隐藏的常量，该常量通常取决于对比度。新的内部边界条件不仅取决于系数的对比度，而且还取决于界面与单元边缘的交角。
We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of low (or high) permeability embedded in a matrix of high (respectively low) permeability. Our method is H^1- conforming, with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise. A key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of O(h) in the energy norm and O(h^2) in the L_2 norm where h is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the “contrast” (i.e. ratio of largest to smallest value) of the PDE coefficient. For standard elements the best estimate in the energy norm would be O(h^(1/2−e)) with a hidden constant which in general depends on the contrast. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges.