Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences
Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences
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DOI：
10.1137/s0036142994262585
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发表时间：
1997-04
影响因子：
2.9
通讯作者：
T. Arbogast;M. Wheeler;I. Yotov
中科院分区：
文献类型：
--
作者：
T. Arbogast;M. Wheeler;I. Yotov
给出了含张量系数二阶椭圆型问题的扩展混合有限元近似。将混合方法扩展为显式逼近三个变量，即未知标量、负梯度及其通量（张量系数乘以负梯度）。得到的线性系统是一个鞍点问题。对于矩形平行六面体上的最低阶Raviart—Thomas元，我们通过引入一定的正交规则来近似这种扩展的混合方法。这使我们能够将该系统写成一个简单的、以细胞为中心的有限差分方法，该方法要求求解未知标量的稀疏、正半定线性系统。对于一般张量系数，未知标量的稀疏模式是二维的9点模板和三维的19点模板。现有理论表明，扩展混合方法在$L^2$-和$H^{-s}$-范数上给出了最优阶逼近（标量变量的$L^2$-投影与其逼近之间获得了超收敛性）。我们证明了这些收敛速率对于有限差分法是保持不变的。如果$h$表示最大网格间距，则最佳速率为$O(h)$。得到了标量未知的超收敛速率$O(h^{2})$，其梯度和通量在一定离散范数下的超收敛速率$O(h^{3/2})$；并且在严格的域内得到了完整的$O(h^{2})$。计算结果验证了这些理论结果。
We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart--Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the $L^2$- and $H^{-s}$-norms (and superconvergence is obtained between the $L^2$-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If $h$ denotes the maximal mesh spacing, then the optimal rate is $O(h)$. The superconvergence rate $O(h^{2})$ is obtained for the scalar unknown and rate $O(h^{3/2})$ for its gradient and flux in certain discrete norms; moreover, the full $O(h^{2})$ is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.