A Modified Finite Volume Approximation of Second-Order Elliptic Equations with Discontinuous Coefficients
A Modified Finite Volume Approximation of Second-Order Elliptic Equations with Discontinuous Coefficients
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DOI：
10.1137/s1064827599353877
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发表时间：
2001-04
期刊：
影响因子：
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通讯作者：
R. Ewing;O. Iliev;R. Lazarov
中科院分区：
文献类型：
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作者：
R. Ewing;O. Iliev;R. Lazarov
对Rn，n=1，2，3中的界面问题提出了一种修正的有限差分近似。修正的实质在于在有限体积的相对面上同时离散化通量的任意两个法向分量。这样，通过界面的通量的连续法向分量用具有二阶一致性的有限差分来近似。对于Rn中的问题，导出的方案具有最小的(2n+1)点模板。证明了一类界面问题关于离散H1范数的二阶收敛。在一维(1-D)、二维(2-D)和三维(3-D)界面问题的一系列数值实验中观察到了二阶逐点收敛。数值实验表明，与已有的采用算术和调和平均不连续扩散系数的格式相比，新格式具有更好的性能。
A modified finite difference approximation for interface problems in Rn, n=1,2,3, is presented. The essence of the modification falls in the simultaneous discretization of any two normal components of the flux at the opposite faces of the finite volume. In this way, the continuous normal component of the flux through an interface is approximated by finite differences with second-order consistency. The derived scheme has a minimal (2n+1)-point stencil for problems in Rn. Second-order convergence with respect to the discrete H1-norm is proved for a class of interface problems. Second-order pointwise convergence is observed in a series of numerical experiments with one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) interface problems. The numerical experiments presented demonstrate advantages of the new scheme compared with the known schemes which use arithmetic and harmonic averaging of the discontinuous diffusion coefficient.