Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation
Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation
复制标题
静态扩散方程的拟态有限差分法的收敛性
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
J. Castillo
中科院分区：
文献类型：
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作者：
J. Guevara;S. Rojas;M. Freites;J. Castillo
使用有限差分模拟方法对偏微分方程进行数值求解，满足连续微分算子的性质并模拟适当积分恒等式的离散版本，更有可能产生更好的近似值。最近，一位作者开发了一种系统方法来获得散度和梯度算子的模拟有限差分离散化，该方法在边界和内部网格点上实现了相同数量级的精度。本文使用这些算子的二阶版本为稳态扩散方程开发了一种新的模拟有限差分方法。提出了这种新方法的完整理论和数值分析，包括这种新方法的二次收敛率的原始和非标准证明。数值结果在所有情况下都与我们的理论分析一致，提供了强有力的证据，证明新方法是比标准有限差分方法更好的选择。
The numerical solution of partial differential equations with finite differences mimetic methods that satisfy properties of the continuum differential operators and mimic discrete versions of appropriate integral identities is more likely to produce better approximations. Recently, one of the authors developed a systematic approach to obtain mimetic finite difference discretizations for divergence and gradient operators, which achieves the same order of accuracy on the boundary and inner grid points. This paper uses the second-order version of those operators to develop a new mimetic finite difference method for the steady-state diffusion equation. A complete theoretical and numerical analysis of this new method is presented, including an original and nonstandard proof of the quadratic convergence rate of this new method. The numerical results agree in all cases with our theoretical analysis, providing strong evidence that the new method is a better choice than the standard finite difference method.