模拟有限差分的自适应一致性
Adaptive Consistency for Mimetic Finite Differences
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中文总结 AI 辅助
研究旨在平衡模拟有限差分（MFD）方法计算成本高和两点通量近似（TPFA）精度受限的问题，提出基于残差一致性指标的自适应MFD框架，能在保持稳定性的同时降低成本并保证精度。
中文摘要 AI 辅助
模拟有限差分（MFD）方法为一般多面体网格上的流动模拟提供了一种稳健的离散化方法，但由于密集的局部算子和降低的全局稀疏性，其计算成本可能会变得很高。两点通量近似（TPFA）提供了一种成本低得多的替代方法，但其精度通常限于K正交网格。为平衡这些相互矛盾的因素，我们提出了一种基于从离散本构方程导出的基于残差的一致性指标的自适应MFD框架。该指标测量局部不一致性，并通过用户规定的容差τ实现自适应TPFA/MFD模板选择。由于在模拟框架内进行自适应，任意TPFA/MFD分区保持稳定且结构保留。理论分析建立了均匀强制性，并证明了相对通量误差的显式容差控制收敛。在具有挑战性的多面体油藏基准上的数值实验表明，其精度与全MFD离散化相当，同时大大降低了矩阵密度和计算成本。
英文摘要
The mimetic finite difference (MFD) method provides a robust discretization for flow simulation on general polyhedral meshes, but its computational cost can become significant due to dense local operators and reduced global sparsity. While the two-point flux approximation (TPFA) offers a substantially cheaper alternative, its accuracy is generally restricted to $K$-orthogonal grids. To balance these competing considerations, we present an adaptive MFD framework based on a residual-based consistency indicator derived from the discrete constitutive equations. The indicator measures local inconsistency and enables adaptive TPFA/MFD stencil selection through a user-prescribed tolerance $τ$. Because the adaptation is performed within a mimetic framework, arbitrary TPFA/MFD partitions remain stable and structure preserving. Theoretical analysis establishes uniform coercivity and proves explicit tolerance-controlled convergence of the relative flux error. Numerical experiments on challenging polyhedral reservoir benchmarks demonstrate accuracy comparable to full MFD discretizations while substantially reducing matrix density and computational cost.
