论文标题
一种新型的等法有限元方法，用于流动和运输的多孔培养基，满足离散的最大原理和保护特性
A novel equi-dimensional finite element method for flow and transport in fractured porous media satisfying discrete maximum principle and conservation properties
论文作者
论文摘要
多孔介质中流量和传输的数值模拟通常依赖于杂交模型，即与嵌入矩阵相比，裂缝被视为较低尺寸的对象。这样的模型通常与不合格的离散化结合在一起，因为它们避免了与明确解析裂缝 - 矩阵接口的网格产生相关的固有困难。但是，不合格的离散要求需要对不同子模型的更复杂的耦合，并且可能需要特殊护理以确保保守的通量。我们提出了一种基于裂缝的等法表示的多孔培养基中流动和转运问题的模拟和转运问题的新方法。这些类型的表示的主要挑战是创建网格，这些网格可以解决裂缝和嵌入矩阵之间的几个复杂接口。为了克服这一困难，我们采用了基于自适应网状精炼（AMR）的策略。所提出的AMR底部的想法是从最初均匀的粗网格开始，并完善与至少一种裂缝具有非空重叠的元素。迭代此过程允许创建不均匀的不合格网格，该网格无法解析接口，但可以以任意精度近似它们。我们证明，适应网格的低阶有限元（FE）离散是全球和局部保守的，我们适当地适应了代数通量校正技术，以确保离散的最大原理。特别是，我们表明，必须将臭名昭著的M型条件适应在不合格网格上定义的基础函数。尽管提出的应用来自地球物理应用，但所获得的结果可以应用于任何扩散和运输问题，无论是符合和不合格的网格。
Numerical simulations of flow and transport in porous media usually rely on hybrid-dimensional models, i.e., the fracture is considered as objects of a lower dimension compared to the embedding matrix. Such models are usually combined with non-conforming discretizations as they avoid the inherent difficulties associated with the generation of meshes that explicitly resolve fractures-matrix interfaces. However, non-conforming discretizations demand a more complicated coupling of different sub-models and may require special care to ensure conservative fluxes. We propose a novel approach for the simulation of flow and transport problems in fractured porous media based on an equi-dimensional representation of the fractures. The major challenge for these types of representation is the creation of meshes which resolve the several complex interfaces between the fractures and the embedding matrix. To overcome this difficulty, we employ a strategy based on adaptive mesh refinement (AMR). The idea at the base of the proposed AMR is to start from an initially uniform coarse mesh and refine the elements which have non-empty overlaps with at least one of the fractures. Iterating this process allows to create non-uniform non-conforming meshes, which do not resolve the interfaces but can approximate them with arbitrary accuracy. We demonstrate that low-order finite element (FE) discretizations on adapted meshes are globally and locally conservative and we suitably adapt an algebraic flux correction technique to ensure the discrete maximum principle. In particular, we show that the notorious conditions on M-matrices have to be adapted to the basis functions defined on non-conforming meshes. Although the proposed applications come from geophysical applications, the obtained results could be applied to any diffusion and transport problems, on both conforming and non-conforming meshes.
