A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers
A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers
复制标题
使用混合有限元局部求解器解决达西问题的多尺度混合方法
DOI：
10.1016/j.cma.2019.05.013
复制
发表时间：
2019
影响因子：
7.2
通讯作者：
F. Valentin
中科院分区：
文献类型：
--
作者：
O. Durán;P. Devloo;S. Gomes;F. Valentin
多尺度混合方法（MHM）是一种求解具有强变解的微分方程组的近似方法。对于流体流动，计算宏观单元边界上的法向通量（乘法器）和每个宏观单元中的粗分段常数势近似（升级）。然后，通过局部问题解决小细节，在宏元素内部使用精细表示，将乘数设置为诺伊曼边界条件（降尺度）。在这项工作中，开发了该方法的一种变体，用MHM-H （div）表示，在降尺度阶段采用混合有限元，而不是在该方法的所有先前出版物中使用的连续有限元。因此，这种替代的MHM方法继承了混合方法的典型改进，如更好的通量精度和宏观元素内部微观尺度上的局部质量守恒，这些都是粗糙非均质介质中多相流的重要特性。考虑了不同的双尺度稳定空间设置。假设向量面函数的法向分量限制在宏观元素边界上定义的给定有限维轨迹空间内。在每个宏元素中，具有消失的法线的内部通量分量和潜在的近似值可以在不同程度上得到丰富：关于内部网格大小，内部多项式度，或两者兼而有之，选择取决于手头的问题。针对上述两种空间场景，给出了MHM-H （div）方法的统一通用误差分析。比较了两个MHM版本的二维测试问题、平滑解、收敛速率验证以及非均质介质中的达西流。MHM-H （div）三维模拟给出了已知的奇异达西解，使用自适应宏观分区，以及振荡渗透率场景。
Abstract Multiscale Hybrid Mixed (MHM) method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions. For fluid flow, normal fluxes (multiplier) over macro element boundaries, and coarse piecewise constant potential approximations in each macro element are computed (upscaling). Then, small details are resolved by local problems, using fine representations inside the macro elements, setting the multiplier as Neumann boundary conditions (downscaling). In this work a variant of the method is developed, denoted by MHM-H (div), adopting mixed finite elements at the downscaling stage, instead of continuous finite elements used in all previous publications of the method. Thus, this alternative MHM method inherits improvements typical of mixed methods, as better flux accuracy, and local mass conservation at the micro scale level inside the macro elements which are important properties for multi-phase flows in rough heterogeneous media. Different two-scale stable space settings are considered. Vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries. In each macro element, the internal flux components, with vanishing normal traces, and the potential approximations, may be enriched in different extents: with respect to internal mesh size, internal polynomial degree, or both, the choice being determined by the problem at hands. A unified general error analysis of the MHM-H (div) method is presented for all these two-scale space scenarios. Both MHM versions are compared for 2D test problems, with smooth solutions, for convergence rates verification, and for Darcy’s flow in heterogeneous media. MHM-H (div) 3D simulations are presented for a known singular Darcy’s solution, using adaptive macro partitions, and for an oscillatory permeability scenario.