Hierarchical Upscaling and Model Reduction Techniques for Multiscale Dual-continuum Systems

Abstract

Simulation in media with multiple interacting continua is often challenging due to distinct properties of the continua, multiple scales and high contrast. Thus, some type of model reduction is required. One of the approaches is a multi-continuum technique, where every process in each continuum is modeled separately and an interaction term is added. Direct numerical simulation in multiscale multi-continuum media is very expensive as it requires a large number of degrees of freedom to completely resolve the micro-scale variation. In this work, we present efficient upscaling and model reduction methods for multiscale dual-continuum systems.

We first consider the numerical homogenization of a multiscale dual-continuum system where the interaction terms between the continua are scaled as O(1/ε²) where ε is the microscopic scale. Computing the effective coefficients of the homogenized equations can be expensive because one needs to solve local cell problems for a large number of macroscopic points. We develop a hierarchical approach for solving these cell problems at a dense network of macroscopic points with an essentially optimal computation cost. The method employs the fact that neighboring representative volume elements (RVEs) share similar features; and effective properties of the neighboring RVEs are close to each other. The hierarchical approach reduces computation cost by using different levels of resolution for cell problems at different macroscopic points. Solutions of the cell problems which are solved with a higher level of resolution are employed to correct the solutions at neighboring macroscopic points that are computed by approximation spaces with a lower level of resolution.

We then consider the case where the interaction terms of the dual-continuum system are scaled as O(1/ε). We derive the homogenized problem that is a dual-continuum system which contains features that are not in the original two scale problem. In particular, the homogenized dual-continuum system contains extra convection terms and negative interaction coefficients while the interaction coefficient between the continua in the original two scale system obtains both positive and negative values. We prove rigorously the homogenization convergence and homogenization convergence rate. Homogenization of dual-continuum system of this type has not been considered before. We present the numerical examples for computing effective coefficients using hierarchical finite element methods.

We assume the above mentioned homogenized equation still possess some degree of multiscale and high contrast features caused by channels in the media. This motivates us to develop the generalized multiscale finite element method (GMsFEM) for an upscaled multiscale dual-continuum equations with general convection and interaction terms. GMs- FEM systematically generates either uncoupled or coupled multiscale basis, via establishing local snapshots and spectral decomposition in the snapshot space. Then the global problem is solved in the constructed multiscale space with a reduced dimensional structure. Convergence analysis of the proposed GMsFEM is accompanied with the numerical results, which support the theoretical results.