There is an abundance of challenging problems in physics and engineering, such as fluid flow in industrial foams and vuggy reservoirs, which are characterized by high porosity and complex material structure. Traditionally, these flows are modeled by Darcy's law relating the macroscopic pressure gradient to the macroscopic fluid velocity. For highly permeable media a more suitable model is given by Brinkman's equations, considered in this project. The internal structure involves multiple spatial scales and permeabilities varying over many orders of magnitude (high contrast). Thus, the development of numerical methods for Brinkman's equations that are robust with respect to the contrast and accurately capture the fine scales is a difficult theoretical problem with important applications to engineering and geosciences. There are numerous methods for efficiently computing multi-scale porous media flows governed by Darcy's law. These include in particular multi-scale finite element methods, subgrid approximation, multi-grid, and domain decomposition methods. Developing an efficient multi-scale approximation for Brinkman is a challenging task, which cannot be addressed using methods available in the Darcy case. The first objective of this project is a development and study of numerical subgrid method for Brinkman's system. The second goal is to represent accurately the fine features of the solution, which cannot be captured by the numerical subgrid approach alone. An extension of the algorithm by alternating Schwarz iterations in order to resolve these features will be developed, theoretically justified and practically tested. The third main objective is to utilize a multi-scale finite element space and an enrichment of the coarse space by adding basis functions resulting from the approximate solution of certain small local generalized eigenvalue problems. This will allow to devise an iterative method that converges independently of the contrast.
Phenomena in engineering such as heat and mass transfer, fluid flows, and materials, are routinely modeled by partial differential equations (PDEs). The solutions of the PDEs are subsequently used to design and develop new materials or new technologies with wide range of applications. These applications require complex models with multiple space scales, which in turn make the solution task challenging and computationally very intensive. The proposed work is motivated by the needs for design and manufacturing of new materials and/or more accurate description of oil reservoirs or aquifers. However, developing efficient and robust solution techniques for computing flows in media with complex internal structures will significantly broaden the areas of application for which the numerical simulation is feasible. Since numerical simulations are key to better understand, control, and optimize the modeled processes the proposed research will eventually benefit other areas in science and engineering.
