Bridging the Gap Between Multiscale Methods and Solvers: Inexact Solvers and Non-Linear Preconditioning

Abstract:

A prevailing challenge in multiscale simulation is to achieve robustness for general problems. In particular, we are interested in problems where the multiscale approximation is of almost sufficient accuracy, but may be unacceptable due to lack of uniform scale separation in the domain. In this setting, it is of interest to find efficient approaches that do not require a completely new analysis of the problem, or a resolved fine-scale approximation.

We thus consider multiscale methods in the framework of domain decomposition and multigrid techniques. Our objective is to formulate linear and non-linear solvers, possessing the following properties: 1) The initial iteration is identical to a multiscale approximation. 2) Further iterations reduce the approximation error with comparable efficiency to dedicated fine-scale solvers. 3) The error can be controlled by explicitly computable error bounds.

We give examples from a range of applications, including large-scale simulation of CO₂ storage, mechanical deformation of heterogeneous poroelastic materials, and flow and heat transfer in highly fractured porous media associated with geothermal energy recovery. The latter application is illustrated in the figures, where we compare the A) Multiscale approximation, B) Approximation with intermediate accuracy, and C) Fine-scale solution for the problem.