Numerical upscaling for heterogeneous materials in fractured domains

Abstract

We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An a priori error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.

A generalized finite element method for problems with sign-changing coefficients

“…Moreover, we believe that a similar result holds if the interface does not cut the elements "too badly". We refer to [19] for a similar discussion in a different context. aŝ︀ ∈ 1 (̂︀ ).…”

A multiscale method for heterogeneous bulk-surface coupling

“…In view of this paper, we particularly mention the application to standard parabolic problems [MP18], thermoelasticity [MP17], and poroelasticity [ACM + 20]. Furthermore, [HMW19] recently discussed the application to fracture problems, where also a bulk problem is coupled to an interface problem. Therein, however, the multiscale features are relevant in the bulk as well and the problems in the volume and on the interface remain completely coupled.…”

A generalized finite element method for problems with sign-changing coefficients

“…Moreover, we believe that a similar result holds if the interface does not cut the elements "too badly". We refer to [17] for a similar discussion in a different context.…”