Assembly of multiscale linear PDE operators

TL;DR

This paper introduces a simple algorithm to assemble multiscale linear PDE operators from single-scale finite element forms, enabling efficient Krylov methods for complex coupled models across different dimensions.

Contribution

The paper presents a novel algorithm for assembling multiscale PDE operators from single-scale forms, facilitating their use in Krylov methods.

Findings

Effective representation of multiscale models as linear operators

Numerical examples demonstrate coupling across dimensional gaps 1 and 2

Discussion of preconditioners for various problems

Abstract

In numerous applications the mathematical model consists of different processes coupled across a lower dimensional manifold. Due to the multiscale coupling, finite element discretization of such models presents a challenge. Assuming that only singlescale finite element forms can be assembled we present here a simple algorithm for representing multiscale models as linear operators suitable for Krylov methods. Flexibility of the approach is demonstrated by numerical examples with coupling across dimensionality gap 1 and 2. Preconditioners for several of the problems are discussed.