Approximation properties of double complexes

TL;DR

This paper compares the simplicial de Rham complex and the Čech-de Rham complex, quantifying their differences and deriving error estimates for mixed-dimensional versus equidimensional formulations.

Contribution

It constructs bounded cochain complexes to measure the approximation between two bigraded Hilbert complexes, providing error estimates for mixed-dimensional problems.

Findings

Quantifies the closeness of the complexes via bounded cochain complexes.

Derives a priori and a posteriori error estimates for the Hodge-Laplace problems.

Shows the error introduced by treating coupled problems as mixed-dimensional.

Abstract

We consider the simplicial de Rham complex and the \v{C}ech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps.