Approximation properties of double complexes
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.
Cite
@article{arxiv.2604.13982,
title = {Approximation properties of double complexes},
author = {Daniel Førland Holmen and Jan Martin Nordbotten and Jon Eivind Vatne},
journal= {arXiv preprint arXiv:2604.13982},
year = {2026}
}