Approximation properties of double complexes
For researchers in numerical analysis and computational geometry, this provides a theoretical framework to quantify the trade-off between mixed-dimensional and equidimensional formulations of spatially coupled problems.
The paper quantifies the approximation error between the simplicial de Rham complex and the Čech-de Rham complex, deriving a priori and a posteriori error estimates for Hodge-Laplace problems. The results show that the error depends on the thickness of overlaps in the equidimensional formulation.
We consider the simplicial de Rham complex and the Čech-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.