Functional analysis and exterior calculus on mixed-dimensional geometries

• DOI: 10.1007/s10231-020-01013-1
• Publication Date: 2020-07-12T10:02:32Z
• Authors: Wietse M. Boon; Jan M. Nordbotten; Jon E. Vatne
• Keywords: Manifold (fluid mechanics); Lemma (botany); Differential form; Operator (biology)

Abstract We are interested in differential forms on mixed-dimensional geometries, the sense of a domain containing sets d -dimensional manifolds, structured hierarchically so that each manifold is contained boundary one or more $$d + 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> manifolds. On any given manifold, we then consider operators tangent to as well discrete (jumps) normal manifold. The combined action these leads notion semi-discrete operator coupling manifolds different dimensions. refer resulting systems equations mixed-dimensional, which have become popular modeling technique for physical applications including fractured and composite materials. establish analytical tools setting, suitable inner products, codifferential operators, Poincaré lemma, Poincaré–Friedrichs inequality. manuscript concluded by defining minimization problem corresponding Hodge Laplacian, show this well-posed.