A theoretical framework for discontinuity capturing: Joining variational multiscale analysis and variation entropy theory

In this paper we show that the variational multiscale method together with the variation entropy concept form the underlying theoretical framework of discontinuity capturing. The variation entropy [M.F.P. ten Eikelder and I. Akkerman, Comput. Methods Appl. Mech. Engrg. 355 (2019) 261-283] is the recently introduced concept that equips total variation diminishing solutions with an entropy foundation. This is the missing ingredient in order to show that the variational multiscale method can capture sharp layers. The novel framework naturally equips the variational multiscale method with a class of discontinuity capturing operators. This class includes the popular YZβ method and methods based on the residual of the variation-entropy. The discontinuity capturing mechanisms do not contain ad hoc devices and appropriate length scales are derived. Numerical results obtained with quadratic NURBS are virtually oscillation-free and show sharp layers, which confirms the viability of the methodology.