Variational multiscale modeling with discontinuous subscales: analysis and application to scalar transport

Abstract

We examine a variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method. We establish stability and convergence results for the methodology as applied to the scalar transport problem, and we prove that the method exhibits optimal convergence rates in the SUPG norm and is robust with respect to the Péclet number if the discontinuous subscale approximation space is sufficiently rich. We apply the method to isogeometric NURBS discretizations of steady and unsteady transport problems, and the corresponding numerical results demonstrate that the method is stable and accurate in the advective limit even when low-order discontinuous subscale approximations are employed.

A discontinuous Galerkin residual‐based variational multiscale method for modeling subgrid‐scale behavior of the viscous Burgers equation

“…To eliminate the fine-scale dependencies in the interface terms of (39), we propose the following fine-scale interface model, in reference to (23):…”

“…Motivated by the individual success of the VMS and DG paradigms, we have developed a general form of the VMS method in a DG framework. In the past, there have been efforts to combine the two approaches, such as the multiscale DG methods introduced in the works of Bochev et al, Buffa et al, and Hughes et al and methods for constructing discontinuous fine‐scale bubble functions . These methods, however, maintain a continuous solution space for the coarse‐scale problem and use discontinuous representations of the fine scales only.…”

“…In the past, there have been efforts to combine the two approaches, such as the multiscale DG methods introduced in the works of Bochev et al, 20 Buffa et al, 21 and Hughes et al 22 and methods for constructing discontinuous fine-scale bubble functions. 23,24 These methods, however, maintain a continuous solution space for the coarse-scale problem and use discontinuous representations of the fine scales only. They thus fundamentally differ from the original VMS idea, that is, the decomposition of the true solution into a discontinuous coarse-scale function space and an accompanying discontinuous fine-scale function space.…”

Preface to: Novel computational approaches to old and new problems in mechanics

“…Contributions address the numerical modeling of material behaviors, i.e., reinforced concrete [14,15,17], composites [1], and shape memory alloys [9], or the numerical simulation of structures, e.g., thin shells [5] and beams [7,10,14,17]. Some of the papers discuss the treatment of multiscale behaviors [4], homogenization techniques [1,6,8], or shape optimization [17]. Advanced discretization approaches include the virtual element method [1] and isogeometric analysis [10,12].…”

“…Mechanics of contact [11], damage [5,7], fracture [2,5,13], phase field [2,16] are other important topics of the SI. Hydro-elastic coupling is considered in [3,12], transport processes are discussed in [4], dynamical applications and seismic problems are tackled in [3,5,17]. The variety of themes makes this volume of particular interest for applied mechanics scholars.…”