The discontinuous Galerkin (DG) method has found widespread application in elliptic problems with rough coefficients, of which the Darcy flow equations are a prototypical example. One of the long-standing issues of DG approximations is the overall computational cost, and many different strategies have been proposed, such as the variational multiscale DG method, the hybridizable DG method, the multiscale DG method, the embedded DG method, and the Enriched Galerkin method. In this work, we propose a mixed dual-scale Galerkin method, in which the degrees-of-freedom of a less computationally expensive coarse-scale approximation are linked to the degrees-of-freedom of a base DG approximation. We show that the proposed approach has always similar or improved accuracy with respect to the base DG method, with a considerable reduction in computational cost. For the specific definition of the coarse-scale space, we consider Raviart–Thomas finite elements for the mass flux and piecewise-linear continuous finite elements for the pressure. We provide a complete analysis of stability and convergence of the proposed method, in addition to a study on its conservation and consistency properties. We also present a battery of numerical tests to verify the results of the analysis, and evaluate a number of possible variations, such as using piecewise-linear continuous finite elements for the coarse-scale mass fluxes.@en

In this article, we develop a dynamic version of the variational multiscale (D-VMS) stabilization for nearly/fully incompressible solid dynamics simulations of viscoelastic materials. The constitutive models considered here are based on Prony series expansions, which are rather common in the practice of finite element simulations, especially in industrial/commercial applications. Our method is based on a mixed formulation, in which the momentum equation is complemented by a pressure equation in rate form. The unknown pressure, displacement, and velocity are approximated with piecewise linear, continuous finite element functions. To prevent spurious oscillations, the pressure equation is augmented with a stabilization operator specifically designed for viscoelastic problems, in that it depends on the viscoelastic dissipation. We demonstrate the robustness, stability, and accuracy properties of the proposed method with extensive numerical tests in the case of linear and finite deformations.

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