Analysis of the HDG method for the stokes–darcy coupling

Abstract

In this article, we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. We consider a fully‐mixed formulation in which the main unknowns in the fluid are given by the stress, the vorticity, the velocity, and the trace of the velocity, whereas the velocity, the pressure, and the trace of the pressure are the unknowns in the porous medium. In addition, a suitable enrichment of the finite dimensional subspace for the stress yields optimally convergent approximations for all unknowns, as well as a superconvergent approximation of the trace variables. To do that, similarly as in previous articles dealing with development of the a priori error estimates, we use the projection‐based error analysis to simplify the corresponding study. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 885–917, 2017

Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems

“…By the local elimination of the unknowns defined in the interior of elements, the HDG method leads to a system where the unknowns are only the globally coupled degrees of freedom describing the introduced Lagrange multipliers. We refer to [3,15,17,20,21,24,27,28,33,37,38,40,47] for some developments and applications for the HDG method. As for a posteriori error analysis of the HDG method, the authors in [2,18,19] established a posteriori error estimates for second-order elliptic equations.…”

An embedded–hybridized discontinuous Galerkin method for the coupled Stokes–Darcy system

“…[5,6,7,8,9,10,11]. Other devised finite element methods include discontinuous Galerkin (DG) methods [12,13,14,15,16,17], hybridizable discontinuous Galerkin (HDG) methods [18,19,20,21], weak Galerkin methods (WG) [22], and weak virtual element methods (WVEM) [23].…”

Simulation of coupled Stokes-Darcy equations using the combination of balanced H(div)-L2 approximation spaces

“…There are several works in the literature dealing with coupled Stokes-Darcy problems, and we refer to [4][5][6] for extensive bibliography review on this topic. Recently, it has 2 regained relevance and is attracting new attention of numerical analysts, and improved numerical methods have been proposed [7].…”