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We introduce and analyze a coupled hybridizable discontinuous Galerkin/discontinuous Galerkin (HDG/DG) method for porous media in which we allow fully and partly immersed faults, and faults that separate the domain into two disjoint subdomains. We prove well-posedness and present an a priori error analysis of the discretization. Numerical examples verify our analysis. 

Hybridizable discretizations allow for the elimination of local degrees-of-freedom leading to reduced linear systems. In this paper, we determine and analyze an approach to construct parameter-robust preconditioners for these reduced systems. Using the framework of Mardal and Winther [Numer. Linear Algebra Appl., 18 (2011), pp. 1–40], we first determine a parameter-robust preconditioner for the full system. We then eliminate the local degrees-of-freedom of this preconditioner to obtain a preconditioner for the reduced system. However, not all reduced preconditioners obtained in this way are automatically robust. We therefore present conditions that must be satisfied for the reduced preconditioner to be robust. To demonstrate our approach, we determine preconditioners for the reduced systems obtained from hybridizable discretizations of the Darcy and Stokes equations. Our analysis is verified by numerical examples in two and three dimensions. 

We introduce and analyze a coupled hybridizable discontinuous Galerkin/discontinuous Galerkin (HDG/DG) method for porous media in which we allow fully and partly immersed faults, and faults that separate the domain into two disjoint subdomains. We prove well-posedness and present an a priori error analysis of the discretization. Numerical examples verify our analysis. 

In this paper we present a hybridizable discontinuous Galerkin method for the time-dependent Navier–Stokes equations coupled to the quasi-static poroelasticity equationsviainterface conditions. We determine a bound on the data that guarantees stability and well-posedness of the fully discrete problem and provea priorierror estimates. A numerical example confirms our analysis. 

We present a strongly conservative and pressure-robust hybridizable discontinuous Galerkin method for the coupled time-dependent Navier–Stokes and Darcy problem. We show existence and uniqueness of a solution and present an optimala priorierror analysis for the fully discrete problem when using Backward Euler time stepping. The theoretical results are verified by numerical examples.