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International audience; This work deals with the discretization of single-phase Darcy flows in fractured and deformable porous media, including frictional contact at the matrix-fracture interfaces. Fractures are described as a network of planar surfaces leading to so-called mixed-dimensional models. Small displacements and a linear poro-elastic behavior are considered in the matrix. One key difficulty to simulate such coupled poro-mechanical models is related to the formulation and discretization of the contact mechanical sub-problem. Our starting point is based on the mixed formulation using facewise constant Lagrange multipliers along the fractures representing normal and tangential stresses. This is a natural choice for the discretization of the contact dual cone in order to account for complex fracture networks with corners and intersections. It leads to local expressions of the contact conditions and to efficient semi-smooth nonlinear solvers. On the other hand, such a mixed formulation requires to satisfy a compatibility condition between the discrete spaces restricting the choice of the displacement space and potentially leading to sub-optimal accuracy. This motivates the investigation of two alternative formulations based either on a stabilized mixed formulation or on the Nitsche's method. These three types of formulations are first investigated theoritically in order to enhance their connections. Then, they are compared numerically in terms of accuracy and nonlinear convergence. The sensitivity to the choice of the formulation parameters is also investigated. Several 2D test cases are considered with various fracture networks using both P1 and P2 conforming Finite Element discretizations of the displacement field and an Hybrid Finite Volume discretization of the mixed-dimensional Darcy flow model.

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp. 283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open source implementation based on the FEniCS Project finite element software is provided as supplementary material.