Analysis of a Hybrid High-Order–discontinuous Galerkin discretization method for nonlinear poroelasticity

Michele Botti (1) , Daniele Antonio Di Pietro (1) , Pierre Sochala (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 BRGM - Bureau de Recherches Géologiques et Minières

In this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

Mots clés: Nonlinear poroelasticity; nonlinear Biot problem; Korn's inequality; Hybrid High-Order methods; discontinuous Galerkin methods; polyhedral meshes

Domaines: Mathématiques [math]; Analyse numérique [math.NA]

Michele Botti, Daniele Antonio Di Pietro, Pierre Sochala. Analysis of a Hybrid High-Order–discontinuous Galerkin discretization method for nonlinear poroelasticity. 2018. ⟨hal-01785810v1⟩