We develop a novel Hybrid High-Order method for the simulation of Darcy flows in fractured porous media. The discretization hinges on a mixed formulation in the bulk region and a primal formulation inside the fracture. Salient features of the method include a seamless treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes. For the version of the method corresponding to a polynomial degree k ě 0, we prove convergence in h k`1 of the discretization error measured in an energy-like norm. In the error estimate, we explicitly track the dependence of the constants on the problem data, showing that the method is fully robust with respect to the heterogeneity of the permeability coefficients, and it exhibits only a mild dependence on the square root of the local anisotropy of the bulk permeability. The numerical validation on a comprehensive set of test cases confirms the theoretical results.
In this work, we propose a model for the passive transport of a solute in a fractured porous medium, for which we develop a Hybrid High-Order (HHO) space discretization. We consider, for the sake of simplicity, the case where the flow problem is fully decoupled from the transport problem. The novel transmission conditions in our model mimic at the discrete level the property that the advection terms do not contribute to the energy balance. This choice enables us to handle the case where the concentration of the solute jumps across the fracture. The HHO discretization hinges on a mixed formulation in the bulk region and on a primal formulation inside the fracture for the flow problem, and on a primal formulation both in the bulk region and inside the fracture for the transport problem. Relevant features of the method include the treatment of nonconforming discretizations of the fracture, as well as the support of arbitrary approximation orders on fairly general meshes.
Abstract. In this work we develop a fully implicit Hybrid High-Order algorithm for the CahnHilliard problem in mixed form. The space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces. The proposed method has several advantageous features: (i) It supports fairly general meshes possibly containing polyhedral elements and nonmatching interfaces; (ii) it allows arbitrary approximation orders; (iii) it has a moderate computational cost thanks to the possibility of locally eliminating element-based unknowns by static condensation. We perform a detailed stability and convergence study, proving optimal convergence rates in energy-like norms. Numerical validation is also provided using some of the most common tests in the literature.
We introduce a three-dimensional Hybrid High-Order method for magnetostatic problems. The proposed method is easy to implement, supports general polyhedral meshes, and allows for arbitrary orders of approximation.
We prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. We then introduce two Hybrid High-Order methods for the approximation of the magnetostatics model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, we perform a comprehensive analysis of the two methods. We finally validate them on a set of test-cases.
We propose a novel Hybrid High-Order method for the Cahn-Hilliard problem with convection. The proposed method is valid in two and three space dimensions, and it supports arbitrary approximation orders on general meshes containing polyhedral elements and nonmatching interfaces. An extensive numerical validation is presented, which shows robustness with respect to the Péclet number.
Cahn-Hilliard equationLet Ω ⊂ R d , d ∈ {2, 3}, denote a bounded connected convex polyhedral domain with Lipschitz boundary ∂ Ω and outward normal n, and let t F > 0. The convective Cahn-Hilliard problem consists in finding the order-parameter c : Ω × (0,t F ] → R and the chemical potential w :
We prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. We then introduce two Hybrid High-Order methods for the approximation of the magnetostatic model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes with star-shaped elements, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, we perform a comprehensive analysis of the two methods. We finally validate them on a set of test-cases.
We propose a new definition of the normal fracture diffusion-dispersion coefficient for a reduced model of passive transport in fractured porous media.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.