Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations

Abstract

International audienceThis paper presents two novel contributions on the recently introduced Mixed High-Order (MHO) methods [D. Di Pietro, A. Ern, hal-00918482]. We first address the hybridization of the MHO method for a scalar diffusion problem and obtain the corresponding primal formulation. Based on the hybridized MHO method, we then design a novel, arbitrary order method for the Stokes problem on general meshes. A full convergence analysis is carried out showing that, when independent polynomials of degree k are used as unknowns (at elements for the pressure and at faces for each velocity omponent), the energy-norm of the velocity and the L2-norm of the pressure converge with order k+1, while the L2-norm of the velocity (super-)converges with order k+2. The latter property is not shared by other methods based on a similar choice of unknowns. The theoretical results are numerically validated in two space dimensions on both standard and polygonal meshes

A Hybrid High-Order Method for Darcy Flows in Fractured Porous Media

“…Our focus is here on the Hybrid High-Order (HHO) methods originally introduced in [22] in the context of linear elasticity, and later applied in [1,24,23,25] to anisotropic heterogeneous diffusion problems. HHO methods are based on degrees of freedom (DOFs) that are broken polynomials on the mesh and on its skeleton, and rely on two key ingredients: (i) physics-dependent local reconstructions obtained by solving small, embarassingly parallel problems and (ii) high-order stabilization terms penalizing face residuals.…”

A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits

“…The size of the linear system corresponding to the discrete problem (39) can be significantly reduced by resorting to static condensation. Following the procedure hinted to in [1] and detailed in [26, Section 6.2], it can be shown that the only globally coupled variables are the face unknowns for the velocity and the mean value of the pressure inside each mesh element. Hence, after statically condensing the other discrete unknowns, the size of the linear system matrix is…”

“…This behaviour is expected, as for the Darcy problem the L 2 -norm of the velocity coincides with the energy norm, and no superconvergent behaviour can be triggered. For the Stokes problem, on the other hand, superconvergence in the L 2 -norm for HHO methods has been proved in, e.g., [1,Theorem 4.5] and [26,Theorem 7], and similar arguments can lead to analogous estimates in the Brinkman case. For the Brinkman and Stokes problems, an inspection of the last lines of Tables 2 and 3 reveals that numerical precision is approached on the finest mesh for k = 4 and, correspondingly, the order of convergence deteriorates (see the starred values in the tables).…”

A Review of Hybrid High-Order Methods: Formulations, Computational Aspects, Comparison with Other Methods

“…For the sake of simplicity, we focus on a model elliptic problem with possibly heterogeneous/anisotropic diffusion tensor. Most of the results contained herein can be derived from relatively straightforward adaptations of the proofs contained in previous works [1,20,25,[27][28][29][30][31]; for the sake of conciseness, we provide bibliographic references for the most technical proofs, while some details are included for those proofs that allow us to highlight the more practical aspects of the method. One novel aspect is that we treat nonhomogeneous mixed Dirichlet-Neumann boundary conditions, while previous work has focused on homogeneous, pure Dirichlet boundary conditions.…”