Hybrid high-order and weak Galerkin methods for the biharmonic problem

Abstract

We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver O(h k+1 ) H 2 -error estimates when using polynomials of order k ≥ 0 to approximate the normal derivative on the mesh (inter)faces. Both HHO methods hinge on a stabilization in the spirit of Lehrenfeld-Schöberl for second-order PDEs. The cell unknowns are polynomials of order (k + 2) that can be eliminated locally by means of static condensation. The face unknowns approximating the trace of the solution on the mesh (inter)faces are polynomials of order (k + 1) in the first HHO method which is valid in dimension two and uses an original stabilization involving the canonical hybrid finite element, and they are of order (k+2) for the second HHO method which is valid in arbitrary dimension and uses only L 2 -orthogonal projections in the stabilization. A comparative discussion with the weak Galerkin methods from the literature is provided, highlighting the close connections and the improvements proposed herein. Additionally, we show how the two HHO methods can be combined with a Nitsche-like boundary-penalty technique to weakly enforce the boundary conditions. An originality in the devised Nitsche's technique is to avoid any penalty parameter that must be large enough. Finally, numerical results showcase the efficiency of the proposed methods, and indicate that the HHO methods can generally outperform discontinuous Galerkin methods and even be competitive with C 0 -interior penalty methods on triangular meshes.

Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D

“…Let T h be a collection of disjoint polyhedral elements K that partition Ω. We assume that all the elements are shape-regular and quasi-uniform in the sense of [10]. We denote by ∂T h the set {∂K : K ∈ T h }.…”

Hybrid high-order method for singularly perturbed fourth-order problems on curved domains

“…Various HHO methods for the biharmonic operator were devised and analyzed recently in [15], including a comparison with existing WG methods for the biharmonic operator. We also refer the reader to [1] for the first HHO method for the biharmonic operator in primal form.…”

“…We also refer the reader to [1] for the first HHO method for the biharmonic operator in primal form. In [15], two HHO methods were proposed (called HHO-A and HHO-B). Both methods use cell unknowns to approximate the solution in each mesh cell, face unknowns to approximate its trace on the mesh faces, and face unknowns to approximate its normal derivatives on the mesh faces.…”

“…HHO-A is restricted to two space dimensions and uses polynomials of degree (k + 1) for the face unknowns related to the trace, whereas HHO-B supports any space dimension but uses polynomials of degree (k + 2) for these face unknowns. Moreover, the HHO-A method was combined in [15] with a Nitsche-type boundary penalty technique, originally introduced in [7] to weakly enforce Dirichlet conditions in HHO methods for second-order PDEs and further developed in [4,3] to handle unfitted meshes in problems with a curved interface or boundary. In particular, one of the advances in [3] is that the weighting parameter in the boundary penalty term does not need to be large enough, but only positive.…”

“…In the present work, our starting point is the HHO-B method from [15]. Consistently with the paradigm considered for singularly perturbed second-order elliptic PDEs, the boundary conditions in (1) are weakly enforced by means of a Nitsche-type boundary penalty technique.…”