A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems

Abstract

We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation k ě 0. For face unknowns that are polynomials of degree k, we devise two versions of the method, depending on the polynomial degree pk´1q or k of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form`ε 1 {2`H k`1`p ε{Hq 1 {2˘, and we illustrate our theoretical findings on some test-cases.

A hybrid high‐order formulation for a Neumann problem on polytopal meshes

“…We have proved the convergence of the method, with the optimal rates of convergence when the exact solution is smooth enough: order k + 1 for the flux error, and k + 2 for potential error, when piecewise polynomial of degree at most k are considered for the corresponding approximations. This technique can deal with hanging nodes, as in the Refined mesh (Figure 1c), and also with triangular, quadrilateral and hexagonal meshes (Figure 1a, b, This library has been used to solve many problems as those described in [21][22][23][24][25][26][27][28][29]. On the other hand, HArDCore (Hybrid Arbitrary Degree::Core, https://github.com/jdroniou/HArDCore) is a C++ code focused on HHO methods, but it can be useful for a wide range of hybrid methods.…”

Bridging the hybrid high-order and virtual element methods

“…As such, we will henceforth refer to them simply as VE methods, and we will exclusively focus on them in the sequel. An example of (nonconforming) method that hinges on a different kind of virtual space is given by the equal-order multiscale HHO method of [22,Section 5.2], for which local virtual functions do solve (oscillatory) PDEs with polynomial data, but the local virtual space does not contain polynomials in general.…”

“…For some insight on the value of c P on more general element shapes, we refer to [47]. In the forthcoming analysis, we will also need (i) the following nonstandard inverse and discrete trace inequalities, whose proofs can be found in [22,Lemma 4.4 (take A ε " I d )]: for all v P H 1 pT q such that v P P q d pT q for some q P N, there holds…”

“…1. In the conforming case with d " 2 (respectively, d " 3), one first computes the operator r k 1,F defined by (22) for all F P F h (respectively, r k 1,e for all e P E h ). This requires to solve a SPD system of size k`1 (cf.…”

“…Proposition 4.2. The operator r k 1,F defined by (22) coincides with the (canonical) reconstruction operator pΣ k 1,F q´1.…”