A Hybrid High-Order method for highly oscillatory elliptic problems

TL;DR

This paper introduces a Hybrid High-Order (HHO) method for solving highly oscillatory elliptic problems on general meshes, offering improved flexibility and complete analysis compared to existing multiscale approaches.

Contribution

The paper presents the first complete analysis of an HHO method for oscillatory elliptic problems on general meshes, with new reconstruction operators and error estimates.

Findings

The method handles arbitrary polynomial orders $k \\geq 0$.

Energy-error estimate of the form $(\\varepsilon^{1/2} + H^{k+1} + (\\\varepsilon/H)^{1/2})$.

Numerical tests confirm theoretical error bounds.

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 $k\geq 0$. For face unknowns that are polynomials of degree $k$, we devise two versions of the method, depending on the polynomial degree $(k-1)$ or $k$ of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form $\left(\varepsilon^{\frac{1}{2}}+H^{k+1}+(\varepsilon/H)^{\frac{1}{2}}\right)$, and we illustrate our theoretical findings on some test-cases.