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

TL;DR

This paper introduces a hybrid high-order method for accurately solving singularly perturbed fourth-order PDEs on curved domains, effectively handling boundary conditions and varying perturbation parameters.

Contribution

The paper presents a novel HHO method combining Nitsche boundary penalties and a scaled stabilization operator for singularly perturbed problems on curved domains.

Findings

Stable and optimal error estimates across all perturbation parameters.

Method effectively handles curved boundaries and boundary conditions.

Numerical results confirm theoretical predictions.

Abstract

We propose a novel hybrid high-order method (HHO) to approximate singularly perturbed fourth-order PDEs on domains with a possibly curved boundary. The two key ideas in devising the method are the use of a Nitsche-type boundary penalty technique to weakly enforce the boundary conditions and a scaling of the weighting parameter in the stabilization operator that compares the singular perturbation parameter to the square of the local mesh size. With these ideas in hand, we derive stability and optimal error estimates over the whole range of values for the singular perturbation parameter, including the zero value for which a second-order elliptic problem is recovered. Numerical experiments illustrate the theoretical analysis.