Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes

TL;DR

This paper derives improved $L^2$-error estimates for wave equation discretizations using hybrid nonconforming methods on simplicial meshes, achieving superclose and optimal bounds.

Contribution

It introduces novel approximation estimates for interpolation operators, enabling superclose and optimal $L^2$-error bounds for hybrid nonconforming methods.

Findings

Achieved superclose $L^2$-error bounds in equal-order case.

Established optimal $L^2$-error bounds in mixed-order case.

Developed new approximation estimates for hybridizable discontinuous Galerkin interpolation.

Abstract

We present improved $L^2$-error estimates on the time-integrated primal variable for the wave equation in its first-order formulation. The space discretization relies on a hybrid nonconforming method, such as the hybridizable discontinuous Galerkin, the hybrid high-order or the weak Galerkin methods. We consider both equal-order and mixed-order settings on simplices, and include the lowest-order case with piecewise constant unknowns on the faces and in the cells. Our main result is a superclose, resp., optimal bound on the above error in the equal-, resp., mixed-order case. A key result of independent interest to achieve these estimates are novel approximation estimates for an interpolation operator inspired from the hybridizable discontinuous Galerkin literature.