Residual-based a posteriori error estimates for $\mathbf{hp}$-discontinuous Galerkin discretisations of the biharmonic problem

TL;DR

This paper develops a new residual-based a posteriori error estimator for an $hp$-discontinuous Galerkin method solving the biharmonic problem, providing reliable bounds and robustness analysis without requiring $ ext{C}^1$-conforming spaces.

Contribution

It introduces the first $hp$-version error estimator for the biharmonic problem in multiple dimensions, with explicit control over polynomial degree effects.

Findings

Error estimator provides both upper and lower bounds on the error.

Lower bounds are robust to mesh size but not polynomial degree.

Numerical experiments confirm practical effectiveness.

Abstract

We introduce a residual-based a posteriori error estimator for a novel $hp$-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a $\mathcal{C}^1$-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the $H^2$ space and a generalised Helmholtz decomposition of the error. This is the first $hp$-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.