A generalized Hessian-based error estimator for an IPDG formulation of the biharmonic problem in two dimensions

TL;DR

This paper introduces a new Hessian-based error estimator for a symmetric interior penalty discontinuous Galerkin method applied to the biharmonic problem in two dimensions, demonstrating its reliability and efficiency through theoretical analysis and numerical tests.

Contribution

It proposes a novel error estimator based on a generalized Hessian split, which is reliable and efficient for polynomial degrees above 3 without requiring DG stabilization.

Findings

Estimator is provably reliable and efficient for polynomial degree > 3.

Numerical results confirm theoretical efficiency for degrees ≥ 2.

Estimator bounds the standard DG residual error estimator from above.

Abstract

We consider a two dimensional biharmonic problem and its discretization by means of a symmetric interior penalty discontinuous Galerkin method. A novel split of an error measure based on a generalized Hessian into two terms measuring the conformity and nonconformity of the scheme is proven. This splitting is the departing point for the design of a new error estimator, which is provably reliable and efficient for polynomial degree larger than~$3$, and does not involve any DG stabilization. Such an error estimator can be bounded from above by the standard DG residual error estimator. Numerical results assess the theoretical predictions, including the efficiency of the proposed estimator, for all polynomial degrees larger than or equal to~$2$.