A generalized Hessian-based error estimator for an IPDG formulation of the biharmonic problem in two dimensions
Théophile Chaumont-Frelet, Joscha Gedicke, Lorenzo Mascotto
TL;DR
This work advances a posteriori error estimation for symmetric IPDG discretizations of the 2D biharmonic problem by introducing a generalized Hessian H_h that combines the broken Hessian with a lifting operator. A regular decomposition of the div-div complex yields a splitting of the Hessian error into conforming and nonconforming parts, enabling a stable, stabilization-free error estimator η. The authors prove reliability and efficiency bounds for η (with p-dependent conditions) and show that η can be controlled by the standard DG residual estimator, while numerical tests confirm matching convergence rates and effective adaptive refinements. The approach promises potential simplifications in adaptive convergence proofs and provides a pathway to p-dependent refinements and nontrivial topology extensions in 2D (with possible extensions to 3D).
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~$2$.