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

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~$2$.

💡 Research Summary

The paper addresses the numerical solution of the two‑dimensional biharmonic equation with clamped boundary conditions using a symmetric interior penalty discontinuous Galerkin (SIPDG) method. Classical a‑posteriori error analysis for DG discretizations of fourth‑order problems typically relies on residual estimators that contain stabilization terms proportional to the jumps of the solution and its gradient. These stabilization parameters must be chosen sufficiently large to guarantee well‑posedness and to obtain reliable and efficient error bounds, but their selection adds complexity to adaptive algorithms.

The authors introduce a novel concept: a generalized Hessian (H_h(u_h)) defined as the sum of the element‑wise Hessian (D^2_h u_h) and a fourth‑order lifting operator (L_h(u_h)). The lifting operator, constructed in (3.4), incorporates the jump information of the discrete solution in a tensorial form, thereby embedding the non‑conformity of the DG solution directly into the generalized Hessian. Consequently, the error measure \