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

2025/07/08 by Théophile Chaumont-Frelet, Chaumont-Frelet, Théophile, Joscha Gedicke +3
Computer Science · Engineering · Mathematics · #65N12 #65N30 #65N50 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2507.05776

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds
• Finite Element Complexes in Two Dimensions
• A finite element elasticity complex in three dimensions
• Residual-based a posteriori error estimates for hp-discontinuous Galerkin discretisations of the biharmonic problem
• Finite elements for divdiv-conforming symmetric tensors
• Convergence of an adaptive C⁰-interior penalty Galerkin method for the biharmonic problem
• Trace operators of the bi-Laplacian and applications
• An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation
• Residual-based a posteriori error analysis for symmetric mixed Arnold-Winther FEM
• A posteriori error estimators for nonconforming finite element methods

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