hp-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem

Zhaonan Dong (1, 2) , Lorenzo Mascotto (3, 4, 5)

1 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 Dipartimento di Matematica e Applicazioni [Milano]
4 IMATI-CNR - Istituto di Matematica Applicata e Tecnologie Informatiche
5 Fakultät für Mathematik [Wien]

We prove hp-optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global H 2 piecewise polynomial approximants with hp-optimal approximation properties over the given meshes. The hp-optimality is also discussed for C 0-IPDG in two and three dimensions and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that p-suboptimality occurs in the presence of singular essential boundary conditions.

Mots clés: Discontinuous Galerkin; Optimal convergence; A priori error estimation; p-version

Domaines: Mathématiques [math]; Analyse numérique [math.NA]

Zhaonan Dong, Lorenzo Mascotto. hp-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem. 2022. ⟨hal-03889072v1⟩