Mollification in strongly Lipschitz domains with application to continuous and discrete de rham complexes

Alexandre Ern (1, 2) , Jean-Luc Guermond (3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 TAMU - Department of Mathematics [Texas]

We construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are L p stable for any real number p ∈ [1, ∞], and commute with the differential operators ∇, ∇×, and ∇·. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields. We use the mollification operators to build projection operators onto general H 1-, H(curl)-and H(div)-conforming finite element spaces, with and without homogeneous boundary conditions. These operators commute with the differential operators ∇, ∇×, and ∇·, are L p-stable, and have optimal approximation properties on smooth functions.

Mots clés: De Rham diagram; Mollification; Finite element approximation

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Jean-Luc Guermond. Mollification in strongly Lipschitz domains with application to continuous and discrete de rham complexes. Computational Methods in Applied Mathematics, 2016, 16 (1), pp.51-75. ⟨10.1515/cmam-2015-0034⟩. ⟨hal-01192941v2⟩