An adaptive stabilized finite element method based on residual minimization

Victor M Calo (1) , Alexandre Ern (2, 3) , Ignacio Muga (4) , Sergio Rojas (1)

1 Department of Applied Geology [Perth]
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 PUCV - Pontificia Universidad Católica de Valparaíso

We devise and analyze a new adaptive stabilized finite element method. We illustrate its performance on the advection-reaction model problem. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf-sup stability. We formulate this residual minimization as a stable saddle-point problem which delivers a stabilized discrete solution and an error representation that drives the adaptive mesh refinement. Numerical results on the advection-reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers.

Domaines: Analyse numérique [math.NA]

Victor M Calo, Alexandre Ern, Ignacio Muga, Sergio Rojas. An adaptive stabilized finite element method based on residual minimization. 2019. ⟨hal-02196242v1⟩