A new goal-oriented formulation of the finite element method

Kenan Kergrene (1) , Serge Prudhomme (1) , Ludovic Chamoin (2) , Laforest Marc (1)

1 Département de mathématiques et de génie industriel
2 LMT - Laboratoire de Mécanique et Technologie

In this paper, we introduce, analyze, and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary-value problem. The objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal-oriented adaptivity, which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement. In the current approach, we first solve the adjoint problem, then use the adjoint information as a minimization constraint for the primal problem. As a result, the constrained finite element solution is enhanced with respect to the quantities of interest, while maintaining near-optimality in energy norm. We describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples.

Mots clés: Goal-oriented formulation; Mixed formulation; Error estimation; Adaptive mesh refinement; Multi-objective goal functionals

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Kenan Kergrene, Serge Prudhomme, Ludovic Chamoin, Laforest Marc. A new goal-oriented formulation of the finite element method. Computer Methods in Applied Mechanics and Engineering, 2017, 327, pp.256-276. ⟨10.1016/j.cma.2017.09.018⟩. ⟨hal-01633411⟩