Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra. (9th August 2015)

• Record Type: Journal Article
• Authors: Apel, Thomas; Pfefferer, Johannes; Winkler, Max
• Is Part Of: Mathematical methods in the applied sciences. Volume 39:Number 5(2016:Mar. 30)
• Page Start: 1206
• Page End: 1232
• Publication Date: 2015-08-09
• Subjects: optimal control -- singularities -- finite element method -- local mesh refinement -- variational discretization -- error estimates; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3566
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H 1 (Ω)‐norm, L 2 (Ω)‐norm and L 2 (Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.