Divergence-free mixed finite elements for the incompressible Navier-Stokes equation [Elektronische Ressource] / vorgelegt von Alexander Linke

Divergence-Free Mixed Finite Elements for the IncompressibleNavier-Stokes EquationDen Naturwissenschaftlichen Fakult¨atender Friedrich-Alexander-Universit¨at Erlangen-Nurn¨ bergzurErlangung des Doktorgradesvorgelegt vonAlexander Linkeaus JugenheimAls Dissertation genehmigt von denNaturwissenschaftlichen Fakult¨aten der Universitat¨ Erlangen-Nur¨ nbergTag der mundlic¨ hen Prufung¨ : 18.12.2007Vorsitzender derPromotionskommission: Prof. Dr. Eberhard B¨anschProf. Dr. Eberhard B¨anschErstberichterstatter:Zweitberichterstatter: Prof. Dr. Lutz TobiskaSummaryThis dissertation is concerned with the numerical approximation of the in-compressible Navier-Stokes equation. As our main contribution, we proposeand analyze a new stabilized mixed finite element scheme for computing in-compressible laminar flows, and investigate several properties of this newscheme. The finite element analysis is presented for the Oseen model prob-lem. The scheme is based on the Scott-Vogelius mixed finite element and onsymmetric stabilization operators for dominant convection, proposed in thelast recentyears. In particular, the scheme delivers divergence-free pointwisevelocityapproximations,withanapproximationqualitythatiscompletelyin-dependent of the pressure. Therefore, the cumbersome grad-div stabilizationdrops out from the discrete equations, and standard multi-grid approachesare possible for the solution of the evolving linear systems.