Adaptive vertex-centered finite volume methods for general second-order linear elliptic PDEs

We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp. 2228--2255] was restricted to symmetric problems, the present analysis also covers non-symmetric problems and hence the important case of present convection

Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general

Conforming multilevel FEM for the biharmonic equation

Der Multigrid V-cycle mit lokalem Glätter liefert einen effizienten iterativen Löser für die adaptive Finite Elemente Methode (AFEM). In Kombination mit einem effizienten und zuverlässigen Schätzer des algebraischen Fehlers ermöglicht dies eine optimale Zeitkomplexität des adaptiven Algorithmus. Diese Arbeit erweitert die a posteriori Analysis der hierarchischen Argyris Finite Elemente Methode (FEM) auf die biharmonische Gleichung mit inhomogenen und gemischten Randbedingungen. Optimale Konvergenzraten der hierarchischen Argyris AFEM folgen aus den Axiomen der Adaptivität unter Beobachtung einer Bestapproximationseigenschaft des Argyris Interpolaten der essenziellen Randdaten. Numerische Experimente bestätigen optimale Konvergenzraten des adaptiven Algorithmus und liefern einen Vergleich zwischen dem direkten Löser und dem iterativen Multigrid Löser. Verschiedene Benchmark-Tests betrachten unterschiedliche Randdaten, Punktlasten und unterstreichen die Stärken der konformen Argyris FEM. Im Fazit ergibt dies die Rehabilitation des Argyris Finiten Elementes in Zusammenhang mit dem erweiterten Argyris Raum.A multigrid V-cycle with local smoothing is considered with an efficient and reliable estimator of the algebraic error. This gives rise to an efficient iterative solver for the adaptive finite element method (AFEM) with optimal time complexity. This thesis extends a posteriori error analysis for the hierarchical Argyris finite element method (FEM) to the biharmonic equation with inhomogeneous and mixed boundary conditions. Optimal convergence of the hierarchical Argyris AFEM with direct solve follows with the axioms of adaptivity by observing a best-approximation property for the Argyris interpolant of the essential boundary data. Numerical validation is presented for optimal rates of AFEM together with a comparison between a direct solver and the local multigrid solver. Model benchmarks include different boundary conditions, point loads and highlight the strength of the lowest-order conforming Argyris FEM. A conclusion underlines the rehabilitation of the Argyris element in conjunction with the extended Argyris space