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

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces

We present a finite volume method for the solution of the two-dimensional Poisson equation r· ((x)ru(x)) = f(x) with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz function on Cartesian grids for the solution u(x) resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the L1 and L2 norm

Eulerian finite element methods for interface problems and fluid-structure interactions

In this thesis, we develop an accurate and robust numerical framework for interface problems involving moving interfaces. In particular, we are interested in the simulation of fluid-structure interaction problems in Eulerian coordinates. Our numerical model for fluid-structure interactions (FSI) is based on the monolithic "Fully Eulerian" approach. With this approach we can handle both strongly-coupled problems and large structural displacements up to contact. We introduce modified discretisation schemes of second order for both space and time discretisation. The basic concept of both schemes is to resolve the interface locally within the discretisation. For spatial discretisation, we present a locally modified finite element scheme that is based on a fixed patch mesh and a local resolution of the interface within each patch. It does neither require any remeshing nor the introduction of additional degrees of freedom. For discretisation in time, we use a modified continuous Galerkin scheme. Instead of polynomials in direction of time, we define polynomial functions on space-time trajectories that do not cross the interface. Furthermore, we introduce a pressure stabilisation technique based on "Continuous Interior Penalty" method and a simple stabilisation technique for the structural equation that increases the robustness of the Fully Eulerian approach considerably. We give a detailed convergence analysis for all proposed discretisation and stabilisation schemes and test the methods with numerical examples. In the final part of the thesis, we apply the numerical framework to different FSI applications. First, we validate the approach with the help of established numerical benchmarks. Second, we investigate its capabilities in the context of contact problems and large deformations. We study contact problems of a falling elastic ball with the ground, an inclined plane and some stairs including the subsequent bouncing. For the case that no fluid layer remains between ball and ground, we use a simple contact algorithm. Furthermore, we study plaque growth in blood vessels up to a complete clogging of the vessel. Therefore, we use a monolithic mechano-chemical fluid-structure-interaction model and include the fast pulsating flow dynamics by means of a temporal two-scale scheme. We present detailed numerical studies for all three applications including a numerical convergence analysis in space and time, as well as an investigation of the influence of different material parameters