Adaptive Spline Finite Element Methods for Fourth Order Elliptic Problems PDF Download

Author: Ibrahim Al Balushi
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Languages : en
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"This thesis concentrates on the error analysis of B-spline based finite-element methods for three fourth-order elliptic partial differential equations subject to essential boundary conditions. The first being the biharmonic equation with square-integrable right-hand side and the second and third are models for quasi-geostrophic equations (QGE) simulating large-scale wind-driven oceanic currents. The goal of this thesis is two-fold. On one hand, we derive and analyze error estimators for the purpose of adaptive h-refinement. The earliest effort was concerned with the linear Stommel-Munk. We note that a second-order treatment has been done in 2009 by Juntunen and Stenberg where the analysis hinges on a so-called saturation assumption to relate the numerical error with the discrete error between two refinements. We carry out a similar analysis for the fourth-order PDE. In the nonlinear SQGE we perform the error analysis without a saturation assumption making this work novel in two ways: The treatment requires dealing with the nonlinear convective term and the reliability proofs are saturation-assumption free. The second goal of this thesis is concerned with the convergence and optimality of Nitschetype adaptive methods for the biharmonic equation. Such a study for general second order elliptic order equations has been extensively studied when essential boundary conditions are prescribed into the discrete space. The first convergence proof for the Poisson problem was given by D ̈orfler in 1996 and improved on by Morin, Nochetto, and Siebert in 2000 where some stringent conditions on the domain partitions were removed. Those ideas were soon to be extended to general second order linear elliptic problems by Mekchay and Nochetto, and finally a convergence analysis in a Hilbert space setting was given by Morin, Siebert and Veeser. The first analysis of convergence rates and quasi-optimality for the Poisson problem is pioneered by Binev, Dahmen and DeVore in 2004 and also by Stevenson where he removed an artificial coarsening step. Those ideas were applied to symmetric second order linear elliptic problems by Casc ́on, Kreuzer, Nochetto and Siebert and further generalized by Feischl, Führer and Praetorius to non-symmetric linear problems as well as to strongly monotone nonlinear operators. We add that all aforementioned literature consider boundary condition conforming finite-element spaces in that those discrete spaces satisfy the boundary conditions. For completeness, we do the same for the biharmonic problem. As far as non-conforming methods are concerned, to the best of our knowledge, no such study has been made for Nitsche’s method before the appearance of our work, not even for the Poisson problem. The closest situation we have is that of discontinuous Galerkin methods for symmetric second order elliptic problems which we draw our inspiration from. The convergence and quasi-optimality of discontinuous Galerkin methods was studied by Bonito, Andrea and Nochetto in 2010"--