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Browsing by Subject "Partial differential equation"

Now showing 1 - 3 of 3
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    Defects and stability of turing patterns
    (2013-05) Wu, Qiliang
    This paper is concerned with formation mechanisms of patterns and contains two main parts. The first one is about defects of patterns. Specifically, we study grain boundaries in the Swift-Hohenberg equation. Grain boundaries arise as stationary interfaces between roll solutions of different orientations. Our analysis shows that such stationary interfaces exist near onset of instability for arbitrary angle between the roll solutions. This extends prior work in [1] where the analysis was restricted to large angles, that is, weak bending near the grain boundary. The main new difficulty stems from possible interactions of the primary modes with other resonant modes. We generalize the normal form analysis in [1] and develop a singular perturbation approach to treat resonances. In the second part, we investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns which exhibits an approximate discrete conservation law. The key ingredients to the normal form is a conjugation of the reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site, we decompose perturbations into neutral phase shifts and normal decaying components. As an application of our normal form construction, we prove nonlinear stability of Turing patterns with respect to perturbations that are small in L1 ∩ L∞, with sharp rates, recovering and slightly improving on results in [2, 3].
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    The variational multiscale method for mixed finite element formulations
    (2018-04) Stoter, Klaas
    In this thesis, the variational multiscale method is explored in the context of mixed formulations of partial differential equations. The domain decomposition variational multiscale method that has recently been introduced by the author is used as a basis. The function spaces of both the primary and the auxiliary variable are decomposed in a coarse-scale and a fine-scale space. The mixed weak formulations are then derived on a per-element basis. The same scale decomposition is used to rewrite the transmission conditions, which are then incorporated into the weak formulations to couple the elements. The result is a mixed finite element formulation that includes all the fine-scale terms that capture the exact scale interaction, irrespective of the order of continuity of the coarse-scale and fine-scale function spaces. A closure model has to be substituted in place of the fine-scale terms. This closure model dictates the nature of the scale decomposition by imposing a constraint on the fine-scale solution. It is shown that, in the context of Poisson's equation, numerous existing discontinuous Galerkin formulations may be interpreted as particular choices of closure models. Due to the mixed origin of the formulation, a broad range of formulations may be retrieved. Also the Raviart-Thomas method, the Brezzi-Douglas-Marini method and hybridized formulations are investigated from this perspective. The associated fine-scale constraints are examined in depth. Similarly, an advection-diffusion problem is considered, and the fine-scale constraint associated with upwind finite element formulations are investigated. Finally, the residual-based modeling of the fine-scale solution is explored in the context of mixed formulations. Incorporation of the model for the one-dimensional advection-diffusion problem leads to a significant accuracy improvement. In particular does it mitigate the overshoot and the oscillation problems that are observed at boundary layers which occur for advection dominated problems.