Browsing by Subject "Discontinuous Galerkin"
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Item Analysis of the DPG method for the Poisson equation(University of Minnesota. Institute for Mathematics and Its Applications, 2010-10) Demkowicz, L.; Gopalakrishnan, JayadeepItem An analysis of the practical DPG method(University of Minnesota. Institute for Mathematics and Its Applications, 2011-07) Gopalakrishnan, Jayadeep; Qiu, W.Item Devising superconvergent HDG methods for partial differential equations(2012-08) Shi, KeThe DG methods are ideally suited for numerically solving hyperbolic problems. However this is not the case for diffusion problems,even though they are ideally suited for hp-adaptivity. Indeed, when compared with the classical continuous Galerkin methods on the same mesh, they have many more global degrees of freedom and they are not easy to implement. When compared with the mixed methods, they do not provide optimally convergent approximations to the flux and do not display superconvergence properties of the scalar variable. As a response to these disadvantages, the HDG methods were introduced in [6]. Therein, it was shown that HDG methods can be implemented as efficiently as the mixed methods. Later in [7] it was proven that the HDG methods do share with mixed methods their superior convergence properties while retaining the advantages typical of the DG methods. Inspired by these results, in this Thesis we are trying to explore HDG methods in a wider circumstance.Item Experimental validation of the Topological Sensitivity approach to elastic-wave imaging(2015-08) Tokmashev, RomanThe focus of this dissertation is on: i) non-invasive imaging of discrete damage in solids by way of the Topological Sensitivity (TS) approach to elastic-wave tomography, ii) experimental verification of the TS imaging technique using non-contact vibration measurements obtained by 3D Scanning Laser Doppler Vibrometer, and iii) upgrade of the Finite Element (FE) elastodynamic computational platform to treat long range wave propagation toward enhancing the imaging performance of TS under the conditions of limited testing aperture.Item Hybridizable discontinuous Galerkin method for curved domains(2012-01) Solano Palma, Manuel EstebanIn this work we present a technique to numerically solve partial differential equations (PDE’s) defined in general domains . It basically consists in approximating the domain by polyhedral subdomains Dh and suitably defining extensions of the solution from Dh to . More precisely, we solve the PDE in Dh by using a numerical method for polyhedral domains. In order to do that, the boundary condition is transferred from ¡ := ∂ to ¡h, the boundary of Dh, by integrating the gradient of the scalar variable along a path. That is why, in principle, any numerical method that provides an accurate approximation of the gradient can be used. In this work we consider a hybridizable discontinuous Galerkin (HDG) method. This technique has two main advantages over other methods in the literature. First of all, it only requires the distance between ¡ and ¡h to be of the order of the meshsize. This allows us to easily mesh the computational domain. Moreover, high degree polynomial approximations can be used and still obtain optimal orders of convergence even though ¡h is “far” from ¡. We numerically explore this approach by considering three types of steady-state equations. As starting point, we deal with Dirichlet boundary problems for second order elliptic equations. For this problem we fully explain how to properly transfer the boundary condition and how to define the paths, as well. We then apply this technique to exterior diffusion problems. Herein, the HDG method is used for solving the so-called interior problem on a bounded region whereas a boundary element method (BEM) is used for solving the problem exterior to that region. Both methods are coupled at the smooth interface that divides the two regions. Finally, we consider convectiondiffusion problems where we explore how the magnitude of the convective field affects the performance of our method.Item Locking-free hp DGP method for linear elasticity with symmetric stresses(University of Minnesota. Institute for Mathematics and Its Applications, 2011-05) Bramwell, J.; Demkowicz, L.; Gopalakrishnan, Jayadeep; Qiu, W.Item A review of unified a posteriori finite element error control(University of Minnesota. Institute for Mathematics and Its Applications, 2010-10) Carstensen, C.; Eigel, M.; Löbhard, C.; Hoppe, Ronald H.W.Item The variational multiscale method for discontinuous Galerkin type finite element formulations(2019-12) Stoter, KlaasIn this dissertation, we develop the foundation for a framework that unifies variational multiscale analysis and discontinuous Galerkin type methods. We adopt the variational multiscale principles while using finite element approximation spaces that are flexible on Dirichlet boundaries, or even allowed to be completely discontinuous from element to element. We show that many classical methods follow as particular choices of scale decomposition in the variational multiscale paradigm. The methods that we derive as such are Nitsche's method for the weak enforcement of essential boundary conditions, Bassi-Rebay's first method, the Interior Penalty method, the Local Discontinuous Galerkin method and the Hybridizable Discontinuous Galerkin method. We derive explicitly the projection operators corresponding to each of these formulations. This illustrates that these `non-conformal' methods and the accompanying penalty terms are in complete agreement with the variational multiscale formalism. At the same time, knowledge of these projectors guides our fine-scale modeling efforts. We develop a residual-based model that incorporates the non-vanishing fine scales at the element boundaries. Our model includes additional boundary terms with new model parameters. For all model parameters, we propose a parameter estimation strategy that is effective for both lower- and higher-order basis functions. Our numerical experiments illustrate that the classical residual-based fine-scale model exhibits overly diffusive behavior at boundaries with weakly enforced conditions. The additional terms in the new augmented model counters this defect and improves the quality of the boundary layer approximation.Item The variational multiscale method for mixed finite element formulations(2018-04) Stoter, KlaasIn 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.