Browsing by Subject "Finite element exterior calculus"
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Item Direct serendipity and mixed finite elements on polygons and cuboidal hexahedra(2023-07-25) Wang, Chuning; Arbogast, Todd James, 1957-; Demkowicz, Leszek; Martinsson, Per-Gunnar; Tsai, Yen-HsiIn recent years, there has been a lot of interest in defining finite elements on nonstandard, polytopal elements, i.e., on polygons and polyhedra. Inspired by Arnold and Awanou's serendipity family of finite elements on cuboidal meshes, the dissertation generalizes the construction of direct serendipity and mixed finite elements to general planar, strictly convex polygons. This work also gives an explicit construction of direct serendipity finite elements on convex cuboidal hexahedra. Direct serendipity and mixed finite elements are H1 and H(div) conforming, respectively, and possess optimal order of accuracy for any order. They have a minimal number of degrees of freedom subject to conformity and accuracy constraints. The finite element shape functions are defined to be the full spaces of scalar or vector polynomials plus a space of supplemental functions, of which the choice is not unique. The direct serendipity elements are the precursors of the direct mixed elements in a de Rham complex. The convergence properties of these new families of finite elements are shown under the assumption of shape regularity, as well as some mild restrictions on the choice of supplemental functions. Numerical experiments on various meshes exhibit their performance, especially their potential value in applications where flexible polytopal meshes are advantageous.Item Stability of dual discretization methods for partial differential equations(2011-05) Gillette, Andrew Kruse; Bajaj, Chandrajit; Demkowicz, Leszek; Gonzalez, Oscar; Luecke, John; Reid, Alan; Vick, JamesThis thesis studies the approximation of solutions to partial differential equations (PDEs) over domains discretized by the dual of a simplicial mesh. While `primal' methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of a simplicial mesh (simplex vertices, edges, faces, etc.), a `dual discretization method' associates DoFs with the geometric duals of these objects. In a tetrahedral mesh, for instance, a primal method might assign DoFs to edges of tetrahedra while a dual method for the same problem would assign DoFs to edges connecting circumcenters of adjacent tetrahedra. Dual discretization methods have been proposed for various specific PDE problems, especially in the context of electromagnetics, but have not been analyzed using the full toolkit of modern numerical analysis as is considered here. The recent and still-developing theories of finite element exterior calculus (FEEC) and discrete exterior calculus (DEC) are shown to be essential in understanding the feasibility of dual methods. These theories treat the solutions of continuous PDEs as differential forms which are then discretized as cochains (vectors of DoFs) over a mesh. While the language of DEC is ideal for describing dual methods in a straightforward fashion, the results of FEEC are required for proving convergence results. Our results about dual methods are focused on two types of stability associated with PDE solvers: discretization and numerical. Discretization stability analyzes the convergence of the approximate solution from the discrete method to the continuous solution of the PDE as the maximum size of a mesh element goes to zero. Numerical stability analyzes the potential roundoff errors accrued when computing an approximate solution. We show that dual methods can attain the same approximation power with regard to discretization stability as primal methods and may, in some circumstances, offer improved numerical stability properties. A lengthier exposition of the approach and a detailed description of our results is given in the first chapter of the thesis.