Exact Smooth Piecewise Polynomial Sequences on Powell-Sabin and Worsey-Farin Splits

Lischke, Anna

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Year:: 2020
Contributor:: Lischke, Anna (creator); Guzman, Johnny (Advisor); Neilan, Michael (Reader); Shu, Chi-Wang (Reader); Brown University. Department of Applied Mathematics (sponsor)
Genre:: theses
Subject:: Scientific Computation; Numerical analysis; Finite element method; Algebra, Homological
Extent:: xi, 163 p.

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Abstract:: The problem of forming exact sequences of finite element spaces that discretize Hilbert complexes is central to the finite element exterior calculus. This framework offers a way of developing and analyzing sequences of finite element spaces that lead to stable discretizations of associated mixed-formulation PDEs, and it has been successfully applied to the de Rham complex with minimal $L^2$ smoothness. In this work, we seek to extend this result to sequences that include smoother spaces in order to develop stable finite element methods for higher order PDEs. The problem of forming exact sequences of finite element spaces that discretize these smoother sequences on general triangulations is yet an open problem. In our approach, we consider a single geometric refinement of a general triangulation on which we are able to solve the problem; in two dimensions, we use the Powell-Sabin split, and in three dimensions, we use the Worsey-Farin split. We prove exactness of (local) sequences of smoother polynomial spaces on these splits and develop commuting projections via degrees of freedom for each space. Furthermore, we demonstrate that these degrees of freedom induce global spaces that also form exact sequences. The finite element spaces proposed in this work may be applied within finite element solvers for a broad class of higher order mixed-formulation PDEs.
Notes:: Thesis (Ph. D.)--Brown University, 2020

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Lischke, Anna, "Exact Smooth Piecewise Polynomial Sequences on Powell-Sabin and Worsey-Farin Splits" (2020). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:7d652s82/

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Applied Mathematics Theses and Dissertations

Theses and Dissertations for the Applied Mathematics department.
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