- Title Information
- Title
- Exact Smooth Piecewise Polynomial Sequences on Powell-Sabin and Worsey-Farin Splits
- Name:
Personal
- Name Part
- Lischke, Anna
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Guzman, Johnny
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Neilan, Michael
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Shu, Chi-Wang
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2020
- Physical Description
- Extent
- xi, 163 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2020
- Genre (aat)
- theses
- 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.
- Subject
- Topic
- Scientific Computation
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01041273")
- Topic
- Numerical analysis
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00924897")
- Topic
- Finite element method
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00804927")
- Topic
- Algebra, Homological
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20210607
- Type of Resource (primo)
- dissertations
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- All rights reserved. Collection is open to the Brown community for research.