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A posteriori error estimates for the Poisson problem on closed, two-dimensional surfaces

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dc.contributor.authorNewton, William F., author
dc.contributor.authorEstep, Donald J., 1959-, advisor
dc.contributor.authorHolst, Michael J., committee member
dc.contributor.authorTavener, Simon, committee member
dc.contributor.authorZhou, Yongcheng, committee member
dc.contributor.authorBreidt, F. Jay, committee member
dc.date.accessioned2007-01-03T04:58:13Z
dc.date.available2007-01-03T04:58:13Z
dc.date.issued2011
dc.description.abstractThe solution of partial differential equations on non-Euclidean Domains is an area of much research in recent years. The Poisson Problem is a partial differential equation that is useful on curved surfaces. On a curved surface, the Poisson Problem features the Laplace-Beltrami Operator, which is a generalization of the Laplacian and specific to the surface where the problem is being solved. A Finite Element Method for solving the Poisson Problem on a closed surface has been described and shown to converge with order h2. Here, we review this finite element method and the background material necessary for defining it. We then construct an adjoint-based a posteriori error estimate for the problem, discuss some computational issues that arise in solving the problem and show some numerical examples. The major sources of numerical error when solving the Poisson problem are geometric error, discretization error, quadrature error and measurement error. Geometric error occurs when distances, areas and angles are distorted by using a flat domain to parametrize a curved one. Discretization error is a result of using a finite-dimensional space of functions to approximate an infinite-dimensional space. Quadrature error arises when we use numerical quadrature to evaluate integrals necessary for the finite element method. Measurement error arises from error and uncertainty in our knowledge of the surface itself. We are able to estimate the amount of each of these types of error and show when each type of error will be significant.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierNewton_colostate_0053A_10662.pdf
dc.identifier.urihttp://hdl.handle.net/10217/46379
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjecta posteriori
dc.subjectPoisson Problem
dc.subjectLaplace-Beltrami
dc.subjecterror quantification
dc.titleA posteriori error estimates for the Poisson problem on closed, two-dimensional surfaces
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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