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Browsing Department of Mathematics by Subject "a posteriori"

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    A posteriori error estimates for the Poisson problem on closed, two-dimensional surfaces
    (Colorado State University. Libraries, 2011) Newton, William F., author; Estep, Donald J., 1959-, advisor; Holst, Michael J., committee member; Tavener, Simon, committee member; Zhou, Yongcheng, committee member; Breidt, F. Jay, committee member
    The 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.