Discontinuous Galerkin Methods for Elliptic Partial Differential Equations with Random Coefficients

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dc.contributor.advisorRiviere, Beatrice M.en_US
dc.creatorLiu, Kunen_US
dc.date.accessioned2013-03-08T00:36:03Zen_US
dc.date.available2013-03-08T00:36:03Zen_US
dc.date.issued2011en_US
dc.description.abstractThis thesis proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients. The stochastic problem is first transformed into a parametrized one by the use of the Karhunen--Loève expansion. This new problem is then discretized by the discontinuous Galerkin (DG) method. A priori error estimate in the energy norm for the stochastic discontinuous Galerkin solution is derived. In addition, the expected value of the numerical error is theoretically bounded in the energy norm and the L2 norm. In the second approach, the Monte Carlo method is used to generate independent identically distributed realizations of the stochastic coefficients. The resulting deterministic problems are solved by the DG method. Next, estimates are obtained for the error between the average of these approximate solutions and the expected value of the exact solution. The Monte Carlo discontinuous Galerkin method is tested numerically on several examples. Results show that the nonsymmetric DG method is stable independently of meshes and the value of penalty parameter. Symmetric and incomplete DG methods are stable only when the penalty parameter is large enough. Finally, comparisons with the Monte Carlo finite element method and the Monte Carlo discontinuous Galerkin method are presented for several cases.en_US
dc.format.extent95 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH.SCI. 2011 LIUen_US
dc.identifier.citationLiu, Kun. "Discontinuous Galerkin Methods for Elliptic Partial Differential Equations with Random Coefficients." (2011) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/70327">https://hdl.handle.net/1911/70327</a>.en_US
dc.identifier.digitalLiuKunen_US
dc.identifier.urihttps://hdl.handle.net/1911/70327en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectApplied sciencesen_US
dc.subjectApplied mathematicsen_US
dc.titleDiscontinuous Galerkin Methods for Elliptic Partial Differential Equations with Random Coefficientsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematical Sciencesen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelMastersen_US
thesis.degree.nameMaster of Artsen_US
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