| dc.identifier.uri | http://hdl.handle.net/11401/77077 | |
| dc.description.sponsorship | This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree. | en_US |
| dc.format | Monograph | |
| dc.format.medium | Electronic Resource | en_US |
| dc.language.iso | en_US | |
| dc.publisher | The Graduate School, Stony Brook University: Stony Brook, NY. | |
| dc.type | Dissertation | |
| dcterms.abstract | The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, especially on complex geometries. One limitation of the classical FEM is its heavy dependence on element shape quality for stability and good performance. We introduce the Adaptive Extended Stencil Finite Element Method (AES-FEM) as a means to overcome this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial (GLP) basis functions, which we construct using local weighted least-squares approximations. The method preserves the theoretical framework of FEM, and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In this dissertation, we describe the formulation and implementation of AES- FEM with quadratic basis functions, and analyze its consistency and stability. Next, we present an extension to high-order AES-FEM, including analysis and implementation details. High-order AES-FEM uses meshes with linear elements, thus avoiding the challenges of isoparametric elements. We present numerical experiments in both 2D and 3D for the Poisson equation and a time-independent convection-diffusion equation, including results on curved boundaries. We demonstrate high-order convergence up to sixth order of accuracy. Since AES-FEM results in a non-symmetric stiffness matrix, we compare the timing results of several combinations of linear solvers and preconditioners. The numerical results demonstrate that high-order AES-FEM is more accurate than high-order FEM, is also more efficient than FEM in terms of error versus run-time on finer meshes, and enables much better stability and faster convergence of iterative solvers than FEM over poor-quality meshes. | |
| dcterms.available | 2017-09-20T16:51:52Z | |
| dcterms.contributor | Glimm, James | en_US |
| dcterms.contributor | Jiao, Xiangmin | en_US |
| dcterms.contributor | Samulyak, Roman | en_US |
| dcterms.contributor | Gu, Xianfeng | en_US |
| dcterms.contributor | Bar-Yoseph, Zvi. | en_US |
| dcterms.creator | Conley, Rebecca | |
| dcterms.dateAccepted | 2017-09-20T16:51:52Z | |
| dcterms.dateSubmitted | 2017-09-20T16:51:52Z | |
| dcterms.description | Department of Applied Mathematics and Statistics | en_US |
| dcterms.extent | 129 pg. | en_US |
| dcterms.format | Application/PDF | en_US |
| dcterms.format | Monograph | |
| dcterms.identifier | http://hdl.handle.net/11401/77077 | |
| dcterms.issued | 2016-12-01 | |
| dcterms.language | en_US | |
| dcterms.provenance | Made available in DSpace on 2017-09-20T16:51:52Z (GMT). No. of bitstreams: 1
Conley_grad.sunysb_0771E_12772.pdf: 3117422 bytes, checksum: 7d53b3b0c58f3668d772b832bc71fb8a (MD5)
Previous issue date: 1 | en |
| dcterms.publisher | The Graduate School, Stony Brook University: Stony Brook, NY. | |
| dcterms.subject | curved boundaries, finite element methods, generalized Lagrange polynomial basis, high-order accuracy, partial differential equations, stability | |
| dcterms.subject | Mathematics | |
| dcterms.title | Overcoming Element Quality Dependence of Finite Element Methods | |
| dcterms.type | Dissertation | |
