Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs
In this thesis we study so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear partial differential equations. The two-grid method is constructed by first solving the nonlinear system of equations stemming from the discontinuous Galerkin...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2014
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| Online Access: | https://eprints.nottingham.ac.uk/13944/ |
| _version_ | 1848791842285420544 |
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| author | Congreve, Scott |
| author_facet | Congreve, Scott |
| author_sort | Congreve, Scott |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | In this thesis we study so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear partial differential equations. The two-grid method is constructed by first solving the nonlinear system of equations stemming from the discontinuous Galerkin finite element method on a coarse mesh partition; then, this coarse solution is used to linearise the underlying problem so that only a linear system is solved on a finer mesh. Solving the complex nonlinear problem on a coarse enough mesh should reduce computational complexity without adversely affecting the numerical error.
We first focus on the a priori and a posteriori error estimation for a scalar second-order quasilinear elliptic PDEs of strongly monotone type with respect to a mesh-dependent energy norm. We then devise an hp-adaptive mesh refinement algorithm, using the a posteriori error estimator, to automatically refine both the coarse and fine meshes present in the two-grid method. We then perform numerical experiments to validate the algorithm and demonstrate the improvements from utilising a two-grid method in comparison to a standard (single-grid) approach.
We also consider deviation of the energy norm based a priori and a posteriori error bounds for both the standard and two-grid discretisations of a quasi-Newtonian fluid flow problem of strongly monotone type. Numerical experiments are performed to validate these bounds. We finally consider the dual weighted residual based a posteriori error estimate for both the second-order quasilinear elliptic PDE and the quasi-Newtonian fluid flow problem with generic nonlinearities. |
| first_indexed | 2025-11-14T18:34:56Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-13944 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T18:34:56Z |
| publishDate | 2014 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-139442025-02-28T11:27:53Z https://eprints.nottingham.ac.uk/13944/ Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs Congreve, Scott In this thesis we study so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear partial differential equations. The two-grid method is constructed by first solving the nonlinear system of equations stemming from the discontinuous Galerkin finite element method on a coarse mesh partition; then, this coarse solution is used to linearise the underlying problem so that only a linear system is solved on a finer mesh. Solving the complex nonlinear problem on a coarse enough mesh should reduce computational complexity without adversely affecting the numerical error. We first focus on the a priori and a posteriori error estimation for a scalar second-order quasilinear elliptic PDEs of strongly monotone type with respect to a mesh-dependent energy norm. We then devise an hp-adaptive mesh refinement algorithm, using the a posteriori error estimator, to automatically refine both the coarse and fine meshes present in the two-grid method. We then perform numerical experiments to validate the algorithm and demonstrate the improvements from utilising a two-grid method in comparison to a standard (single-grid) approach. We also consider deviation of the energy norm based a priori and a posteriori error bounds for both the standard and two-grid discretisations of a quasi-Newtonian fluid flow problem of strongly monotone type. Numerical experiments are performed to validate these bounds. We finally consider the dual weighted residual based a posteriori error estimate for both the second-order quasilinear elliptic PDE and the quasi-Newtonian fluid flow problem with generic nonlinearities. 2014-07-10 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/13944/1/Congreve_PhD.pdf Congreve, Scott (2014) Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs. PhD thesis, University of Nottingham. hp-adaptivity a priori a posteriori non-Newtonian fluids discontinuous Galerkin fine element methods two-grid dual weighted residual |
| spellingShingle | hp-adaptivity a priori a posteriori non-Newtonian fluids discontinuous Galerkin fine element methods two-grid dual weighted residual Congreve, Scott Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs |
| title | Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs |
| title_full | Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs |
| title_fullStr | Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs |
| title_full_unstemmed | Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs |
| title_short | Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs |
| title_sort | two-grid hp-version discontinuous galerkin finite element methods for quasilinear pdes |
| topic | hp-adaptivity a priori a posteriori non-Newtonian fluids discontinuous Galerkin fine element methods two-grid dual weighted residual |
| url | https://eprints.nottingham.ac.uk/13944/ |