Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs

Congreve, Scott (2014) Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear PDEs. PhD thesis, University of Nottingham.

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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.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Houston, P.
Cliffe, K. A.
Keywords: hp-adaptivity, a priori, a posteriori, non-Newtonian fluids, discontinuous Galerkin fine element methods, two-grid, dual weighted residual
Subjects: Q Science > QA Mathematics > QA299 Analysis
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 13944
Depositing User: EP, Services
Date Deposited: 23 Jan 2015 13:03
Last Modified: 15 Dec 2017 05:55

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