Full Record

Author | Capodaglio, Giacomo |

Title | Multigrid methods for finite element applications with arbitrary-level hanging node configurations |

URL | http://hdl.handle.net/2346/73836 |

Publication Date | 2018 |

Date Accessioned | 2018-06-01 20:57:20 |

Degree | PhD |

Discipline/Department | Mathematics |

Degree Level | doctoral |

University/Publisher | Texas Tech University |

Abstract | In this dissertation, multigrid methods for finite element applications with arbitrary-level hanging nodes are considered. When a local midpoint refinement procedure is carried out on the finite element grid, hanging nodes are introduced. The presence of hanging nodes complicates the way the problem has to be addressed for several reasons. For instance, if a continuous finite element solution is sought, extra effort has to be made to enforce continuity. In this work, we propose two different strategies to achieve the desired continuity. Chapter I lays out the first strategy, which relies on the introduction of modified basis functions that are continuous by construction. Finite element spaces are the defined as the spanning sets of these modified basis functions, and the continuity of the finite element solution immediately follows. A detailed computational analysis is presented, where a multigrid algorithm defined on the continuous finite element spaces is used either as a solver, or as a preconditioner for other iterative solvers. Specifically, the conjugate gradient (CG) and the generalized minimal residual (GMRES) will be considered. The numerical results aim to investigate the convergence properties of the multigrid algorithm proposed in this chapter. In Chapter II, a theoretical analysis of multigrid algorithms with successive subspace correction (SSC) smoothers is presented. Here, we obtain convergence estimates under no regularity assumptions on the solution of the underlying partial differential equation (PDE), highlighting a dependence of the convergence bound on the number of smoothing iterations. In this framework, the second strategy to enforce continuity is described. Such a strategy relies on a particular choice of subspaces for the SSC smoother, made according to a multilevel approach that exploits the multigrid hierarchy. Continuity is recovered by decomposing functions on the finite element spaces at finer levels as linear combinations of continuous functions at coarser levels. In this context, the introduction of modified basis functions is not necessary. On the other hand, this second strategy is tied to the multigrid method, since it relies on the multigrid hierarchy and on the SSC smoother. It is important to note that, once continuous finite element spaces are obtained with the approach in Chapter I, a multigrid solver with SSC smoother can be defined also on such spaces. In this case, the choice of subspaces for the space decomposition should be made according to a domain decomposition strategy rather than a multilevel strategy, since continuity is already guaranteed by the modified basis functions, so exploiting the multigrid hierarchy is not necessary. Both the multilevel approach and the domain decomposition approach for the choice of subspaces in the SSC smoother are investigated theoretically in Chapter II. The chapter is concluded with numerical results that compare the convergence performances of the two approaches. In Chapter III, a thorough computational analysis of a multigrid… |

Subjects/Keywords | Multigrid; Finite Element Method; Hanging Nodes; Local Refinement; Iterative Methods; Successive Subspace Correction |

Contributors | Bornia, Giorgio (committee member); Heister, Timo (committee member); Howle, Victoria (committee member); Parameswaran, Siva (committee member); Aulisa, Eugenio (Committee Chair) |

Language | en |

Rights | Unrestricted. |

Country of Publication | us |

Record ID | handle:2346/73836 |

Repository | ttu |

Date Retrieved | 2018-12-03 |

Date Indexed | 2018-12-06 |

Grantor | Texas Tech University |

Sample Search Hits | Sample Images | Cited Works

…*Texas* *Tech* *University*, Giacomo Capodaglio, May 2018
results obtained with existing strategies. Global smoothing provides better convergence properties, especially when the solution of the underlying PDE lacks regularity.
vii
*Texas* *Tech* *University*…

…subspace solver. . . . . . . . .
viii
93
*Texas* *Tech* *University*, Giacomo Capodaglio, May 2018
III.3 Spectral radius of EJ for the 3-dimensional L-shaped geometry, with
different type of preconditioners for the subspace solver. . . . . . . . .
96
III.4…

…equation with discontinuous coefficients. . . . . . . . . . . . . . . . . 105
ix
*Texas* *Tech* *University*, Giacomo Capodaglio, May 2018
LIST OF FIGURES
1.1
An irregular grid in 2D with hanging nodes. . . . . . . . . . . . . . .
1.2
Example of a two…

…dimensional driven-cavity flow, with ν = 0.001. . . . . . . . . . .
40
1.17 Three-dimensional buoyancy driven flow with P r = 1 and Ra = 10000.
44
x
*Texas* *Tech* *University*, Giacomo Capodaglio, May 2018
2.1
Example of a subdomains involved in the uniform…

…corresponds to wedge elements, dark gray to tetrahedra and gray
to hexahedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
xi
*Texas* *Tech* *University*, Giacomo Capodaglio, May 2018
CHAPTER I
CONSTRUCTION OF CONTINUOUS BASIS FUNCTIONS FOR…

…freedom are associated with the hanging nodes. We refer to [35, 45] for
1
*Texas* *Tech* *University*, Giacomo Capodaglio, May 2018
works that, on the contrary, assign degrees of freedom to the hanging nodes. In
constrained approximation, most…

…multigrid methods. The
2
*Texas* *Tech* *University*, Giacomo Capodaglio, May 2018
continuity of our finite element spaces allows the multigrid smoothing to be performed
on all degrees of freedom. Global smoothing guarantees an arbitrary improvement in
the…

…elements of Tk−1 that
lie on Θk .
We next define new sequences of subsets of Ω. These are introduced to set our formulation in a multilevel framework, in order to easily apply our analysis to multigrid
methods.
3
*Texas* *Tech* *University*, Giacomo Capodaglio…