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 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

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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…