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

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## 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 method with SSC smoothers and multilevel strategy for the subspaces is presented. The analysis is motivated by a desire to test the performances of the method for a wider range of settings than those addressed in Chapter II. While a symmetric smoothing procedure is assumed in the theoretical convergence analysis in Chapter II, non-symmetric smoothers may be also used for practical applications, since they require fewer operations. The subspace solver assumed in the theoretical analysis in Chapter II was an unpreconditoned Richardson’s method, however, preconditioners such as Jacobi or Incomplete LU (ILU) factorization are often used in practice. In the computational analysis of Chapter III, we consider symmetric and nonsymmetric smoothing procedures as well as several types of preconditioners for the subspace solver. With the methods studied in this dissertation, the smoothing process is carried out on the entire multigrid space. We refer to this situation as global smoothing. When hanging nodes are present, the usual approach in the literature consists of smoothing only on a subspace of the multigrid space, that does not contain hanging nodes. We refer to this second situation as local smoothing. In Chapter III, we compare the global smoothing convergence results of the proposed method with local smoothing results obtained with existing strategies. Global smoothing provides better convergence properties, especially when the solution of the underlying PDE lacks regularity.