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Capodaglio, Giacomo.
Multigrid methods for finite element applications with arbitrary-level *hanging* node configurations.

Degree: PhD, Mathematics, 2018, Texas Tech University

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

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…
*Advisors/Committee Members: Bornia, Giorgio (committee member), Heister, Timo (committee member), Howle, Victoria (committee member), Parameswaran, Siva (committee member), Aulisa, Eugenio (Committee Chair).*

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

…May 2018
LIST OF FIGURES
1.1
An irregular grid in 2D with *hanging* *nodes*… …special *nodes* called *hanging* *nodes* are introduced [23], and the finite
element… …degrees of freedom are associated with the *hanging*
*nodes*. Our analysis fits in the framework of… …constrained approximation, where no
degrees of freedom are associated with the *hanging* *nodes*. We… …that, on the contrary, assign degrees of freedom to the *hanging* *nodes*. In
constrained…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Capodaglio, G. (2018). Multigrid methods for finite element applications with arbitrary-level hanging node configurations. (Doctoral Dissertation). Texas Tech University. Retrieved from http://hdl.handle.net/2346/73836

Chicago Manual of Style (16^{th} Edition):

Capodaglio, Giacomo. “Multigrid methods for finite element applications with arbitrary-level hanging node configurations.” 2018. Doctoral Dissertation, Texas Tech University. Accessed October 26, 2020. http://hdl.handle.net/2346/73836.

MLA Handbook (7^{th} Edition):

Capodaglio, Giacomo. “Multigrid methods for finite element applications with arbitrary-level hanging node configurations.” 2018. Web. 26 Oct 2020.

Vancouver:

Capodaglio G. Multigrid methods for finite element applications with arbitrary-level hanging node configurations. [Internet] [Doctoral dissertation]. Texas Tech University; 2018. [cited 2020 Oct 26]. Available from: http://hdl.handle.net/2346/73836.

Council of Science Editors:

Capodaglio G. Multigrid methods for finite element applications with arbitrary-level hanging node configurations. [Doctoral Dissertation]. Texas Tech University; 2018. Available from: http://hdl.handle.net/2346/73836

University of Texas – Austin

2. -1155-8213. Addressing challenges in modeling of coupled flow and poromechanics in deep subsurface reservoirs.

Degree: PhD, Engineering Mechanics, 2018, University of Texas – Austin

URL: http://dx.doi.org/10.26153/tsw/2120

In coupled flow and poromechanics phenomena representing hydrocarbon production or CO₂ sequestration in deep subsurface non-fractured reservoirs, the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs. The vertical extent of the poromechanical domain can be two orders of magnitude more than the characteristic thickness of the flow domain (reservoir). The lateral extent of the poromechanical domain should also be allowed to be substantially larger than that of the flow domain to enable the imposition of far-field boundary conditions on the poromechanical domain. The typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock. The former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs. The flow domain requires an areal resolution fine enough to be able to capture the underlying nonlinearities in the multiphase flow equations. If the same grid resolution is employed for the poromechanical domain, the simulator would crash for lack of memory and computing resource. With that in mind, it is imperative to establish a framework in which fluid flow is resolved on a finer grid and poromechanical deformation is resolved on a coarse grid. In addition, the geometry of the flow domain necessitates the use of non-nested grids which allows for freedom of choice of the poromechanical grid resolution. Furthermore, to achieve the goal of rendering realistic simulations of subsurface phenomena, we cannot ignore the heterogeneity in flow and poromechanical properties, as well as the lack in accuracy of the poromechanical calculations if the grid for the poromechanics domain is too coarse. This dissertation is a rendition of how we invoke concepts in computational geometry, parallel computing, applied mathematics and convex optimization in designing and implementing algorithms that tackle all the aforementioned challenges.
*Advisors/Committee Members: Wheeler, Mary F. (Mary Fanett) (advisor), Landis, Chad (committee member), Huang, Rui (committee member), Balhoff, Matthew (committee member).*

Subjects/Keywords: Fixed-stress split iterative scheme; Overlapping nonmatching hexahedral grids; Upscaling and downscaling; Singular value decompositions; Surface intersections; Delaunay triangulations; Mandel’s problem; Biot system; Heterogeneous poroelastic medium; Nested two-grid approach; Contraction mapping; Anisotropic poroelastic medium; Computational homogenization; Hanging nodes; Augmented Lagrangian

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

-1155-8213. (2018). Addressing challenges in modeling of coupled flow and poromechanics in deep subsurface reservoirs. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://dx.doi.org/10.26153/tsw/2120

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

Chicago Manual of Style (16^{th} Edition):

-1155-8213. “Addressing challenges in modeling of coupled flow and poromechanics in deep subsurface reservoirs.” 2018. Doctoral Dissertation, University of Texas – Austin. Accessed October 26, 2020. http://dx.doi.org/10.26153/tsw/2120.

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

MLA Handbook (7^{th} Edition):

-1155-8213. “Addressing challenges in modeling of coupled flow and poromechanics in deep subsurface reservoirs.” 2018. Web. 26 Oct 2020.

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

Vancouver:

-1155-8213. Addressing challenges in modeling of coupled flow and poromechanics in deep subsurface reservoirs. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2018. [cited 2020 Oct 26]. Available from: http://dx.doi.org/10.26153/tsw/2120.

Author name may be incomplete

Council of Science Editors:

-1155-8213. Addressing challenges in modeling of coupled flow and poromechanics in deep subsurface reservoirs. [Doctoral Dissertation]. University of Texas – Austin; 2018. Available from: http://dx.doi.org/10.26153/tsw/2120

Author name may be incomplete