@PhDThesis{4529/THESES,
   address     = {Lausanne},
   affiliation = {EPFL},
   author      = {Pena, Gon\c{c}alo},
   details     = {http://infoscience.epfl.ch/record/141938},
   oai-id      = {oai:infoscience.epfl.ch:thesis-4529},
   oai-set     = {thesis},
   publisher   = {EPFL},
  abstract={In this thesis we address the numerical approximation of the incompressible Navier-Stokes equations evolving in a moving domain with the spectral element method and high order time integrators. First, we present the spectral element method and the basic tools to perform spectral discretizations of the Galerkin or Galerkin with Numerical Integration (G-NI) type. We cover a large range of possibilities regarding the reference elements, basis functions, interpolation points and quadrature points. In this approach, the integration and differentiation of the polynomial functions is done numerically through the help of suitable point sets. Regarding the differentiation, we present a detailed numerical study of which points should be used to attain better stability (among the choices we present). Second, we introduce the incompressible steady/unsteady Stokes and Navier-Stokes equations and their spectral approximation. In the unsteady case, we introduce a combination of Backward Differentiation Formulas and an extrapolation formula of the same order for the time integration. Once the equations are discretized, a linear system must be solved to obtain the approximate solution. In this context, we consider the solution of the whole system of equations combined with a block type preconditioner. The preconditioner is shown to be optimal in terms of number of iterations used by the GMRES method in the steady case, but not in the unsteady one. Another alternative presented is to use algebraic factorization methods of the Yosida type and decouple the calculation of velocity and pressure. A benchmark is also presented to access the numerical convergence properties of this type of methods in our context. Third, we extend the algorithms developed in the fixed domain case to the Arbitrary Lagrangian Eulerian framework. The issue of defining a high order ALE map is addressed. This allows to construct a computational domain that is described with curved elements. A benchmark using a direct method to solve the linear system or the \yosida{q} methods is presented to show the convergence orders of the method proposed. Finally, we apply the developed method with an implicit fully coupled and semi-implicit approach, to solve a fluid-structure interaction problem for a simple 2D hemodynamics example.},
   thesis-note = {Th\`{e}se Ecole polytechnique f\'{e}d\'{e}rale de Lausanne EPFL, no
                 4529 (2009), Programme doctoral Math\'{e}matiques, Facult\'{e}
                 des sciences de base SB, Institut d'analyse et calcul
                 scientifique IACS (Chaire de mod\'{e}lisation et calcul
                 scientifique CMCS). Dir.: Alfio Quarteroni},
   title       = {Spectral element approximation of the incompressible
                 {N}avier-{S}tokes equations in a moving domain and
                 applications},
   unit        = {CMCS},
   url         = {http://library.epfl.ch/theses/?nr=4529},
   year        = 2009
}

@PHDTHESIS{Stamm2008,
  author = {Stamm, Benjamin},
  title = {Stabilization strategies for discontinuous {G}alerkin methods},
  year = {2008},
  address = {Lausanne},
  affiliation = {EPFL},
  details = {http://infoscience.epfl.ch/record/124839},
  oai-id = {oai:infoscience.epfl.ch:thesis-4135},
  oai-set = {thesis},
  publisher = {EPFL},
  status = {SUBMITTED},
  thesis-note = {Thèse Ecole polytechnique fédérale de Lausanne EPFL, no 4135 (2008),
	Faculté des sciences de base SB, Programme doctoral Mathématiques,
	Institut d'analyse et calcul scientifique IACS (Chaire de modélisation
	et calcul scientifique CMCS). Dir.: Alfio Quarteroni, Erik Burman},
  unit = {CMCS},
  url = {http://library.epfl.ch/theses/?nr=4135}
}

@PHDTHESIS{Winkelmann2007,
  author = {Winkelmann, Christoph},
  title = {Interior penalty finite element approximation of {N}avier-{S}tokes
	equations and application to free surface flows},
  year = {2007},
  address = {Lausanne},
  abstract={In the present work, we investigate mathematical and numerical aspects of interior penalty finite element methods for free surface flows. We consider the incompressible Navier-Stokes equations with variable density and viscosity, combined with a front capturing model using the level set method. We formulate interior penalty finite element methods for both the Navier-Stokes equations and the level set advection equation. For the two-fluid Stokes equations, we propose and analyze an unfitted finite element scheme with interior penalty. Optimal a priori error estimates for the velocity and the pressure are proved in the energy norm. A preconditioning strategy with adaptive reuse of incomplete factorizations as preconditioners for Krylov subspace methods is introduced and applied for solving the linear systems. Different and complementary solutions for reducing the matrix assembly time and the memory consumption are proposed and tested, each of which is applicable in general in the context of either multiphase flow or interior penalty stabilization. As level set reinitialization method, we apply a combination of the interface local projection and a fast marching scheme. We provide for the latter a reformulation of the distance computation algorithm on unstructured simplicial meshes in any spatial dimension, allowing for both an efficient implementation and geometric insight. We present and discuss numerical solutions of reference problems for the one-fluid Navier-Stokes equations and for the level set advection problem. Solutions of benchmark problems in two and three dimensions involving one or two fluids are then approximated, and the results are compared to literature values. Finally, we describe software design techniques and abstractions for the efficient and general implementation of the applied methods.},
  affiliation = {EPFL},
  details = {http://infoscience.epfl.ch/record/112720},
  documenturl = {http://library.epfl.ch/theses/?nr=3971},
  keywords = {Navier-Stokes equations; free surface; finite elements; interior penalty
	stabilization; unfitted elements; front capturing; level set method;
	reinitialization; preconditioning},
  oai-id = {oai:infoscience.epfl.ch:thesis-3971},
  oai-set = {thesis; thesis:fulltext; fulltext},
  pagecount = {147 p.},
  publisher = {EPFL},
  status = {PUBLISHED},
  thesis-note = {Thèse sciences Ecole polytechnique fédérale de Lausanne EPFL, no 3971
	(2007), Faculté des sciences de base SB, Section de mathématiques,
	Programme doctoral Mathématiques, Institut d'analyse et calcul scientifique
	IACS (Chaire de modélisation et calcul scientifique CMCS). Dir.:
	Alfio Quarteroni},
  unit = {CMCS},
  url = {http://library.epfl.ch/theses/?nr=3971}
}