Special Issue "Modern Finite Element Methods"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 June 2018).

Special Issue Editor

Prof. Dr. Olivier Pironneau
Website
Guest Editor
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France
Interests: scientific computing; finite element methods for fluids; turbulence modelling; optimal shape design; numerical methods for quantitative finance

Special Issue Information

Dear Colleagues,

The numerical approximation of partial differential systems of equations is a very lively field with the Finite Element Method (FEM) at its core, especially with the development of discontinuous Galerkin approximations and a-posteriori error estimations for mesh refinements. Multiscale problems require special finite element methods, such as Xfem, multiscale elements and mimetic methods. Large industrial applications lead also to research on 3-dimensional time dependent problems with uncertainties on the data, optimization and control. For these domain decomposition algorithm and level sets based methods are being investigated and moving mesh techniques, model coupling, sparse grids isoparametric high degree elements and isogeometric elements, to name a few. Finally, any tool which makes the computer implementation easier is a useful research as well; it covers high level dedicated languages like Fenics and FreeFem but also C++ toolboxes or others.

Prof. Dr. Olivier Pironneau
Guest Editor

Manuscript Submission Information

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Keywords

  • Innovative Finite Element Method
  • Galerkin discontuous methods
  • Mimetic methods with polygonal elements
  • Time dependent meshes and remeshing
  • Iso parametric high degree elements
  • Multiscale elements
  • Domain decomposition FEM
  • Sparse Grid FEM
  • Isogeometric elements
  • Level sets based FEM optimization
  • C++ FEM Toolbox

Published Papers (3 papers)

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Research

Open AccessArticle
High-Order Finite-Element Framework for the Efficient Simulation of Multifluid Flows
Mathematics 2018, 6(10), 203; https://doi.org/10.3390/math6100203 - 15 Oct 2018
Cited by 1
Abstract
In this paper, we present a comprehensive framework for the simulation of Multifluid flows based on the implicit level-set representation of interfaces and on an efficient solving strategy of the Navier-Stokes equations. The mathematical framework relies on a modular coupling approach between the [...] Read more.
In this paper, we present a comprehensive framework for the simulation of Multifluid flows based on the implicit level-set representation of interfaces and on an efficient solving strategy of the Navier-Stokes equations. The mathematical framework relies on a modular coupling approach between the level-set advection and the fluid equations. The space discretization is performed with possibly high-order stable finite elements while the time discretization features implicit Backward Differentation Formulae of arbitrary order. This framework has been implemented within the Feel++ library, and features seamless distributed parallelism with fast assembly procedures for the algebraic systems and efficient preconditioning strategies for their resolution. We also present simulation results for a three-dimensional Multifluid benchmark, and highlight the importance of using high-order finite elements for the level-set discretization for problems involving the geometry of the interfaces, such as the curvature or its derivatives. Full article
(This article belongs to the Special Issue Modern Finite Element Methods)
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Open AccessFeature PaperArticle
About Revisiting Domain Decomposition Methods for Poroelasticity
Mathematics 2018, 6(10), 187; https://doi.org/10.3390/math6100187 - 02 Oct 2018
Abstract
In this paper, we revisit well-established domain decomposition (DD) schemes to perform realistic simulations of coupled flow and poroelasticity problems on parallel computers. We define distinct solution schemes to take into account different transmission conditions among subdomain boundaries. Indeed, we examine two different [...] Read more.
In this paper, we revisit well-established domain decomposition (DD) schemes to perform realistic simulations of coupled flow and poroelasticity problems on parallel computers. We define distinct solution schemes to take into account different transmission conditions among subdomain boundaries. Indeed, we examine two different approaches, i.e., Dirichlet-Neumann (DN) and the mortar finite element method (MFEM), and we recognize their advantages and disadvantages. The MFEM significantly lessens the computational cost of reservoir compaction and subsidence calculations by dodging the conforming Cartesian grids that arise from the pay-zone onto its vicinity. There is a manifest necessity of producing non-matching interfaces between the reservoir and its neighborhood. We thus employ MFEM over nonuniform rational B-splines (NURBS) surfaces to stick these non-conforming subdomain parts. We then decouple the mortar saddle-point problem (SPP) using the Dirichlet-Neumann domain decomposition (DNDD) scheme. We confirm that this procedure is proper for calculations at the field level. We also carry comprehensive comparisons between the conventional and non-matching solutions to prove the method’s accuracy. Examples encompass linking finite element codes for slightly compressible single-phase and poroelasticity. We have used this program to a category of problems ranking from near-borehole applications to whole field subsidence estimations. Full article
(This article belongs to the Special Issue Modern Finite Element Methods)
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Open AccessArticle
Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow
Mathematics 2018, 6(5), 71; https://doi.org/10.3390/math6050071 - 03 May 2018
Cited by 3
Abstract
In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution [...] Read more.
In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b ( x ) , and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P 1 -discontinuous, P 2 -continuous and P 1 -continuous finite elements respectively. Full article
(This article belongs to the Special Issue Modern Finite Element Methods)
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