scholarly journals COUPLING HETEROGENEOUS MULTISCALE FEM WITH RUNGE–KUTTA METHODS FOR PARABOLIC HOMOGENIZATION PROBLEMS: A FULLY DISCRETE SPACETIME ANALYSIS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250002 ◽  
Author(s):  
ASSYR ABDULLE ◽  
GILLES VILMART
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Convergence Rates ◽  
Implicit Methods ◽  
Fully Discrete ◽  
Discrete Analysis ◽  
Discrete Spacetime ◽  
Runge Kutta ◽  
Heterogeneous Multiscale ◽  
Homogenization Problems

Numerical methods for parabolic homogenization problems combining finite element methods (FEMs) in space with Runge–Kutta methods in time are proposed. The space discretization is based on the coupling of macro and micro finite element methods following the framework of the Heterogeneous Multiscale Method (HMM). We present a fully discrete analysis in both space and time. Our analysis relies on new (optimal) error bounds in the norms L2(H1), [Formula: see text], and [Formula: see text] for the fully discrete analysis in space. These bounds can then be used to derive fully discrete spacetime error estimates for a variety of Runge–Kutta methods, including implicit methods (e.g. Radau methods) and explicit stabilized method (e.g. Chebyshev methods). Numerical experiments confirm our theoretical convergence rates and illustrate the performance of the methods.

Semi-Discrete and Fully Discrete Hybrid Stress Finite Element Methods for Elastodynamic Problems

2015 ◽  
Vol 8 (4) ◽  
pp. 582-604
Author(s):  
Zhengqin Yu ◽  
Xiaoping Xie
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Stability Result ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Quadrilateral Finite Element ◽  
Hybrid Stress Finite Element ◽  
Hybrid Stress

AbstractThis paper proposes and analyzes semi-discrete and fully discrete hybrid stress finite element methods for elastodynamic problems. A hybrid stress quadrilateral finite element approximation is used in the space directions. A second-order center difference is adopted in the time direction for the fully discrete scheme. Error estimates of the two schemes, as well as a stability result for the fully discrete scheme, are derived. Numerical experiments are done to verify the theoretical results.

scholarly journals Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

2019 ◽  
Vol 65 (3) ◽  
pp. 807-838 ◽  
Author(s):  
F. de Prenter ◽  
C. V. Verhoosel ◽  
E. H. van Brummelen ◽  
J. A. Evans ◽  
C. Messe ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Isogeometric Analysis ◽  
Convergence Rates ◽  
Computational Cost ◽  
Iterative Solvers ◽  
System Matrix ◽  
B Splines ◽  
Immersed Finite Element ◽  
Immersed Finite Element Methods

AbstractIll-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

scholarly journals Convergence Analysis of H(div)-Conforming Finite Element Methods for a Nonlinear Poroelasticity Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yuping Zeng ◽  
Zhifeng Weng ◽  
Fen Liang
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
A Priori ◽  
Mixed Finite Elements ◽  
Elastic Displacement ◽  
Fully Discrete ◽  
Conforming Finite Element ◽  
Priori Error Estimates ◽  
Galerkin Formulation ◽  
Flow Variables

In this paper, we introduce and analyze H(div)-conforming finite element methods for a nonlinear model in poroelasticity. More precisely, the flow variables are discretized by H(div)-conforming mixed finite elements, while the elastic displacement is approximated by the H(div)-conforming finite element with the interior penalty discontinuous Galerkin formulation. Optimal a priori error estimates are derived for both semidiscrete and fully discrete schemes.

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