scholarly journals A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations

2018 ◽  
Vol 40 (1) ◽  
pp. 322-357 ◽  
Author(s):  
David Mora ◽  
Gonzalo Rivera
Keyword(s):  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Optimal Order ◽  
Error Estimator ◽  
Computational Domain ◽  
A Posteriori ◽  
Order Error ◽  
A Posteriori Error ◽  
Virtual Element

AbstractWe present a priori and a posteriori error analyses of a virtual element method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyse a variational formulation relying only on the solid displacement and propose an $H^{1}(\Omega )$-conforming discretization by means of the VEM. Under standard assumptions on the computational domain, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal–order error estimate for the eigenfunctions and a double order for the eigenvalues. Since the VEM has the advantage of using general polygonal meshes, which allows efficient implementation of mesh refinement strategies, we also introduce a residual-type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests that allow us to assess the performance of this approach.

A Priori and a Posteriori Error Estimates for H(div)-Elliptic Problem with Interior Penalty Method

2013 ◽  
Vol 14 (3) ◽  
pp. 753-779
Author(s):  
Yuping Zeng ◽  
Jinru Chen
Keyword(s):  
Elliptic Problem ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Energy Norm ◽  
Error Estimator ◽  
A Posteriori ◽  
Interior Penalty ◽  
A Posteriori Error ◽  
A Priori Error Estimate

AbstractIn this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.

scholarly journals A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions

2016 ◽  
Vol 8 (2) ◽  
pp. 1
Author(s):  
Rola Ali Ahmad ◽  
Toufic El Arwadi ◽  
Houssam Chrayteh ◽  
Jean-Marc Sac-Epee
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Time Discretization ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Dynamical Boundary Conditions ◽  
Error Indicators ◽  
Crank Nicolson

In this article we claim that we are going to give a priori and a posteriori error estimates for a Crank Nicolson type scheme. The problem is discretized by the finite elements in space. The main result of this paper consists in establishing two types of error indicators, the first one linked to the time discretization and the second one to the space discretization.

scholarly journals A priori and a posteriori error analysis for a hybrid formulation of a prestressed shell model.

2021 ◽  
Author(s):  
Serge Nicaise ◽  
Ismail Merabet ◽  
Rayhana REZZAG BARA
Keyword(s):  
Shell Model ◽  
A Priori ◽  
Functional Space ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
A Priori Error Estimate

This work deals with the finite element approximation of a prestressed shell model using a new formulation where the unknowns (the displacement and the rotation of fibers normal to the midsurface) are described in Cartesian and local covariant basis respectively. Due to the constraint involved in the definition of the functional space, a penalized version is then considered. We obtain a non robust a priori error estimate of this penalized formulation, but a robust one is obtained for its mixed formulation. Moreover, we present a reliable and efficient a posteriori error estimator of the penalized formulation. Numerical tests are included that confirmthe efficiency of our residual a posteriori estimator.

scholarly journals Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

2019 ◽  
Vol 144 (3) ◽  
pp. 585-614
Author(s):  
Joscha Gedicke ◽  
Arbaz Khan
Keyword(s):  
Discontinuous Galerkin ◽  
Eigenvalue Problems ◽  
Convergence Rates ◽  
A Priori ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Posteriori Error Estimator ◽  
A Posteriori Error

AbstractIn this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient. We finally present some numerical examples that verify the a priori convergence rates and the reliability and efficiency of the residual based a posteriori error estimator.

scholarly journals Multigoal-oriented error estimates for non-linear problems

2019 ◽  
Vol 27 (4) ◽  
pp. 215-236 ◽  
Author(s):  
Bernhard Endtmayer ◽  
Ulrich Langer ◽  
Thomas Wick
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Nonlinear Problems ◽  
Posteriori Error ◽  
Stopping Rules ◽  
Nonlinear Pdes ◽  
Error Estimator ◽  
Mesh Adaptivity ◽  
A Posteriori ◽  
A Posteriori Error

Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

scholarly journals A Class of Spectral Element Methods and Its A Priori/A Posteriori Error Estimates for 2nd-Order Elliptic Eigenvalue Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Jiayu Han ◽  
Yidu Yang
Keyword(s):  
Spectral Methods ◽  
Eigenvalue Problems ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Spectral Element ◽  
A Posteriori ◽  
Spectral Element Methods ◽  
A Posteriori Error ◽  
Elliptic Eigenvalue Problems

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derivedp-version,h-version, andhp-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.

scholarly journals A posteriori error estimates for the virtual element method

2017 ◽  
Vol 137 (4) ◽  
pp. 857-893 ◽  
Author(s):  
Andrea Cangiani ◽  
Emmanuil H. Georgoulis ◽  
Tristan Pryer ◽  
Oliver J. Sutton
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
Virtual Element Method ◽  
A Posteriori Error ◽  
Virtual Element ◽  
Element Method

A Posteriori Error Estimates for a Steklov Eigenvalue Problem

2012 ◽  
Vol 557-559 ◽  
pp. 2081-2086 ◽  
Author(s):  
Ling Ling Sun ◽  
Yi Du Yang
Keyword(s):  
Eigenvalue Problem ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
Validity And Reliability ◽  
A Posteriori Error ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem

This paper discusses the finite element approximation for a Steklov eigenvalue problem. Based on the work of Armentano and Padra ( Appl Numer Math 58 (2008) 593-601), an a posteriori error estimator is provided and its validity and reliability are proved theoretically. Finally, numerical experiments with Matlab program are carried out to confirm the theoretical analysis.

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