Errata to “Approximation classes for adaptive higher order finite element approximation”

2016 ◽  
Vol 86 (305) ◽  
pp. 1525-1526
Author(s):  
Fernando D. Gaspoz ◽  
Pedro Morin
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Higher Order ◽  
Element Approximation ◽  
Approximation Classes

scholarly journals Higher-order finite element approximation of the dynamic Laplacian

2020 ◽  
Vol 54 (5) ◽  
pp. 1777-1795
Author(s):  
Nathanael Schilling ◽  
Gary Froyland ◽  
Oliver Junge
Keyword(s):  
Nonlinear Dynamics ◽  
Finite Element ◽  
Laplace Operator ◽  
Finite Time ◽  
Fluid Flows ◽  
Finite Element Approximation ◽  
Higher Order ◽  
Element Approximation ◽  
Eigenvalues And Eigenvectors ◽  
Order Element

The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in Froyland and Junge [SIAM J. Appl. Dyn. Syst. 17 (2018) 1891–1924]. In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally. We also prove the numerically computed eigenvalues and eigenvectors converge to the true objects for both schemes under certain assumptions. We provide an efficient implementation of the higher-order element schemes in an accompanying Julia package.

scholarly journals Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters

2014 ◽  
Vol 14 (4) ◽  
pp. 509-535 ◽  
Author(s):  
Dietmar Gallistl
Keyword(s):  
Finite Element ◽  
Convergence Rates ◽  
Nonlinear Approximation ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
Eigenvalues Of The Laplacian ◽  
Optimal Convergence Rates ◽  
Approximation Classes ◽  
Adjoint Operators

AbstractThis paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system.

scholarly journals A Posteriori Error Estimates for the Finite Element Approximation of Eigenvalue Problems

2003 ◽  
Vol 13 (08) ◽  
pp. 1219-1229 ◽  
Author(s):  
Ricardo G. Durán ◽  
Claudio Padra ◽  
Rodolfo Rodríguez
Keyword(s):  
Finite Element ◽  
Eigenvalue Problems ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Higher Order ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Higher Order Terms

This paper deals with a posteriori error estimators for the linear finite element approximation of second-order elliptic eigenvalue problems in two or three dimensions. First, we give a simple proof of the equivalence, up to higher order terms, between the error and a residual type error estimator. Second, we prove that the volumetric part of the residual is dominated by a constant times the edge or face residuals, again up to higher order terms. This result was not known for eigenvalue problems.

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