\(C^0\mathrm{IPG}\) adaptive algorithms for the biharmonic eigenvalue problem
type: Item
instance of: scholarly article
title: \(C^0\mathrm{IPG}\) adaptive algorithms for the biharmonic eigenvalue problem
zbMATH Open document ID: 1402.65146
author: Y. Aharonov
published in: Numerical Algorithms
publication date: 2018-06-25T00:00:00Z
review text: The \(C^0\mathrm{IPG}\) refers to \(C^0\) interior penalty Galerkin methods as initiated in [\textit{S. C. Brenner}, Lect. Notes Comput. Sci. Eng. 85, 79--147 (2012; Zbl 1248.65120)] and further elaborated in subsequent papers. Here the problem is to study the biharmonic eigenvalue problem \(\Delta^2 u=\lambda u\) in \(\Omega\), a bounded polygonal domain in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) while \(u\) and its outward normal derivative vanishes on the boundary \(\partial \Omega\). Using Lagrange finite elements on a shape-regular triangularization, Brenner [loc. cit.] also provides a posteriori error indicators for \(C^0\)IPG, which are proved in this paper to be reliable and efficient for the problem at hand. Moreover, the standard adaptive refinement algorithm, which shows an increase in the ratio \(h^{k-1}/h^\alpha_{\min}\) (\(h\) is the mesh size, \(k\) the degree of the method, and \(\alpha\in(0.5,1]\) the regularity index). This ratio should be \(O(1)\). By preventing an increase in this ratio above some threshold, a more uniform refinement algorithm is proposed. This is tested numerically on a unit square with a slit, and on an L-shaped domain.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 31A30
zbMATH DE Number: 6894100
zbMATH Keywords: biharmonic eigenvalue problem
zbMATH Keywords: \(C^0\) interior penalty Galerkin method
zbMATH Keywords: Lagrange finite element method
zbMATH Keywords: adaptive refinement
zbMATH Keywords: a posteriori error analysis
zbMATH Keywords: adaptive algorithm
describes a project that uses: iFEM
MaRDI profile type: Publication
full work available at URL: s11075-017-0388-8
OpenAlex ID: W2744058931
cites work: A posteriori error estimation in finite element analysis
cites work: Direct and inverse error estimates for finite elements with mesh refinements
cites work: Q3126801
cites work: Error Estimates for Adaptive Finite Element Computations
cites work: On the boundary value problem of the biharmonic operator on domains with angular corners
cites work: C 0 Interior Penalty Methods
cites work: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains
cites work: A $\mathcal{C}^0$ Interior Penalty Method for a Fourth Order Elliptic Singular Perturbation Problem
cites work: A Quadratic $C^0$ Interior Penalty Method for Linear Fourth Order Boundary Value Problems with Boundary Conditions of the Cahn--Hilliard Type
cites work: Q4531674
cites work: Poincaré–Friedrichs Inequalities for PiecewiseH2Functions
cites work: Guaranteed lower eigenvalue bounds for the biharmonic equation
cites work: \(C^0\)IP methods for the transmission eigenvalue problem
cites work: Q3126804
cites work: Convergence and optimal complexity of adaptive finite element eigenvalue computations
cites work: Q1858456
cites work: A new error analysis for discontinuous finite element methods for linear elliptic problems
cites work: Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods
cites work: Lower bounds of the discretization error for piecewise polynomials
cites work: Convergence of Adaptive Finite Element Methods
cites work: Q4104278
cites work: A posteriori error estimates of the Morley element for the fourth order elliptic eigenvalue problem
cites work: Nonconforming finite element methods for eigenvalue problems in linear plate theory
cites work: Q4886148
cites work: A \(C^0\) discontinuous Galerkin formulation for Kirchhoff plates
cites work: The lower bound property of the Morley element eigenvalues
cites work: Eigenvalue approximation from below using non-conforming finite elements
cites work: Q5647993
DOI: 10.1007/S11075-017-0388-8
Recommended article: C 0 Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems
Recommended article: A new method using \(C^0\)IPG for the biharmonic eigenvalue problem
Recommended article: Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the biharmonic problem
Recommended article: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains
Recommended article: A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations
Recommended article: A C0 interior penalty method for mth-Laplace equation
Recommended article: Quasi-Optimal Nonconforming Methods for Symmetric Elliptic Problems. III---Discontinuous Galerkin and Other Interior Penalty Methods
Recommended article: C 0 Interior Penalty Methods
Recommended article: Multigrid Algorithms for C0 Interior Penalty Methods
Recommended article: An adaptive \(C^0\)IPG method for the Helmholtz transmission eigenvalue problem
description: scientific article; zbMATH DE number 6894100