A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models

DOI10.1090/S0025-5718-2013-02708-1zbMath1327.65101OpenAlexW2088485015MaRDI QIDQ5326495

Kun Zhou, Qiang Du, Li Tian, Lili Ju

Publication date: 6 August 2013

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0025-5718-2013-02708-1

zbMATH Keywords

finite elementa posteriori error estimatenonlocal diffusionperidynamic models

Mathematics Subject Classification ID

Numerical methods for integral equations (65R20) Error bounds for boundary value problems involving PDEs (65N15) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Numerical solutions to equations with nonlinear operators (65J15)

Related Items (44)

A Nonlocal Stokes System with Volume Constraints ⋮ A Dual-Horizon Nonlocal Diffusion Model and Its Finite Element Discretization ⋮ Numerical methods for nonlocal and fractional models ⋮ Identification of the diffusion parameter in nonlocal steady diffusion problems ⋮ A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems ⋮ Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains ⋮ Quadrature rules for finite element approximations of 1D nonlocal problems ⋮ A Convergent Point Integral Method for Isotropic Elliptic Equations on a Point Cloud ⋮ On a class of nonlocal wave equations from applications ⋮ Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations ⋮ An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems ⋮ A fast discontinuous Galerkin method for a bond-based linear peridynamic model discretized on a locally refined composite mesh ⋮ Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems ⋮ Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems ⋮ A Mixed-Type Circulant Preconditioner for a Nonlocal Elastic Model ⋮ Moving interfaces in peridynamic diffusion models and the influence of discontinuous initial conditions: numerical stability and convergence ⋮ Prediction-Correction Scheme for Decoupled Forward Backward Stochastic Differential Equations with Jumps ⋮ Quasi-nonlocal Coupling of Nonlocal Diffusions ⋮ A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model ⋮ Peridynamics review ⋮ Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains ⋮ Localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for nonlocal diffusion problems ⋮ Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion ⋮ Nonlocal convection-diffusion problems and finite element approximations ⋮ Nonlocal and Mixed-Locality Multiscale Finite Element Methods ⋮ A reproducing kernel enhanced approach for peridynamic solutions ⋮ Analysis of fully discrete approximations for dissipative systems and application to time-dependent nonlocal diffusion problems ⋮ Asymptotically Compatible Fourier Spectral Approximations of Nonlocal Allen--Cahn Equations ⋮ A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions ⋮ Convergence studies in meshfree peridynamic simulations ⋮ Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs ⋮ Robust Discretization of Nonlocal Models Related to Peridynamics ⋮ Fractional diffusion on bounded domains ⋮ Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation ⋮ Convergence of the point integral method for Laplace-Beltrami equation on point cloud ⋮ A spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions ⋮ Finite element convergence for state-based peridynamic fracture models ⋮ A Multiscale Implementation Based on Adaptive Mesh Refinement for the Nonlocal Peridynamics Model in One Dimension ⋮ Asymptotically Compatible Schemes for Robust Discretization of Parametrized Problems with Applications to Nonlocal Models ⋮ Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes ⋮ A fast collocation method for a static bond-based linear peridynamic model ⋮ A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization ⋮ A preconditioned fast collocation method for a linear bond-based peridynamic model ⋮ A conforming DG method for linear nonlocal models with integrable kernels

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Analysis of the volume-constrained peridynamic Navier equation of linear elasticity
• Variational theory and domain decomposition for nonlocal problems
• Continuous and discontinuous finite element methods for a peridynamics model of mechanics
• Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis
• A posteriori error analysis for a class of integral equations and variational inequalities
• Force flux and the peridynamic stress tensor
• Convergence of peridynamics to classical elasticity theory
• A feedback finite element method with a posteriori error estimation. I: The finite element method and some basic properties of the a posteriori error estimator
• Multiscale dynamics of heterogeneous media in the peridynamic formulation
• Reformulation of elasticity theory for discontinuities and long-range forces
• On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity
• Mathematical analysis for the peridynamic nonlocal continuum theory
• Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions
• Results on Nonlocal Boundary Value Problems
• A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems
• A Nonlocalp-Laplacian Evolution Equation with Nonhomogeneous Dirichlet Boundary Conditions
• Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems
• Can the Nonlocal Characterization of Sobolev Spaces by Bourgain et al. Be Useful for Solving Variational Problems?
• Convergence, adaptive refinement, and scaling in 1D peridynamics
• Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity
• A Posteriori Error Analysis of Finite Element Solutions for One-Dimensional Problems
• A‐posteriori error estimates for the finite element method
• Error Estimates for Adaptive Finite Element Computations
• A Posteriori Error Control for Finite Element Approximations of the Integral Equation for Thin Wire Antennas
• All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable
• Efficiency of a posteriori BEM–error estimates for first-kind integral equations on quasi–uniform meshes
• Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints
• A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS
• Adaptive Finite Element Methods for Elliptic PDEs Based on Conforming Centroidal Voronoi–Delaunay Triangulations

This page was built for publication: A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models