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Postprocessing-Based a Posteriori Error Estimation for Spectral Galerkin Approximations of Fourth-Order Boundary Value Problems

Postprocessing-Based a Posteriori Error Estimation for Spectral Galerkin Approximations of Fourth-Order Boundary Value Problems

Year:    2025

Author:    Zhe Li, Tao Sun, Lijun Yi

East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 2 : pp. 314–343

Abstract

A postprocessing-based a posteriori error estimates for the spectral Galerkin approximation of one-dimensional fourth-order boundary value problems are developed. Our approach begins by introducing a novel postprocessing technique aimed at enhancing the accuracy of the spectral Galerkin approximation. We prove that this post-processing step improves the convergence rate in both $L^2$- and $H^2$-norm. Using post-processed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact as the polynomial degree increases. We further extend the postprocessing technique and error estimators to more general one-dimensional even-order equations and to multidimensional fourth-order equations. The results of numerical experiments illustrate the efficiency of the error estimators.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.2023-268.120324

East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 2 : pp. 314–343

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    30

Keywords:    Spectral Galerkin method fourth-order boundary value problem superconvergent post-processing a posteriori error estimation.

Author Details

Zhe Li

Tao Sun

Lijun Yi