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A posteriori error analysis of an ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

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Abstract

In this paper, we present and analyze a posteriori error estimates for the ultra-weak local discontinuous Galerkin (UWLDG) method applied to nonlinear fourth-order boundary-value problems for ordinary differential equations of the form \(-u^{(4)}=f(x,u)\). Building upon the superconvergence results established in Baccouch (Numer Algor 92(4):1983–2023, 2023), we demonstrate the convergence of the UWLDG solution, in the \(L^2\)-norm, towards a special p-degree interpolating polynomial when piecewise polynomials of degree at most \(p\ge 2\) are employed. The convergence order is proven to be \(p+2\). Additionally, we decompose the UWLDG error on each element into two components. The dominant component is proportional to a special \((p+1)\)-degree polynomial, represented as a linear combination of Legendre polynomials with degrees \(p-1\), p, and \(p+1\). The second component converges to zero with an order of \(p+2\) in the \(L^2\)-norm. These findings enable the construction of computationally efficient a posteriori error estimates for the UWLDG method. These estimates are obtained by solving a local problem on each element without imposing boundary conditions. Furthermore, we establish that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the \(L^2\)-norm as the mesh is refined, with a convergence order of \(p+2\). In addition, we prove that the global effectivity index converges to unity at a rate of \(\mathcal {O}(h)\). Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Numerical results are provided to illustrate the reliability and efficiency of the proposed error estimator.

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Acknowledgements

The author would like to thank the anonymous reviewers for their valuable comments and suggestions which improved the quality of the paper.

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  1. Department of Mathematics, University of Nebraska at Omaha, Omaha, NE, 68182, USA

    Mahboub Baccouch

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Baccouch, M. A posteriori error analysis of an ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01773-4

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