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A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

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Abstract

In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form − u(4) = f(x, u). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the L2-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order p + 1 in the L2-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most p and mesh size h are used. We then show that the UWLDG solutions are superconvergent with order p + 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.

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Acknowledgements

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

Funding

This research was supported by the NASA Nebraska Space Grant (Federal Grant/Award Number 80NSSC20M0112).

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

    Mahboub Baccouch

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This paper is dedicated to my mother, Rebah Jetlaoui, who passed away from COVID in July 31, 2021, while I was completing this work.

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Baccouch, M. A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems. Numer Algor 92, 1983–2023 (2023). https://doi.org/10.1007/s11075-022-01374-z

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