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Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids

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Abstract

This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin (LDG) method for two-dimensional semilinear second-order elliptic problems of the form \(-\Delta u=f(x,y,u)\) on Cartesian grids. By introducing special Gauss-Radau projections and using duality arguments, we obtain, under some suitable choice of numerical fluxes, the optimal convergence order in \(L^2\)-norm of \({O}(h^{p+1})\) for the LDG solution and its gradient, when tensor product polynomials of degree at most p and grid size h are used. Moreover, we prove that the LDG solutions are superconvergent with an order \(p+ 2\) toward particular Gauss-Radau projections of the exact solutions. Finally, we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves \((p+1)\)-th order superconvergence. Some numerical experiments are performed to illustrate the theoretical results.

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Acknowledgements

The author would like to thank the anonymous reviewers for the valuable comments and suggestions which improved the quality of the paper. 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 research was supported by the NASA Nebraska Space Grant (Federal Grant/Award Number 80NSSC20M0112).

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Baccouch, M. Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids. Commun. Appl. Math. Comput. 4, 437–476 (2022). https://doi.org/10.1007/s42967-021-00123-8

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