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An hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems

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Abstract

We present an hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. We derive a priori error bound in the H1-norm that is fully explicit with respect to the local time steps and the local approximation orders. Moreover, we prove that the hp-version of the C0-continuous Petrov-Galerkin time stepping method based on geometrically refined time steps and on linearly increasing approximation orders yields exponential rates of convergence for solutions with start-up singularities. Numerical examples confirm the theoretical results.

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Acknowledgments

The authors would like to thank the anonymous referees for many constructive and valuable suggestions, which considerably improved the presentation of the paper.

Funding

The work of this author is supported in part by the National Natural Science Foundation of China (Nos. 11771298, 11871043, 11301343).

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  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Yichen Wei & Lijun Yi

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  1. Yichen Wei
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  2. Lijun Yi
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Correspondence to Lijun Yi.

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Communicated by: Lourenco Beirao da Veiga

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Wei, Y., Yi, L. An hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. Adv Comput Math 46, 56 (2020). https://doi.org/10.1007/s10444-020-09800-3

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