Abstract
This paper is concerned with the unconditionally optimal H1-error estimate of a fast second-order scheme for solving nonlinear subdiffusion equations on the nonuniform mesh. We use the Galerkin finite element method (FEM) to discretize the spacial direction, the Newton linearization method to approximate the nonlinear term and the sum-of-exponentials (SOE) approximation to speed up the evaluation of Caputo derivative. Our analysis of the unconditionally optimal H1-error estimate involves the temporal-spatial error splitting approach, an improved discrete fractional Grönwall inequality and error convolution structure. In order to find a suitable test function to estimate H1-error, we here consider two cases: linear and high-order FEM space, using the time-discrete operator and Laplace operator in the test function respectively. Numerical tests are provided demonstrate the effectiveness and the unconditionally optimal H1-error convergence of our scheme.
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Acknowledgements
The numerical simulations in this work have been done on the supercomputing system in the Supercomputing Center of Wuhan University.
Funding
Jiwei Zhang is partially supported by NSFC under grants Nos. 11771035 and 12171376, 2020-JCJQ-ZD-029. Yanping Chen is partially supported by the State Key Program of National Natural Science Foundation of China (No. 11931003) and National Natural Science Foundation of China (No. 41974133). Yanmin Zhao is partially supported by NSFC (No. 11971416) and the Scientific Research Innovation Team of Xuchang University (No. 2022CXTD002).
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Appendix: . The proof of Lemma 3.4
Appendix: . The proof of Lemma 3.4
Proof 4
By the Taylor expansion, we obtain
By the condition (1.5) of v, we have
It can be further obtained that
where Cv in different places represents different constant. The proof is completed. □
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Liu, N., Chen, Y., Zhang, J. et al. Unconditionally optimal H1-error estimate of a fast nonuniform L2-1σ scheme for nonlinear subdiffusion equations. Numer Algor 92, 1655–1677 (2023). https://doi.org/10.1007/s11075-022-01359-y
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DOI: https://doi.org/10.1007/s11075-022-01359-y