Abstract
A time-fractional initial-boundary value problem is considered, where the differential equation has a sum of fractional derivatives of different orders, and the spatial domain lies in \({\mathbb {R}}^d\) with \(d\in \{1,2,3\}\). A priori bounds on the solution and its derivatives are stated; these show that typical solutions have a weak singularity at the initial time \(t=0\). A standard finite element method with mapped piecewise bilinears is used to discretise the spatial derivatives, while for each time derivative we use the L1 scheme on a temporal graded mesh. Our analysis reveals the optimal grading that one should use for this mesh. A novel discrete fractional Gronwall inequality is proved: the statement of this inequality and its proof are different from any previously published Gronwall inequality. This inequality is used to derive an optimal error estimate in \(L^\infty (H^1)\). It is also used to show that, if each mesh element is rectangular in the case \(d=2\) or cubical in the case \(d=3\), with the sides of the element parallel to the coordinate axes, then a simple postprocessing of the computed solution will yield a higher order of convergence in the spatial direction. Numerical results are presented to show the sharpness of our theoretical results.
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Acknowledgements
The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under Grant NSAF-U1930402. The research of Chaobao Huang is supported in part by the National Natural Science Foundation of China (Grant Nos. 11801332 and 11971259).
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Huang, C., Stynes, M. Superconvergence of a Finite Element Method for the Multi-term Time-Fractional Diffusion Problem. J Sci Comput 82, 10 (2020). https://doi.org/10.1007/s10915-019-01115-w
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DOI: https://doi.org/10.1007/s10915-019-01115-w