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Error estimates of a continuous Galerkin time stepping method for subdiffusion problem
Yan, Yubin Yan, Yuyuan Liang, Zongqi Egwu, Bernard
Yan, Yubin
Yan, Yuyuan
Liang, Zongqi
Egwu, Bernard
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2021-07-29
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Abstract
A continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \, \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and L-type methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.
Citation
Yan, Y., Egwu, B. A., Liang, Z., Yan, Y. (2021). Error estimates of a continuous Galerkin Time Stepping Method for subdiffusion problem. Journal of Scientific Computing, 88, 68. https://doi.org/10.1007/s10915-021-01587-9
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Springer
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Journal of Scientific Computing
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10.1007/s10915-021-01587-9
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0885-7474
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1573-7691