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Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. (English) Zbl 1381.65082
The authors analyse two schemes for subdiffusion and diffusion-wave equations. The schemes use the Galerkin finite element method in space and the convolution quadrature, developed in [E. Cuesta et al., Math. Comput. 75, No. 254, 673–696 (2006; Zbl 1090.65147)], generated by the backward Euler method and second-order backward difference in time. Optimal error estimates for both smooth and nonsmooth initial data are derived. In particular, it is shown that the schemes achieve first-order and second-order convergence in time. Several numerical experiments are presented to support the theoretical results.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35K05 Heat equation 35R11 Fractional partial differential equations 35L05 Wave equation 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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