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High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation
Author(s) -
Ren Jincheng,
Shi Dongyang,
Vong Seakweng
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22428
Subject(s) - mathematics , superconvergence , finite element method , nonlinear system , galerkin method , diffusion equation , discontinuous galerkin method , approximation error , computation , norm (philosophy) , mixed finite element method , projection (relational algebra) , mathematical analysis , algorithm , physics , economy , quantum mechanics , political science , law , economics , thermodynamics , service (business)
In this work, an effective and fast finite element numerical method with high‐order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two‐level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H 1 ‐norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.