TY - JOUR
T1 - Semidiscrete finite element analysis of time fractional parabolic problems
T2 - A unified approach
AU - Karaa, Samir
N1 - Funding Information:
∗Received by the editors June 13, 2017; accepted for publication (in revised form) April 23, 2018; published electronically June 20, 2018. http://www.siam.org/journals/sinum/56-3/M113416.html Funding: This research was supported by the Research Council of Oman, grant ORG/CBS/15/001. †Department of Mathematics and Statistics, Sultan Qaboos University, Al-Khod 123, Muscat, Oman (skaraa@squ.edu.om).
Funding Information:
This research was supported by the Research Council of Oman, grant ORG/CBS/15/001. The author thanks Prof. Amiya K. Pani for valuable comments and suggestions.
Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.
PY - 2018
Y1 - 2018
N2 - In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order a, 0 < a < 1. We derive optimal error estimates for semidiscrete Galerkin finite element (FE) type approximations for problems with smooth and nonsmooth initial data. Our analysis relies on energy arguments and exploits the properties of the inverse of the associated elliptic operator. We present the analysis in a general setting so that it is easily applicable to various spatial approximations such as conforming and nonconforming FE methods (FEMs) and FEM on nonconvex domains. The finite element approximation in mixed form is also presented and new error estimates are established for smooth and nonsmooth initial data. Finally, an extension of our analysis to a multiterm time-fractional model is discussed.
AB - In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order a, 0 < a < 1. We derive optimal error estimates for semidiscrete Galerkin finite element (FE) type approximations for problems with smooth and nonsmooth initial data. Our analysis relies on energy arguments and exploits the properties of the inverse of the associated elliptic operator. We present the analysis in a general setting so that it is easily applicable to various spatial approximations such as conforming and nonconforming FE methods (FEMs) and FEM on nonconvex domains. The finite element approximation in mixed form is also presented and new error estimates are established for smooth and nonsmooth initial data. Finally, an extension of our analysis to a multiterm time-fractional model is discussed.
KW - Mixed method
KW - Multiterm fractional diffusion
KW - Nonsmooth initial data
KW - Optimal error estimates
KW - Semidiscrete finite element scheme
KW - Time-fractional parabolic equation
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U2 - 10.1137/17M1134160
DO - 10.1137/17M1134160
M3 - Article
AN - SCOPUS:85049458929
SN - 0036-1429
VL - 56
SP - 1673
EP - 1692
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 3
ER -