Abstract
In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in (Formula presented.)-, (Formula presented.)-norms, and a quasi-optimal bound in (Formula presented.)-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a (Formula presented.) type of weights to take care of the singular behavior of the continuous solution at (Formula presented.). The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.
| Original language | English |
|---|---|
| Pages (from-to) | 1-17 |
| Number of pages | 17 |
| Journal | Journal of Scientific Computing |
| DOIs | |
| Publication status | Accepted/In press - May 19 2017 |
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Keywords
- Energy argument error analysis
- Finite elements
- Fractional diffusion equation
- Nonsmooth data
ASJC Scopus subject areas
- Theoretical Computer Science
- Software
- Engineering(all)
- Computational Theory and Mathematics
Cite this
Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments. / Karaa, Samir; Mustapha, Kassem; Pani, Amiya K.
In: Journal of Scientific Computing, 19.05.2017, p. 1-17.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments
AU - Karaa, Samir
AU - Mustapha, Kassem
AU - Pani, Amiya K.
PY - 2017/5/19
Y1 - 2017/5/19
N2 - In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in (Formula presented.)-, (Formula presented.)-norms, and a quasi-optimal bound in (Formula presented.)-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a (Formula presented.) type of weights to take care of the singular behavior of the continuous solution at (Formula presented.). The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.
AB - In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in (Formula presented.)-, (Formula presented.)-norms, and a quasi-optimal bound in (Formula presented.)-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a (Formula presented.) type of weights to take care of the singular behavior of the continuous solution at (Formula presented.). The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.
KW - Energy argument error analysis
KW - Finite elements
KW - Fractional diffusion equation
KW - Nonsmooth data
UR - http://www.scopus.com/inward/record.url?scp=85019589472&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85019589472&partnerID=8YFLogxK
U2 - 10.1007/s10915-017-0450-7
DO - 10.1007/s10915-017-0450-7
M3 - Article
AN - SCOPUS:85019589472
SP - 1
EP - 17
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
SN - 0885-7474
ER -