TY - JOUR
T1 - Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier functions
AU - Kopteva, Natalia
AU - Meng, Xiangyun
N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.
PY - 2020
Y1 - 2020
N2 - An initial-boundary value problem with a Caputo time derivative of fractional order \alpha \in (0, 1) is considered, solutions of which typically exhibit a singular behavior at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semidiscretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.
AB - An initial-boundary value problem with a Caputo time derivative of fractional order \alpha \in (0, 1) is considered, solutions of which typically exhibit a singular behavior at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semidiscretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.
KW - Alikhanov scheme
KW - Arbitrary degree of grading
KW - Fractional-order parabolic equation
KW - Graded temporal mesh
KW - L1 method
KW - Pointwise-in-time error bounds
UR - http://www.scopus.com/inward/record.url?scp=85084758654&partnerID=8YFLogxK
U2 - 10.1137/19M1300686
DO - 10.1137/19M1300686
M3 - Article
AN - SCOPUS:85084758654
SN - 0036-1429
VL - 58
SP - 1217
EP - 1238
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 2
ER -