An α-Robust Semidiscrete Finite Element Method for a Fokker–Planck Initial-Boundary Value Problem with Variable-Order Fractional Time Derivative
A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain Ω × [ 0 , T ] , where Ω is an open bounded domain in R d for some d ≥ 1 , and the order α ( x ) of the Riemann-Liouville fractional derivative may vary in space with 1 / 2 < α ( x ) <...
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| Veröffentlicht in: | Journal of scientific computing Jg. 86; H. 2; S. 22 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.02.2021
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0885-7474, 1573-7691 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain
Ω
×
[
0
,
T
]
, where
Ω
is an open bounded domain in
R
d
for some
d
≥
1
, and the order
α
(
x
)
of the Riemann-Liouville fractional derivative may vary in space with
1
/
2
<
α
(
x
)
<
1
for all
x
. Such problems appear naturally in the formulation of certain continuous-time random walk models. Uniqueness of any solution
u
of the problem is proved under reasonable hypotheses. A semidiscrete numerical method, using finite elements in space to yield a solution
u
h
(
t
)
, is constructed. Error estimates for
‖
(
u
-
u
h
)
(
t
)
‖
L
2
(
Ω
)
and
∫
0
t
∂
t
1
-
α
(
u
-
u
h
)
(
s
)
1
2
d
s
are proved for each
t
∈
[
0
,
T
]
under the assumptions that the following quantities are finite:
‖
u
(
·
,
0
)
‖
H
2
(
Ω
)
,
|
u
(
·
,
t
)
|
H
1
(
Ω
)
for each
t
, and
∫
0
t
[
‖
u
(
·
,
t
)
‖
H
2
(
Ω
)
2
+
|
∂
t
1
-
α
u
|
H
2
(
Ω
)
2
]
, where
u
(
x
,
t
) is the unknown solution. Furthermore, these error estimates are
α
-robust: they do not fail when
α
→
1
, the classical Fokker–Planck problem. Sharper results are obtained for the special case where the drift term of the problem is not present (which is of interest in certain applications). |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0885-7474 1573-7691 |
| DOI: | 10.1007/s10915-020-01375-x |