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An initial-boundary value problem of the form
D
t
α
u
+
Δ
2
u
−
c
Δ
u
=
f
is considered, where
D
t
α
is a Caputo temporal derivative of order
α
∈ (0,1) and
c
is a nonnegative constant. The spatial domain
Ω
⊂
ℝ
d
for some
d
∈{1,2,3}, with
Ω
bounded and convex. The boundary conditions are
u
=
Δ
u
= 0 on
∂
Ω
. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time
t
= 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of
D
t
α
on a graded temporal mesh. The numerical method computes approximations
u
h
n
and
p
h
n
of
u
(⋅,
t
n
) and
Δ
u
(⋅,
t
n
) at each time level
t
n
. The stability of the method (i.e. a priori bounds on
∥
u
h
n
∥
L
2
(
Ω
)
and
∥
p
h
n
∥
L
2
(
Ω
)
) is established by means of a new discrete Gronwall inequality that is
α
-robust, i.e. remains valid as
α
→ 1
−
. Error bounds on
∥
u
(
⋅
,
t
n
)
−
u
h
n
∥
L
2
(
Ω
)
and
∥
Δ
u
(
⋅
,
t
n
)
−
p
h
n
∥
L
2
(
Ω
)
are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of
α
, and they are
α
-robust if one considers
α
→ 1
−
.