AbstractA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time t and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $$O(\tau ^{1+ \alpha }), \, \alpha \in (0, 1)$$
O
(
τ
1
+
α
)
,
α
∈
(
0
,
1
)
for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $$\tau $$
τ
is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich’s convolution methods) and L-type methods (e.g., L1 method), which have only $$O(\tau )$$
O
(
τ
)
convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.