A 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 \(O(\tau ^{1+ \alpha }), \, \alpha \in (0, 1)\) for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where \(\tau \) 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 \(O(\tau )\) convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

介绍了一种连续伽辽金时间步长方法并分析了抽象设置中的子扩散问题。近似解将作为时间t 中的连续分段线性函数来寻找，测试空间基于不连续分段常数函数。我们证明了所提出的时间步进方法对于非光滑数据的一般扇形椭圆算子具有收敛阶数\(O(\tau ^{1+ \alpha }), \, \alpha \in (0, 1)\)拉普拉斯变换方法，其中\(\tau \)是时间步长。这种收敛阶次高于流行的卷积正交方法（例如 Lubich 卷积方法）和 L 型方法（例如 L1 方法）的收敛阶数，它们只有\(O(\tau )\)非光滑数据的收敛。给出了数值例子来验证时间离散化方案在数据规律性方面的鲁棒性。