This paper considers the semidiscrete Galerkin finite element approximation for time fractional diffusion equations with memory in a bounded convex polygonal domain. We use novel energy arguments in conjunction with repeated applications of time integral operators to study the error analysis. Our error estimates cover both smooth and nonsmooth initial data cases. Since the continuous solution u of such models has singularity at t = 0 even for smooth initial data, we use t^j, j = 1,2, type of weights to overcome the singular behavior at t = 0. Optimal order error estimates in both L²(Ω)- and H¹(Ω)-norms are proved with respect to both convergence order of the approximate solution and regularity of the initial data. Moreover, a quasi-optimal pointwise in time error bound in the maximum norm is shown to hold for smooth initial data. In the end, we provide numerical results to support our theoretical findings.

本文考虑时间分数扩散方程的半离散Galerkin有限元逼近，在有界凸多边形域中具有记忆。我们使用新的能量参数结合时间积分算子的重复应用来研究误差分析。我们的误差估计涵盖了平滑和非平滑的初始数据情况。由于此类模型的连续解u在t = 0处具有奇异性，即使对于平滑的初始数据，我们也使用t ^j , j = 1,2 类型的权重来克服t = 0 处的奇异行为。两者中的最优阶误差估计L ² ( Ω )-和H ¹ (Ω )-范数证明了近似解的收敛顺序和初始数据的规律性。此外，最大范数中的准最优逐点时间误差界限被证明适用于平滑的初始数据。最后，我们提供了数值结果来支持我们的理论发现。