An initial-boundary value problem of the form \(\sum _{i=1}^{l}q_i D _t ^{\alpha _i} u(x,t)- \Delta u =f\) is considered, where each \(D _t ^{\alpha _i}\) is a Caputo fractional derivative of order \(\alpha _i\in (0,1)\) and the spatial domain \(\Omega \) lies in \(\mathbb {R}^d\) for \(d\in \{1,2,3\}\). To solve the problem numerically, we apply the L1 discretisation to each fractional derivative on a graded temporal mesh, together with a standard finite element method for the spatial derivatives on a quasiuniform spatial mesh. A new proof of the stability of this method, which is more complicated than the \(l=1\) case of a single fractional derivative, is given using barrier functions; this powerful new technique leads to sharp error estimates in \(L^2(\Omega )\) and \(H^1(\Omega )\) at each time level \(t_m\) that show precisely the improvement in accuracy of the method as one moves away from the initial time \(t=0\). Consequently, while for global optimal accuracy one needs a mesh that is strongly graded when all the \(\alpha _i\) are near zero, for local optimal accuracy away from \(t=0\) one needs a much less severe mesh grading. Numerical experiments show the sharpness of our theoretical results.

考虑形式为\（\ sum _ {i = 1} ^ {l} q_i D _t ^ {\ alpha _i} u（x，t）-\ Delta u = f \）的初边值问题，其中每个\（D _t ^ {\ alpha _i} \）是阶\（\ alpha _i \ in（0,1）\）的Caputo分数导数，而空间域\（\ Omega \）位于\（\ mathbb {R} ^ d \）代表\（d \ in \ {1,2,3 \} \）。为了从数值上解决问题，我们将L1离散化应用于渐变时空网格上的每个分数导数，以及用于准均匀空间网格上空间导数的标准有限元方法。此方法稳定性的新证明，它比\（l = 1 \）更复杂使用分数函数给出单个分数导数的情况；这项功能强大的新技术可在每个时间水平 \（t_m \）上对\（L ^ 2（\ Omega）\）和\（H ^ 1（\ Omega）\）进行清晰的误差估计，从而精确地显示出精度的提高。该方法远离初始时间\（t = 0 \）。因此，虽然对于全局最佳精度，当所有\（\ alpha _i \）都接近零时，需要一个网格进行强分级 ，但对于远离\（t = 0 \）的局部最优精度，则需要的网格分级要低得多。数值实验表明了我们理论结果的敏锐性。