An initial-boundary value problem of the form \({D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f\) is considered, where \({D}_{t}^{\alpha }\) is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain \({\varOmega } \subset \mathbb {R}^{d}\) for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 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 \({D}_{t}^{\alpha }\) on a graded temporal mesh. The numerical method computes approximations \({u_{h}^{n}}\) and \({{p}_{h}^{n}}\) of u(⋅,t_n) and Δu(⋅,t_n) at each time level t_n. The stability of the method (i.e. a priori bounds on \(\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}\) and \(\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}\)) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → 1⁻. Error bounds on \(\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}\) and \(\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}\) 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⁻.

考虑形式为\（{D} _ {t} ^ {\\ alpha} u + {\ varDelta} ^ {2} uc {\ varDelta} u = f \）的初边值问题，其中\（{D } _ {吨} ^ {\阿尔法} \）是顺序的时间卡普托衍生物α＆Element;（0,1）和C ^是一个非负常数。空间域\（{\ varOmega} \子集\ mathbb {R} ^ {d} \）一段d ∈{1,2,3}，具有Ω界和凸。边界条件是Ù = Δ Ù = 0上＆PartialD;＆Ω。给定足够的数据规律性和兼容性，就可以确定解决方案的先验界限。典型的解决方案在初始时具有弱的奇点t =0。将问题重写为两个二阶微分方程的系统，然后使用空间中的标准有限元以及\（{D} _ {t} ^ {\ alpha} \的L1离散化离散该问题。渐变的临时网格。数值方法计算近似值\（{U_ {H} ^ {N}} \）和\（{{P} _ {H} ^ {N}} \）的ü（⋅，吨_Ñ）和Δ ù（⋅ ，t _n）在每个时间级别t _n。方法的稳定性（即\（\ | {{u} _ {h} ^ {n}} \ | _ {L ^ {2}（{\ varOmega}}} \）和\（\ | {p_ {h} ^ {n}} \ | _ {L ^ {2}（{\ varOmega}}} \））通过一个新的离散Gronwall不等式即来建立α -robust，即保持为有效α →交通1 ^- 。上误差界限\（| U（\ CDOT，T_ {N}） - {U_ {H} ^ {N}} \ | \ _ {L ^ {2}（{\ varOmega}）} \）和\（\ | {\ varDelta} u（\ cdot，t_ {n}）-{{p} _ {h} ^ {n}} \ | _ {L ^ {2}（{\ varOmega}}} \\）然后得出; 这些边界是在为每个固定值的空间和时间网格参数最佳顺序的α，并且它们是α -robust如果考虑α →交通1 ^- 。