We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components \((u_1,\textbf{u}_2)=(u,-\nabla _\textbf{x} u)\). The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of \(L_2\)-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides \(L_2\)-norms of \(\nabla _\textbf{x} u_1\) and \(\textbf{u}_2\), the (graph) norm of U contains the \(L_2\)-norm of \(\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2\). When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of \(\textbf{u}_2\). In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of \(\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2\), i.e., of the forcing term \(f=(\partial _t-\Delta _x)u\). Numerical results show significantly improved convergence rates.

我们考虑由 Bochev 和 Gunzburger 引入的热方程的一阶系统时空公式（见：Bochev 和 Gunzburger (eds) 应用数学科学，第 166 卷，Springer，纽约，2009 年），并由 Führer 和 Karkulik 进行分析(Comput Math Appl 92:27–36, 2021) 和 Gantner 和 Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021)，解决方案组件 \ ((u_1,\textbf{u}_2)=( u,-\nabla _\textbf{x} u)\)。相应的算子在希尔伯特空间U和\(L_2\)型空间的笛卡尔积之间有界可逆，这有利于简单的一阶系统最小二乘 (FOSLS) 离散化。除了\ (\nabla _\textbf{x} u_1\)和\(\textbf{u}_2\)的 \(L_2\) -范数之外， U的（图）范数还包含\(L_2\) -范数的\(\partial _t u_1 +{{\,\textrm{div}\,}}_\textbf{x} \textbf{u}_2\)。当将标准有限元应用于时空圆柱的单纯分区时，后一范数的近似误差估计需要\(\textbf{u}_2\)的高阶平滑度。在均匀划分和自适应细化划分的实验中，这表现为非光滑解u的收敛速度极低，令人失望。在本文中，我们用棱柱形分区构造有限元空间。它们带有一个准插值，满足近通勤图，从某种意义上说，除了一些无害的项之外，上述误差完全取决于\(\partial _t u_1 +{{\,\textrm{div}\ ,}}_\textbf{x} \textbf{u}_2\)，即强迫项\(f=(\partial _t-\Delta _x)u\)。数值结果显示收敛速度显着提高。