We consider the first-order system space-time formulation of the heat equation introduced in [Bochev, Gunzburger, Springer, New York (2009)], and analyzed in [F\"uhrer, Karkulik, Comput. Math. Appl. 92 (2021)] and [Gantner, Stevenson, ESAIM Math. Model. Numer. Anal.} 55 (2021)], with solution components $(u_1,{\bf u}_2)=(u,-\nabla_{\bf x} u)$. 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 $\nabla_{\bf x} u_1$ and ${\bf u}_2$, the (graph) norm of $U$ contains the $L_2$-norm of $\partial_t u_1 +{\rm div}_{\bf x} {\bf u}_2$. 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 ${\bf u}_2$. 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 $\partial_t u_1 +{\rm div}_{\bf x} {\bf u}_2$, i.e., of the forcing term $f=(\partial_t-\Delta_x)u$. Numerical results show significantly improved convergence rates.
翻译:我们考虑在[Bochev, Gunzburger, Springer, New York(2009)] 推出的热方程式的第一阶系统空间-时间配制,并在[F\"元首, Karkulik, compuut. Math. Appl. 92 (2021)] 和[Gantner, Steveson, ESAM Math. 模型. Num. Num. Anal.} 55 (2021)] 和[Ganteral system sy-time 配制的热方程式。 相应的操作员在Hilbert 空间的美元与 libertal $2, dkulukulik 类的递解(美元) 等值之间无法忽略。 平滑度的值 2 flif =2 美元, 低值的运算值在平滑度中含有 美元- 平面 平面 平面 平面 平面 平流 平流 平流 的计算结果 。