发表机构
University of Twente; University of Amsterdam（特温特大学; 阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对线性抛物型初值问题的时空有限元离散化，提出双自适应求解器，推导后验条件与误差估计子，通过1+1和2+1维实验验证其有效性。
AI 中文摘要
我们研究了以标准时空变分形式表示的线性抛物型初值问题的极小残差时空有限元离散化。为处理产生的对偶范数，我们引入残差的Riesz提升作为额外变量。混合系统原始变量的拟最优性源于一致的inf-sup条件，已知该条件对时空柱体的棱柱形分区对应的有限元空间成立，这类分区可分解为时间层。我们证明在其他情况下无法期望该条件成立。为恢复一般分区和给定数据下的稳定性，我们推导了残差的精确Riesz提升与其Galerkin近似（即系统的次变量）之间误差的后验条件，在此条件下原始变量具有拟最优性。我们为两个变量推导了后验误差估计子，并将其用于交替测试空间和试空间加密的双自适应循环中。我们用1+1和2+1维的数值实验说明上述结果。
英文摘要
We study minimal residual space-time finite element discretizations of linear parabolic initial value problems in canonical space-time variational form. To deal with the arising dual norm, we introduce the Riesz lift of the residual as an additional variable. Quasi-optimality of the primal variable of the mixed system follows from a uniform inf-sup condition. This condition is known to be satisfied for finite element spaces w.r.t. prismatic partitions of the space-time cylinder that allow for a decomposition into time-slabs. We prove that this condition cannot be expected to hold otherwise. To recover stability for general partitions and the data at hand, we derive an a posteriori condition on the error between the exact Riesz lift of the residual and its Galerkin approximation -- being the secondary variable of our system -- under which the primal variable is quasi-optimal. We derive a posteriori error estimators for both variables, and use them in a double-adaptive loop that alternates test-space with trial-space enrichment. We illustrate our findings with numerical experiments in $1+1$ and $2+1$ dimensions.
Comments36 pages, code and data available at https://github.com/Rsmeets99/DoubleAdapParabolicFEM
