抛物型偏微分方程时空FOSLS的Aubin-Nitsche型估计
Aubin--Nitsche-type estimates for space-time FOSLS for parabolic PDEs
AI总结：
本文针对求解热传导方程的FOSLS发展了Aubin-Nitsche型估计，证明了在特定条件下标量场变量$L^2$误差具有更高收敛速度，并展示了高阶守恒性质与弱范数拟最优性。
摘要：
我们针对最近提出的求解热传导方程的一阶系统最小二乘有限元方法（FOSLS）发展了Aubin-Nitsche型估计。在特定假设下（若空间区域为凸且热源与初始数据足够光滑即可满足），我们证明了标量场变量近似的$L^2$误差比总体误差具有更高的收敛速度。此外，展示了高阶守恒性质。另外，我们讨论了较弱范数下的拟最优性。数值实验证实了理论发现。
英文摘要：
We develop Aubin--Nitsche-type estimates for recently proposed first-order system least-squares finite element methods (FOSLS) for the heat equation. Under certain assumptions, which are satisfied if the spatial domain is convex and the heat source and initial datum are sufficiently smooth, we prove that the $L^2$ error of approximations of the scalar field variable converges at a higher rate than the overall error. Furthermore, a higher-order conservation property is shown. In addition, we discuss quasi-optimality in weaker norms. Numerical experiments confirm our theoretical findings.
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