后验误差分析：直至临界指数的演化非线性薛定谔方程
A posteriori error analysis for evolution nonlinear Schrödinger equations up to the critical exponent
AI总结：
本文基于后验误差分析，针对直至临界指数的演化非线性薛定谔方程，提出松弛时间离散和完全离散方案，在$ L^\infty(L^2) $-范数下提供误差估计，结合重构技术与连续问题的非线性稳定性论证，利用Gagliardo-Nirenberg不等式和守恒律，验证了估计的最优收敛阶。
摘要：
我们为一类直至临界指数的演化非线性薛定谔方程的松弛时间离散和完全离散方案，在$ L^\infty(L^2) $-范数下提供后验误差估计。特别地，在时间离散化中使用Besse引入的松弛Crank-Nicolson型方案，在空间离散化中使用允许在时间步之间变化的有限元空间。通过重构技术，将问题转化为原偏微分方程的扰动问题，从而能够使用连续问题中的非线性稳定性论证。主要使用的工具包括Gagliardo-Nirenberg不等式和连续问题的两个守恒律（质量守恒和能量守恒）。数值结果表明，估计确实具有最优的收敛阶。
英文摘要：
We provide a posteriori error estimates in the $L^\infty(L^2)-$norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank-Nicolson-type scheme introduced by Besse in \cite{Besse}. For the discretization in space we use finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. In particular, main ingredients we use in our analysis are the Gagliardo-Nirenberg inequality and the two conservation laws (mass and energy conservation) of the continuous problem. Numerical results illustrate that the estimates are indeed of optimal order of convergence.
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