界面问题协调逼近的局部可计算误差估计子不可能具有鲁棒性
Locally computable error estimators for conforming approximations of interface problems cannot be robust
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中文总结 AI 辅助
该研究证明，对于界面问题的协调有限元方法，任何局部可计算的误差估计子（残差型、均衡型或恢复型）都无法同时具备与系数对比度无关的可靠性和效率性，其下界为$C_{\rm eff}(M)C_{\rm rel}(M)\gtrsim M^{1/4}$。
中文摘要 AI 辅助
我们证明了一个不可能性结果，针对棋盘交叉点处椭圆界面问题的协调方法，关于扩展局部后验估计子。我们构造了两个扩散问题实例，它们位于相同的界面拟合网格上，但其系数$M$分支的远程延续不同，然而每个满足效率性的扩展局部估计子必须为它们赋予相同的值。它们共享一个分片常数载荷，对于该载荷，有限元解和数据振荡均为零。它们的精确能量误差之比至少以系数对比度$M$的四次方根增长。因此，局部性迫使对于误差变得日益不同的问题给出相同的估计值，从而得到中心下界$C_{\rm eff}(M)C_{\rm rel}(M)\gtrsim M^{1/4}$，其中$C_{\rm rel}(M)$和$C_{\rm eff}(M)$分别是可靠性和效率常数。因此，任何局部可计算的误差估计子，无论是残差型、均衡型还是恢复型，都不能同时具有与对比度无关的常数下的可靠性和效率性，无论其代数形式如何。
英文摘要
We prove an impossibility result for finite-range locally computable a posteriori error estimators for the conforming finite element discretization of an elliptic interface problem. For a class of interface problems with a checkerboard cross-point and $\{1,M\}$-valued coefficients, we construct two problem instances on the same interface-fitted mesh. The two instances share a piecewise-constant load for which the finite element solution and the data oscillation both vanish. The ratio of their exact energy errors grows at least proportionally to $M^{1/4}$. Locality together with efficiency forces identical estimator values for these instances. The product of reliability and efficiency constants is therefore bounded below by a constant multiple of $M^{1/4}$. Consequently, any locally computable error estimator for interface problems, whether of residual, equilibrated, or recovery type, cannot be simultaneously reliable and efficient with contrast-independent constants.
发表机构
- Zhejiang University（浙江大学）
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