最坏情况下的多目标误差估计与自适应性
Worst-case multi-objective error estimation and adaptivity
AI总结：
本文提出了一种新的计算方法，用于确定有限元近似中的后验多目标误差估计，并构建相应的（准）最优自适应细化。与传统目标导向方法不同，该方法适用于一般闭凸集的对偶空间子集，并构造了有限元近似误差的最坏情况误差估计。
摘要：
本文介绍了一种新的计算方法，用于确定有限元近似中的后验多目标误差估计，并构建相应的（准）最优自适应细化。与传统目标导向方法不同，该方法适用于一般闭凸集的对偶空间子集，并构造了有限元近似误差的最坏情况误差估计。该最坏情况多目标误差估计符合一个对偶加权残差，其中对偶解与近似支撑函数在近似误差处相关联。我们考虑了标准近似误差和与边界数据不兼容相关的数据不相容误差。数值实验展示了所提出最坏情况多目标误差在自适应细化过程中的有效性。
英文摘要：
This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multi-objective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multi-objective error in adaptive refinement procedures.
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