AI 中文总结
该研究为一阶系统最小二乘有限元方法构建面向目标的误差估计框架，基于物理偏微分方程伴随，通过原对偶校正泛函得出误差恒等式，构造校正近似并证明误差估计，数值实验验证了收敛性和标记有效性。
AI 中文摘要
我们基于物理偏微分方程伴随而非最小二乘公式诱导的伴随，为一阶系统最小二乘（FOSLS）有限元方法开发了一个面向目标的误差估计框架。从微分算子、其混合边界条件和输出中明确识别出的物理伴随具有自身的一阶通量系统，从而有一个原生的内置最小二乘估计器，而最小二乘诱导的伴随则没有。我们的误差恒等式基于原对偶校正泛函：它们仅从连续的原方程和伴随方程得出，对任意协调近似都成立，且不需要伽辽金正交性。对于包含加权狄利克雷边界通量的输出，其权重成为伴随的本质数据，我们构造了两个校正近似，一个来自势，另一个也使用通量，并证明了原误差和伴随误差的乘积型误差估计。然后，最小二乘泛函产生可计算的后验界和一个平衡标记指标，其元素和恰好等于两个估计器的乘积。数值实验证实了预测的收敛性和标记的有效性。
英文摘要
We develop a goal-oriented error-estimation framework for first-order system least-squares (FOSLS) finite element methods based on the physical PDE adjoint rather than the adjoint induced by the least-squares formulation. Identified explicitly from the differential operator, its mixed boundary conditions, and the output, the physical adjoint admits its own first-order flux system and hence a native, built-in least-squares estimator, which the least-squares-induced adjoint does not. Our error identities rest on a primal-dual corrected functional: they follow from the continuous primal and adjoint equations alone, hold for arbitrary conforming approximations, and require no Galerkin orthogonality. For outputs containing a weighted Dirichlet-boundary flux, whose weight becomes the essential datum of the adjoint, we construct two corrected approximations, one from the potentials and one also using the fluxes, and prove product-type error estimates in the primal and adjoint errors. The least-squares functionals then yield computable a posteriori bounds and a balanced marking indicator whose element sum equals the product of the two estimators exactly. Numerical experiments confirm the predicted convergence and the effectiveness of the marking.
Comments25 pages
