AI 中文总结
研究具有粗糙系数椭圆方程的系数自适应方法，针对经典MsFEM不足，提出类似FEM的结构可扩展性概念，证明满足固定界的确定性构造不能一致收敛，其最坏情况误差有与\(H\)无关的下界，为相关研究提供理论参考。
AI 中文摘要
几种系数自适应方法为具有粗糙系数的椭圆方程提供了最优阶逼近，如局部正交分解等，但它们的精度是通过让局部化半径或局部谱维随着粗尺度\(H\)趋于零而增长得到的。经典MsFEM有类似FEM的局部构造，但其现有分析未给出在全有界对比度可测系数类上系数一致的\(\bigO(H)\)能量估计。受此差距启发，我们形式化了一种类似FEM的结构可扩展性概念。我们证明，在支撑半径、系数信息半径和局部重数上满足固定界的确定性构造不能在系数类上一致收敛，其最坏情况的\(L^2\)到能量伽辽金误差保持由一个与\(H\)无关的正常数下界界定。该下界通过固定的有限光滑周期系数族和光滑右边项建立，证明结合了在局部补丁上重合的系数、校正场的有限维逼近下界、正密度网格论证和强周期校正收敛。因此，均匀最优精度至少需要一个局部构造参数增长或需要超出固定可见性模型的系数信息。
英文摘要
Several coefficient-adapted methods provide optimal-order approximation for elliptic equations with rough coefficients. Prominent examples include localized orthogonal decomposition, multiscale spectral GFEM, and constraint energy-minimizing GMsFEM. Their proven accuracy, however, is obtained by allowing the localization radius or the local spectral dimension to grow as the coarse scale \(H\) tends to zero. Classical MsFEM has an FEM-like local construction, but its available analysis does not give a coefficient-uniform \(\bigO(H)\) energy estimate over the full bounded-contrast measurable coefficient class. Motivated by this gap, we formalize an FEM-like notion of structural scalability. A chosen spatially local basis has uniformly bounded overlap, hence \(\bigO(1)\) stiffness entries per row, and every anchored local span uses coefficient information from only \(\bigO(1)\) coarse-element layers. We prove that no deterministic construction satisfying fixed bounds on the support radius, coefficient-information radius, and local multiplicity can converge uniformly over the coefficient class. In fact, its worst-case \(L^2\)-to-energy Galerkin error remains bounded below by a positive constant independent of \(H\). The lower bound is established using a fixed finite family of smooth periodic coefficients and smooth right-hand sides. The proof combines coefficients that coincide on local patches, a finite-dimensional approximation lower bound for corrector fields, a positive-density mesh argument, and strong periodic corrector convergence. Thus uniform optimal accuracy requires at least one local construction parameter to grow or requires coefficient information beyond the fixed-visibility model.
