超越多项式的自适应有限元空间的变分富集
Schur-Riesz Variational Enrichment: A Generalized Refinement Framework for Finite Elements
AI总结：
本文提出变分富集方法，引入问题驱动的紧凑函数补充常规h、p加密，经商舒尔分析优化，在SPE10模型测试中最大提升多项式富集效果41.6%，可选择有用的算子兼容结构或退化为常规hp加密。
AI中文摘要：
自适应有限元方法通过加密网格（h加密）或提高多项式次数（p加密）来优化解，但在两种方式下，与偏微分方程（PDE）算子相关的紧凑非多项式结构仍可能带来高昂计算成本。本文提出变分富集方法，即引入由问题信息驱动的紧凑函数，补充常规的h加密和p加密。该框架适用于Galerkin方法和最小残差公式，只要可计算容许函数及其残差，且不局限于某一类PDE。商舒尔（Quotient-Schur）分析可去除冗余的补充函数，并衡量其变分价值。对于对称强制问题，该理论还能给出精确的能量误差减少量及与参考无关的上界。在异质介质、随机系数、不规则域、奇点、界面、输运及振荡测试中，当紧凑函数捕捉到与PDE算子对齐的未解析结构时，hc方法（系数适配函数）可发挥作用；若该结构冗余或不匹配，则增益有限。在SPE10模型1和模型2中，等维度下，系数适配函数在全部11个规定案例中均提升了多项式富集效果，最大提升达41.6%。正负两方面的证据共同支持同一结论：hc方法能选择有用的算子兼容结构，否则会退化为常规hp加密。
英文摘要：
Width theory identifies economical spaces for compact PDE solution families, but does not provide a stable adaptive selection rule. We introduce Schur-Riesz refinement, which compares ordinary h/p refinement with operator-informed functions on a common variational scale. Projection removes content already represented by the incumbent, Riesz bounds test coefficient stability, and exact Schur gain ranks the surviving directions. New non-regression, bulk-contraction, and mixed near-oracle results provide finite-run error, dimension, and work certificates for coercive Galerkin and noncoercive minimum-residual formulations. We applied Schur-Riesz refinement across coercive and noncoercive PDEs. At matched dimension, automatic modes reduce held-out error by $50.2\%$ for 2D heterogeneous Helmholtz and $30.9\%$ for 2D Darcy. At matched Darcy error, the selected space uses $38.2\%$ fewer coordinates, reducing deployment memory by $38.9\%$ and online time by $30.1\%$. The additional offline construction is reusable and can be amortized across multiple right-hand sides for a fixed operator. On locked 3D heterogeneous Helmholtz tests, the method reduces mean error by $25.1$-$47.3\%$ against matched spectral-polynomial spaces and meets target errors with 24-96 coordinates, versus 1,331-6,859 for native MFEM hp-refinement. In an adaptive audit, 48 coordinates attain mean error $.1092$, compared with $.0967$ using 878 polynomial coordinates, while making online solves $6.8\times$ faster. Thus, the experiments demonstrate that Schur-Riesz refinement can construct smaller stable spaces with lower deployment memory and repeated-solve cost, while preserving a certified incumbent when richer functions do not help.
