Schur-Riesz Variational Enrichment: A Generalized Refinement Framework for Finite Elements

Abstract

Width theory identifies low-dimensional spaces that are optimal for compact solution families, but does not turn them into a stable adaptive method. We introduce Schur-Riesz refinement, a variational framework for combining ordinary polynomial finite-element refinement with functions derived from known PDE structure, thereby bridging width-optimal spaces and stable, computable adaptive approximation. Projection removes represented content, the conditional Schur spectrum measures the remaining width, Riesz bounds certify coefficient stability, and an exact residual identity measures variational gain. Additionally, we develop non-regression and bulk-coverage results that retain the incumbent unless transfer clears an independent margin and otherwise certify contraction, finite-run error, dimension, and work. The practical rule is to compare h/p and operator-adapted refinements on the same scale: projection defines novelty, the Riesz bound rejects unstable blocks, and certified variational gain ranks those that remain. The method covers coercive Galerkin and noncoercive minimum-residual formulations. At matched dimension, automatic modes reduce held-out error from $.865$ to $.431\pm.039$ against a frozen expert Helmholtz incumbent and from $.476\pm.043$ to $.329\pm.035$ against an h/p/compositional Darcy incumbent. Against conforming hp-FEM at matched Darcy error, refinement uses 68 rather than 110 coordinates, reducing deployment memory by $38.9\%$ and online time by $30.1\%$, with $28.4\%$ greater offline cost. Controlled DPG studies test the noncoercive formulation, while an external MFEM 4.8 Maxwell comparison reproduces the qualitative behavior beyond the prototype assembly; the safe rule rejects every inferior fine-grid proposal. The resulting approach provides a width-guided, non-regressing route to constructing and testing operator-adapted approximation spaces.