Nodal auxiliary a posteriori error estimates
Yuwen Li, Ludmil T. Zikatanov
TL;DR
The paper develops a unified, operator-theoretic route to a posteriori error estimation by linking adaptive error control with subspace correction preconditioning in the FEEC setting. Central to the approach is a nodal auxiliary space preconditioner that reduces complex $H(\mathrm{curl})$, $H(\mathrm{div})$, and general $H(\mathrm{d})$ problems to simpler $H^1$-type or Poisson-like subproblems, yielding two-sided, robust error bounds via computable residual indicators. It constructs regular and robust decompositions of de Rham-type spaces, introduces weighted norms for singularly perturbed regimes, and develops both explicit and implicit estimators based on local Dirichlet problems and Riesz representations. The framework applies to curl-curl, grad-div, Hodge Laplacian, and linear elasticity with weak symmetry, including parameter-insensitive estimators and practical resolvent-based residuals. Overall, it provides a comprehensive methodology for reliable a posteriori error control and robust adaptivity across a broad class of PDEs on de Rham complexes.
Abstract
We introduce and explain key relations between a posteriori error estimates and subspace correction methods viewed as preconditioners for problems in infinite dimensional Hilbert spaces. We set the stage using the Finite Element Exterior Calculus and Nodal Auxiliary Space Preconditioning. This framework provides a systematic way to derive explicit residual estimators and estimators based on local problems which are upper and lower bounds of the true error. We show the applications to discretizations of $δd$, curl-curl, grad-div, Hodge Laplacian problems, and linear elasticity with weak symmetry. We also provide a new regular decomposition for singularly perturbed H(d) norms and parameter-independent error estimators. The only ingredients needed are: well-posedness of the problem and the existence of regular decomposition on continuous level.