Convergence and quasi-optimality of adaptive finite element methods for harmonic forms
Provides theoretical guarantees for adaptive finite element methods in computing harmonic forms, benefiting researchers in computational electromagnetics and computer graphics.
This work proves that an adaptive finite element method for computing harmonic forms is contractive and achieves optimal convergence rate from any initial mesh, unlike elliptic eigenvalue problems which require a sufficiently fine initial mesh.
Numerical computation of harmonic forms (typically called harmonic fields in three space dimensions) arises in various areas, including computer graphics and computational electromagnetics. The finite element exterior calculus framework also relies extensively on accurate computation of harmonic forms. In this work we study the convergence properties of adaptive finite element methods (AFEM) for computing harmonic forms. We show that a properly defined AFEM is contractive and achieves optimal convergence rate beginning from any initial conforming mesh. This result is contrasted with related AFEM convergence results for elliptic eigenvalue problems, where the initial mesh must be sufficiently fine in order for AFEM to achieve any provable convergence rate.