Convergence and quasi-optimality of adaptive finite element methods for harmonic forms

Abstract

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.1991 Mathematics Subject Classification. 65N15, 65N30.

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

“…The approximation of harmonic functions is out of the aims of this paper. We point the reader to possible approaches for the approximation of H: a direct discretization of the space has been proposed in [1]; an adaptive algorithm has been presented in [25]; another indirect approach may be the use of the following alternative mixed formulation known as Kikuchi formulation (see [31,6]): find λ ∈ R such that for u ∈ H 0 (curl; Ω) and p ∈ H 1 0 (Ω), with u = 0, it holds…”

Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors

“…First, the a posteriori error analysis hinges on delicate decomposition results and possibly bounded commuting quasi-interpolations onto a sequence of finite elements spaces, see, e.g., [1,39,22]. In addition, those quasi-interpolations are even required to locally preserve finite element functions when deriving discrete reliability, see, e.g., [21,46]. Second, the exact solution U of a system of equations is generally only a critical point of some variational principle.…”

“…At the same time, Falk and Winther [24] constructed a technical local bounded commuting interpolation connecting the de Rham complex (2.10) and its finite element subcomplex. Using these ingredients, [21,19,30,27] recently developed quasi-optimal AMFEMs for problems posed on the de Rham complex. For the Hodge Laplace equation, we [30] developed an AMFEM for controlling σ − σ V with quasi-optimal convergence rate and another AMFEM for controlling σ −σ V + p−p + d(u−u ) without convergence rate.…”

Some Convergence and Optimality Results of Adaptive Mixed Methods in Finite Element Exterior Calculus

“…To localize the upper bound, we use the local bounded cochain projection developed by Falk and Winther [19] and the classical Scott-Zhang interpolation [33]. Demlow (see Lemma 3 in [16]) originally used this technique to prove a localized upper bound in the AFEM for computing harmonic forms.…”

“…To prove the quasi-optimality of AMFEM2, we need a localized upper bound, which can be viewed as a discrete and local version of the global upper bound η σ (T h ) for σ − σ h HΛ k−1 . Following [16], we apply the locally defined V -bounded cochain projectionπ h given by Falk and Winther [19]. Let T H be a refinement of T h and R H = R T H →T h be the set of refined elements in T H .…”