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

Transcription

1 Noname manuscript No. (wi be inserted by the editor) Convergence and quasi-optimaity of adaptive finite eement methods for harmonic forms Aan Demow 1 the date of receipt and acceptance shoud be inserted ater Abstract Numerica computation of harmonic forms (typicay caed harmonic fieds in three space dimensions) arises in various areas, incuding computer graphics and computationa eectromagnetics. The finite eement exterior cacuus framework aso reies extensivey on accurate computation of harmonic forms. In this work we study the convergence properties of adaptive finite eement methods (AFEM) for computing harmonic forms. We show that a propery defined AFEM is contractive and achieves optima convergence rate beginning from any initia conforming mesh. This resut is contrasted with reated AFEM convergence resuts for eiptic eigenvaue probems, where the initia mesh must be sufficienty fine in order for AFEM to achieve any provabe convergence rate. Keywords Finite eement methods, exterior cacuus, a posteriori error estimates, adaptivity, harmonic forms, harmonic fieds Mathematics Subject Cassification (2000) 65N15, 65N30 1 Introduction In this paper we prove convergence and rate-optimaity of adaptive finite eement methods (AFEM) for computing harmonic forms. Let Ω R 3 be a poyhedra domain with Lipschitz boundary Ω. The space H 1 of harmonic forms is H 1 = {v L 2 (Ω) 3 cur v = 0, div(v) = 0, v n = 0 on Ω}. (1.1) Partiay supported by Nationa Science Foundation grant DMS A. Demow Department of Mathematics, Texas A&M University, Coege Station, TX E-mai: demow@math.tamu.edu

2 2 A. DEMLOW Here n is the outward unit norma on Ω. β 1 := dim H 1 < is equa to the number of handes of the domain Ω, so β 1 = 0 if Ω is simpy connected. We denote by {q 1,..., q β1 } a fixed L 2 -orthonorma basis for H 1. Computation of harmonic forms arises in appications incuding computer graphics and surface processing [35], [18] and numerica soution of PDE reated to probems having Hodge-Lapace structure posed on domains with nontrivia topoogy. Important practica exampes of the atter type arise in boundary and finite eement methods for eectromagnetic probems. There computation of harmonic fieds may arise as a necessary precursor step which in essence factors nontrivia topoogy out of subsequent computations. We refer to [22], [23], [1], [28], [34], [5] for genera discussion, topoogicay motivated agorithms for efficient computation of H 1, and specific appications requiring such a step. In addition, harmonic fieds on spherica subdomains pay an important roe in understanding singuarity structure of soutions to Maxwe s equations on poyhedra domains [11, Section 6.4]. Computation of harmonic forms is aso an important part of the finite eement exterior cacuus (FEEC) framework. The FEEC framework provides a systematic exposition of the toos needed to staby sove Hodge-Lapace probems reated to the de Rham compex and other differentia compexes [2], [3]. It thus has provided a broader exposition of many of the toos needed for the numerica anaysis of Maxwe s equations. The mixed methods used in these works sove the Hodge-Lapace probem moduo harmonic forms, so as above their accurate computation is a prerequisite for accurate computation of Hodge-Lapace soutions. This is true of adaptive computation of soutions to Hodge-Lapace probems aso [15], [26]. Our resuts thus serve as a necessary precursor to study of convergence of AFEM for soving the fu Hodge-Lapace probem as posed within the FEEC framework. Whie keeping in mind the concrete representation (1.1), outside of the introduction we wi use the more genera notation and toos that have been deveoped within the FEEC framework. The resuts described in the introduction beow are vaid essentiay verbatim in arbitrary space dimension n and for arbitrary cohomoogy group H k, 1 k n 1. We briefy describe our setting. Let T, 0, be a set of nested, adaptivey generated simpicia decompositions of Ω. Let aso V 0 H 1 (Ω) and V 1 H(cur; Ω) be conforming finite eement spaces which together satisfy a compex property described in more detai beow. For (1.1) we may take V 0 to be H1 -conforming piecewise inear functions and V 1 to be owest-order Nédéec edge eements. Higher-degree anaogues aso may be used. The space H 1 of discrete harmonic fieds corresponding to H1 is then those fieds q V 1 satisfying q, τ = 0, τ V 0, cur q, cur v = 0, v V 1. (1.2) Note that whie cur q = 0, q is not generay in H(div; Ω). The first equation in (1.2) instead impies that div h q = 0, where div h is a weaky defined discrete divergence operator which is not generay the same as the restriction of div to

3 AFEM for harmonic forms 3 V 1. We denote by {qj }β1 a (computed) orthonorma basis for H1. Let aso P H 1 be the L 2 projection onto H 1. Given q H1, a posteriori error estimates for controing q P H 1 q L2(Ω) were given in [15]. From these we may generate an adaptive finite eement method having the standard form sove estimate mark refine for controing the defect between H 1 and H 1. There are important technica and conceptua paraes between AFEM for controing harmonic forms and AFEM for eiptic eigenvaue probems, as both consider approximation of finite-dimensiona (invariant) subspaces. Consider the eigenvaue probem u = λu with Dirichet boundary conditions. Assume that λ is the j-th eigenvaue of the continuous probem. AFEM empoying residua-type error indicators based on the j-th discrete eigenvaue λ on the -th mesh eve yied an approximating sequence (λ, u ) for the j-th continuous eigenpair (λ, u). Anayses of AFEM for eiptic eigenvaue probems have focused on two convergence regimes. The first is a preasymptotic regime in which the method converges, but with no provabe rate. In [20] it was proved that starting from any initia mesh T 0, (λ, u ) (λ, u) in R H 1 0 (Ω), but with no guaranteed convergence rate. In the second convergence regime the eigenvaue probem is effectivey inearized, and the convergence behavior is that of a source probem. More precisey, it the initia mesh T 0 is sufficienty fine, then the AFEM is contractive and achieves an optima convergence rate [13], [21], [12], [19], [6]. The pain convergence resuts of [20] guarantee that such a state is initiay reached from any initia mesh T 0. Our resuts beow indicate that the convergence behavior of harmonic forms essentiay begins in a transition region in which the AFEM contracts, but with contraction constant that may improve as the overap between H 1 and H 1 increases. A regime in which AFEM contracts with constants independent of essentia quantities is then eventuay reached, as for AFEM for eigenvaue probems. It is to be expected that AFEM for harmonic forms contract from any initia mesh as computation of harmonic forms reduces to soving Au = 0 with A a inear operator. That is, harmonic forms are eigenfunctions with known eigenvaue and their computation is essentiay a inear probem. On the other hand, we have set as our main goa the production of an orthonorma basis for H 1 and do not assume any particuar aignment or method of production for the basis. The probem of producing an orthonorma basis is midy noninear, so it is reasonabe that the contraction constant can improve as the mesh is refined. As we discuss in 7.4, aternate methods for producing and aigning the discrete and continuous bases may ead to AFEM with different properties. Our framework has the advantage of being competey generic with respect to the method used to compute H 1. There are two main chaenges in proving contraction of AFEM for computing eigenspaces. These are ack of orthogonaity caused by non-nestedness of the discrete eigenspaces, and ack of aignment of computed eigenbases at adjacent discrete eves. In the context of eiptic eigenvaue probems, suitabe nestedness and orthogonaity is recovered ony on sufficienty fine meshes as the noninearity of the probem is resoved. In the case of harmonic forms a nove

4 4 A. DEMLOW situation arises. The discrete spaces {H 1 } of harmonic forms are not themseves nested. However, as we show beow the Hodge decomposition nonetheess guarantees sufficient nestedness and orthogonaity uniformy starting from any initia mesh. In essence, topoogica resoution of Ω by T 0 is sufficient to ensure some anaytica resoution of H 1 by H 1 0. Lack of aignment between discrete bases at adjacent mesh eves occurs in the case of muti-dimensiona target subspaces, incuding mutipe or custered eigenvaues and harmonic forms when β > 1. Standard AFEM convergence proofs empoy indicator continuity arguments which in our case woud require comparing discrete basis members q j and qj +1 on adjacent mesh eves. When β 1 > 1, q j and qj +1 may not be meaningfuy reated. To overcome the difficuty of mutipe eigenvaues, we foow [12], [19], [6] in using a noncomputabe error estimator µ cacuated with respect to projections P q j of a fixed basis for the continuous harmonic forms. Indicator continuity arguments appy to these theoretica error indicators, which must in turn be compared to the practica ones based on {q j }. We foow [6] in estabishing an equivaence between the theoretica and practica indicators with constants asymptoticay independent of essentia quantities. We now briefy describe our resuts. We first show that there exists γ > 0 and 0 < ρ < 1 such that β 1 β 1 q j P +1 q j 2 + γµ 2 +1 ρ q j P q j 2 + γµ 2. (1.3) Whie we may take γ, ρ above independent of mesh eve, our proof indicates that ρ may in fact improve (decrease) as the overap between H 1 and H 1 improves. Because a contraction occurs from the initia mesh with fixed constant ρ < 1, this improvement in overap is guaranteed to occur. Thus our resut ies between standard resuts for eiptic source probems in which contraction occurs from the initia mesh with fixed contraction constant, and AFEM convergence resuts for eiptic eigenvaue probems for which sufficient overap between discrete and continuous eigenspaces is not guaranteed uness the initia mesh is sufficienty fine. We additionay prove rate-optimaity, or more precisey that β 1 q j P q j 2 1/2 C(#T #T 0 ) s (1.4) whenever systematic bisection is abe to produce a sequence of meshes that simiary approximates H 1 with rate s. As is typica for AFEM resuts, our proof of (1.4) requires that the Dörfer (buk) marking parameter that specifies the fraction of eements to be refined in each step of the agorithm be sufficienty sma. We show that the threshod vaue for the Dörfer parameter is independent of a essentia quantities, incuding the initia resoution of H 1 by H 1. This situation is typica of eiptic source probems; cf. [7]. In contrast to

5 AFEM for harmonic forms 5 eiptic source probems, however, the constant C may depend on the quaity of approximation of H 1 by H 1. Finay, proof of rate optimaity requires estabishing a ocaized a posteriori upper bound for the defect between the target spaces on nested meshes. This step is substantiay more invoved for genera probems of Hodge-Lapace type than for standard scaar probems. Proof of ocaized upper bounds has been previousy given for Maxwe s equations [36], but the necessary toos have not previousy appeared in a form suitabe for our purposes. Our approach beow is vaid for arbitrary form degree and space dimension and genericay for probems of Hodge-Lapace type. In addition to being more genera, our proof is aso sighty simper due to its use of the recenty-defined quasi-interpoant of Fak and Winther [17]. The remainder of the paper is organized as foows. In 2 we describe continuous and discrete spaces of differentia forms, interpoants into discrete spaces of differentia forms, and existing a posteriori estimates for harmonic forms. In 3 we give a number of preiminary resuts concerning the Hodge decomposition and L 2 projections described above. Section 5 and 6 contain statements, discussion, and proofs of our contraction and optimaity resuts. Section 7 contains brief discussion of essentia boundary conditions and harmonic forms in the presence of coefficients, and aso of the effects of methods for computing harmonic fieds on the resuting AFEM. Finay, in 8 we present numerica resuts iustrating our theory. 2 The de Rham compex and its finite eement approximation In this section we generaize the cassica function space setting from the introduction to arbitrary space dimension and form degree and introduce corresponding finite eement spaces and toos. We for the most part foow [3] in our notation and refer to that work for more detai. 2.1 The de Rham compex Let Ω be a bounded Lipschitz poyhedra domain in R n, n 2. Let Λ k (Ω) represent the space of smooth k-forms on Ω. The natura L 2 inner product is denoted by,, the L 2 norm by, and the corresponding space by L 2 Λ k (Ω). We et d be the exterior derivative, and HΛ k (Ω) be the domain of d k consisting of L 2 forms ω for which dω L 2 Λ k+1 (Ω). We denote by H the associated graph norm; here one may concretey think of H as H(cur), H(div), or H 1. We denote by W r p Λ k (Ω) the corresponding Soboev spaces of forms and set H r Λ k (Ω) = W r 2 Λ k (Ω). Finay, for ω R n, we et ω = L2Λ k (ω) and H,ω = HΛ k (ω); in both cases we omit ω when ω = Ω. Denote by δ the codifferentia, that is, the adjoint of the exterior derivative d with respect to,.the space of harmonic k-forms is given by H k = {q HΛ k (Ω) : dq = 0, δq = 0, tr q = 0}. (2.1)

6 6 A. DEMLOW Here tr is the trace operator and the Hodge star operator. We denote by β k = dim H k (2.2) the k-th Betti number of Ω. When n = 3, β 0 = 1, β 1 is the number of hoes in Ω, β 2 the number of voids, and β 3 = 0. In genera, β <. We additionay et B k = dhλ k 1 be the range of d k 1, Z k = kerned k the kerne of d k, and by Z k the orthogona compement of Z k in HΛ k. We have Z k = H k B k, and the Hodge decomposition is given by HΛ k = B k H k Z k. Note that B k Z k, that is, d d = Meshes and mesh properties We empoy, and now briefy review, the conforming simpicia mesh refinement framework commony empoyed in AFEM convergence theory; cf. [33] for detais. Let T 0 be a conforming, shape reguar simpicia decomposition of Ω. By fixing a oca numbering of a vertices of a T T 0, a possibe descendants T of T 0 that can be created by newest vertex bisection are uniquey determined. By newest vertex bisection we mean either the refinement procedure as it was deveoped in two space dimensions or its generaization to any space dimension. The simpices in any of those partitions are uniformy shape reguar, dependent ony on the shape reguarity parameters of T 0 and the dimension n. Generay a descendant of T 0 is non-conforming. Our AFEM wi generate a nested sequence T 0 T 1 T 2... of conforming meshes which we index by. Given marked sets M T, conforming bisection thus refines a T M and then additiona eements in order to ensure that T +1 is conforming. With a suitabe numbering of the vertices in the initia partition, the tota number of refinements needed to make the sequence {T } conforming can be bounded by the number of marked eements. More precisey, for any 0, there is a constant C ref, such that for >, 1 #T #T C ref, #M i (2.3) with constant depending on T. Such an initia numbering aways exists, possiby after an initia uniform refinement of T 0. Assuming such a numbering, we denote the set of a conforming descendants T of T 0 by T. For T, T T, we write T T when T is a refinement of T. Finay, for T T T we et h T = T 1/n. i= 2.3 Approximation of the de Rham compex and interpoants Given T T, et...v k 1 T VT k V k+1 T... be an approximating subcompex of the de Rham compex with underying mesh T. That is, VT k HΛk (Ω), v VT k k 1 is poynomia on each T T, and dv VT k. When T = T we use T

7 AFEM for harmonic forms 7 the abbreviation V T = V (where here we aso depress the dependence on k). We refer to [3] for descriptions of the reevant spaces. In three space dimensions, we may think for exampe of standard Lagrange spaces approximating HΛ 0 = H 1, Nédeéc spaces approximating H(cur), and Raviart-Thomas or BDM spaces approximating H(div), with appropriate reationships between poynomia degrees enforced to ensure commutation properties. In addition to the subcompex property, we aso require the existence of a commuting interpoant (or more abstracty, commuting cochain projection) with certain properties. Desirabe properties incude commutativity, boundedness on the function spaces L 2 Λ and/or HΛ that are natura for our setting, oca definition of the operator, and projectivity, and a oca reguar decomposition propertyl. Severa recent papers have achieved constructions possessing some of these properties [29], [30], [9], [17], though no one operator has been shown to possess a of them. We use two different such constructions beow aong with a standard Scott-Zhang interpoant. First, the commuting projection operator π CW : L 2 Λ(Ω) V T of Christiansen and Winther [9] has the foowing usefu properties: d k π k CW = π k+1 CW dk, π 2 CW = π CW, π CW v L2(Ω) v L2(Ω) ω L 2 Λ(Ω). (2.4) The Christiansen-Winther interpoant aso can be modified to preserve homogeneous essentia boundary conditions with no change in its other properties. The commuting projection operator π F W defined by Fak and Winther in [17] has the foowing properties. First, it commutes: d k π k F W = π k+1 F W dk. (2.5) Second, given T T, there is a patch of eements surrounding T such that and for v HΛ(Ω), #(T ω T ) 1. (2.6) π F W v T v ωt + h T dπ F W v ωt, π F W v HΛ(T ) v HΛ(ωT ). (2.7) The patch ω T is not necessariy the standard patch of eements sharing a vertex with T, and its configuration may depend on form degree. The reationship (2.6) is however the essentia one for our purposes. Given v H 1 Λ(ω T ), there aso foows the interpoation estimate h 1 T v π F W v T + v π F W v H1 Λ(T ) v H1 Λ(ω T ). (2.8) Finay, π F W is a oca projection in the sense that u ωt V T ωt (u π F W u) T 0. (2.9) In summary, π F W and π CW are both commuting projection operators. π F W is however ocay defined, but ony stabe in HΛ, whie π CW is gobay defined but stabe in L 2 Λ. Aso, to our knowedge there is no detaied discussion in

8 8 A. DEMLOW the iterature of the modifications needed to construct a version of π F W which preserves essentia boundary conditions, athough a comment in [17] indicates that such an adaptation is natura. We sha aso empoy an L 2 -stabe Scott-Zhang operator I SZ : L 2 Λ k (Ω) ṼT k H1 Λ k (Ω) [31]. Here Ṽ T k is the smaest space of forms containing a forms with continuous piecewise inear coefficients. I SZ may be obtained by empoying the standard scaar Scott-Zhang interpoant coefficientwise. We then have for u L 2 Λ(Ω) I SZ u L2Λ(T ) u L2Λ(ω T ), h 1 T u I SZu L2Λ(T ) + u I SZ u H 1 Λ(T ) u H 1 Λ(ω T ). (2.10) We sha finay use a reguar decomposition property, which is given in [15, Lemma 5] (this emma is argey a reformuation of more genera resuts given in [27]). Given a form v HΛ k (Ω), there are ϕ H 1 Λ k 1 (Ω) and z H 1 Λ k (Ω) such that v = dϕ + z, ϕ H1 Λ k 1 (Ω) + z H1 Λ k (Ω) v HΛk (Ω). (2.11) 2.4 Discrete harmonic forms Let H k T V T k be the set of discrete harmonic forms, which is more precisey defined as those q T VT k satisfying q T, dτ T = 0, τ T V k 1 T, dq T, dv T = 0, v T V k T. (2.12) The discrete Hodge decomposition VT k = Bk T Hk T Zk, T (Z k T = Bk T H k T = kern d V ) is defined entirey anaogousy to the continuous version. T k Note however that whie there hods Z k T Z k and B k T B k, (2.13) H k T H k and Z k, T Z k,. (2.14) We briefy summarize the index system we use. The integer n is the dimension of the domain Ω. The integer k, 0 k n, is the order of differentia form. The subscript, = 0, 1, 2... is used for a sequence of trianguations of the domain Ω and the corresponding finite eement spaces. For a fixed k, 0 k n, we consider the convergence of the sequence {q, = 0, 1, 2,...}. Therefore we may suppress the index k when no confusion can arise. j is used to index the bases of H and H T.

9 AFEM for harmonic forms 9 3 Properties of an L 2 Projection We first specify the error quantity that we seek to contro. Let {q j } β be an orthonorma basis for H, and et {q j T }β be an orthonorma basis for H T. Denote by P the L 2 projection onto H and by P T = P HT the L 2 projection onto H T. As above we use the abbreviation P = P T and simiary for q j. We seek to contro the subspace defect Proposition 1 E T := q j P T q j 2 q j P T q j 2 1/2 1/2 = q j T P qj T 2. (3.1) 1/2. (3.2) Proof There hods P T q j = β m=1 qj, qt m qm T and P T q j 2 = β m=1 qj, qt m 2, and simiary P qt m 2 = β qm T, qj 2. Aso, q j P T q j 2 = q j P T q j, q j P T q j = 1 P T q j 2 and simiary for qt m P qm T 2. Thus ( ) q j P T q j 2 = 1 q j, qt m 2 = qt m P qt m 2. (3.3) m=1 m=1 Previous error estimates for errors in harmonic forms have instead controed the gap between H and H. Given two finite-dimensiona subspaces A, B of the same ambient Hibert space with dim A = dim B, we define δ(a, B) = sup x P B x, gap(a, B) = max{δ(a, B), δ(b, A)}. (3.4) x A, x =1 There aso hods in this case Let P ZT be the L 2 projection onto Z T. Lemma 1 δ(a, B) = δ(b, A). (3.5) P T q = P ZT q, for a q H. (3.6) Proof Because Z T = B T H T, we have for q H that P ZT q = P BT q + P HT q. But B T B and H B, so P BT q = 0. Proposition 2 Given T T and q H, we have P T q P T q. In particuar, P q P +1 q. Proof For q H, P T q = P ZT q. Noting that Z T Z T competes the proof.

10 10 A. DEMLOW We next show that there is overap between H and H on any conforming mesh T ; cf. [26, Theorem 2.7]. Proposition 3 The projection P is injective. Namey, given q H and q 0, P q 0. Proof Let π be a commuting projection operator (either π F W or π CW wi suffice). Suppose there exists a 0 q H with P q = 0. By (3.5), δ(h, H ) = 1 = δ(h, H), so there is a non-zero form q H with P H q = 0. Note that H Z Z = B H, so P H q = 0 impies q B. That is, q = dv for some v V k 1. But q = π k q = π k dv = dπ k 1 v B. But B H impies that q = 0, which is a contradiction. Combining the above two propositions yieds the foowing emma. Lemma 2 P : H H is an isomorphism. In addition, P 1 P0 1 <. Proof Since β = β < and P is injective, we concude P is an isomorphism. Consider the constant c 0 = inf q H, q =1 P 0 q, which is positive because it is the infimum over a compact set of a continuous and positive function. Then P0 1 c 1 0. By Proposition 2, P 0q P 1 q P 2 q..., and so we have P P... P0 1 c A posteriori error estimates 4.1 Previous estimates Our first goa is to contro the error quantity E defined in (3.1), which is a measure of the distance between H and H. We foow [15], where a posteriori error estimates were given for measuring the gap between H and H. Let T be the mesh on eve, and et h T = T 1/n for T T. Given q H, et and η (T ) 2 = η (T ; q ) = h T δ T q T + h 1/2 T tr q T (4.1) ( ) 1/2 η (T ; q j )2, η (T ) = η (T ) 2, T T. (4.2) T T Here denotes the jump in the given quantity across eement interfaces. In the concrete case of the cassica space given in (1.1), we have η (T ; q ) = h T div q T + h 1/2 T q n T. A sight modification of Lemma 9 and Lemma 13 of [15] yieds η (T ) C 1 E η (T ). (4.3)

11 AFEM for harmonic forms 11 with C 1 and the constant hidden in depending on the shape reguarity properties of T 0, but independent of essentia quantities incuding especiay the dimension β of H and H. From [15] we aso have gap(h, H ) η (T ) βgap(h, H ). (4.4) Empoying the error notion E thus aows us to obtain estimates with entirey nonessentia constants. 4.2 Non-computabe error estimators Convergence of adaptive FEM for mutipe and custered eigenvaues has been studied in [12], [19], [6]. Our probem is simiar in that our AFEM approximates a subspace rather than a singe function. The estimators defined above with respect to {q j } are probematic when viewed from the standpoint of standard proofs of AFEM contraction, which require continuity between estimators at adjacent mesh eves. Because the bases {q j } and {qj +1 } are not generay aigned, such continuity resuts are not meaningfu. Foowing [12], we empoy a non-computabe intermediate estimator which soves this aignment probem and is equivaent to η (T ). Let {q j } β be a fixed orthonorma basis for H. We define µ (T ) 2 = ( ) 1/2 η (T ; P q j ) 2, µ (T ) = µ (T ) 2, T T. (4.5) We next estabish that approximation of H is sufficienty good on the initia mesh to guarantee that the estimators η (T ) and µ (T ) are equivaent. Theorem 1 µ (T ) P η (T ) P P 1 µ (T ) P 1 µ (T ), T T, 0. Proof Reca that µ is defined using {P q j } with {q j } β a fixed orthonorma basis for H and η is defined using the orthonorma basis {q j }β. Let q = (q 1, q 2,, q β ) T and q = (q 1, q2,, qβ )T. The matrix [ q i, q j ] =: M : Rβ R β satisfies P q = Mq. Foowing the proof of [6, Lemma 3.1], et B := MM T = [ P q i, P q j ]. MM T and M T M are isospectra and thus have the same 2-norm and M T 2 = M 2 = M T M = B. We thus compute M T. Given v R β with v = 1, et ṽ = β v jq j so that ṽ H with ṽ = 1. Then M T v = [ β qi, v jq j ] = [ q i, ṽ ] = P ṽ P. Thus M = M T P T T

12 12 A. DEMLOW 1. Simiary, M 1 = (M T ) 1 = = sup v R β, v =1 = P 1. sup y R β \0 1 M T v = M T y y ( M T M T v = sup v R β \0 M T v ( 1 inf M v ) T = v R β, v =1 Thus M 1 = P 1. As δ is inear and commutes with M, we have Simiary δ T P q j 2 T = (Mδ T q ) j 2 T M 2 P 2 δ T q 2 T M 1 2 =1 =1 δ T q j 2 T. δ T P q 2 T P 1 2 Simiar inequaities hod for the boundary term. inf P ṽ ṽ H, ṽ =1 δ T q j 2 T δ T P q 2 T. Theorem 1 and (4.3) immediatey impy an a posteriori bound using µ. Coroary 1 =1 ) 1 E 2 C 1 P 1 2 µ (T ) 2 C 1 P 1 2 µ (T ) 2, 0, (4.6) where C 1 is independent of essentia quantities. 4.3 Locaized a posteriori estimates As is common in AFEM optimaity proofs, we require a ocaized upper bound for the difference between discrete soutions on nested meshes. More precisey, et R T T be the set of eements refined in passing from T to T. A standard estimate for eiptic probems with finite eement soutions u and ũ on T, T is u ũ ( T R ξ(t ) 2 ) 1/2, where ξ(t ) is a standard eiptic residua T T indicator and is the energy norm. The estimate we prove is not as sharp but suffices for our purposes.

13 AFEM for harmonic forms 13 Lemma 3 Let T be a refinement of T so that V V T exists a set ˆR T T with R T T ˆR T T and HΛ. Then there # ˆR T T #R T T (4.7) such that (P P T )q j 2 C2 2 T ˆR T T where C 2 is independent of essentia quantities. η (T ) 2, (4.8) Proof Foowing the notation used in the proof of Theorem 1, denote by q, q, and q T the coumn vectors of orthonorma basis functions for H, H, and H T, respectivey. Let M = (q i, q j T ) be the matrix satisfying P T q = Mq T ; reca that M P T 1, with M the matrix (operator) 2-norm. Reca aso from (3.6) that P q j = P Z q j for harmonic forms q j H, so that P q = P Z q and P T q = P ZT q. Because V V T and Z Z T we aso have P Z q T = P q T, q T H T. Thus P q = P P T q. We then compute (P P T )q j 2 = (P P T )q 2 = (P P T P T )q 2 = P Mq T Mq T 2 = M(P q T q T ) 2 M 2 P q T q T 2 q P T q 2, (4.9) where in the ast inequaity above we empoy M 1 aong with (3.2). Foowing [15, (2.12) and Lemma 9], we note that q j P T q j B T H T, and q j B. Thus for 1 j β and some v T V k 1 T with v T HΛ(Ω) 1, q j P T q j sup q j P T q j, dv qj P T q j, v T. v V k 1 T, v HΛ(Ω) =1 (4.10) We next appy the reguar decomposition (2.11) to find z H 1 Λ k 1 (Ω) with dz = dv T (note that ϕ as in (2.11) pays no roe here since v T = dϕ + z impies dv T = dz). We now denote by π T and π the Fak-Winther interpoants π F W on T and T, respectivey. The commutativity of π T impies that dv T = π T dv T = dπ T z, so that using orthogonaity properties as above we have q j P T q j qj, dπ T z = q j, d(i π )π T z. (4.11) In addition, by (2.9) we have for some ˆR T T satisfying (4.7) supp(d(i π )π T z) T T. (4.12) ˆRT T

14 14 A. DEMLOW Integrating by parts eementwise the ast inner product in (4.11) and carrying out standard manipuations yieds q j P T q j T ˆR T T η (T ; q j ) (h 1 T (I π )π T z T + h 1/2 T (I π )π T z T ). (4.13) A standard scaed trace inequaity may be appied to the term d(i π )π T z T ony if extra care is taken. We first write (I π )π T z = [π T z I SZ z] + [I SZ z π I SZ z] + [π (I SZ z π T z)] := I + II + III, where I SZ is the Scott-Zhang interpoant on T. For T T, we appy a standard scaed trace inequaity v 2 L 2( T ) h 1 T v T + h T v H1 (T ), an inverse inequaity, and the approximation properties (2.8) and (2.10) to find h 1 T I 2 T T T,T T T T,T T z 2 H 1 (ω T ), (h T h T ) 1 I 2 T (h T h T ) 1 ( z π T z T + z I SZ z T ) 2 (4.14) where we have used the fact that h T h T when T T as above. Next, III V k 1, so we appy the trace inequaity on T T, the stabiity estimate (2.7) and an inverse inequaity, and then approximation properties as before to find h 1 T III 2 T h 2 T π (π T I SZ )z 2 T (h T h T ) 1 (π T I SZ z) 2 ω T T T (4.15) z 2 H 1 Λ(ω T ) z 2 H 1 Λ(ω T ). T T Here we et ω T = T ωt ω T. Finay, we have II H 1 Λ(T ), T T, so we directy appy a scaed trace inequaity, approximation properties of π, and H 1 stabiity of I SZ on T to find that Simiar arguments yied h 1 T III 2 T z 2 H 1 Λ(ω T ). (4.16) h 2 T (I π )π T z L2(T ) z 2 H 1 Λ(ω T ). (4.17) Inserting the previous inequaities into (4.13), appying the Cauchy-Schwartz inequaity over T, empoying finite overap of the patches ω T, and finay using

15 AFEM for harmonic forms 15 (2.11) aong with v T HΛ(Ω) 1, we find that q j P T q j ( η (T ; q j )2 ) 1/2 z H 1 (Ω) ( ( T ˆR T T T ˆR T T η (T ; q j )2 ) 1/2 v HΛ(Ω) η (T ; q j )2 ) 1/2. (4.18) Summing over j yieds (4.8). T ˆR T T Remark 1 In [36], the authors empoy the oca reguar decomposition resuts of [30], severa different quasi-interpoants, and a discrete reguar decomposition as in [24] to prove a ocaized a posteriori upper bound for time-harmonic Maxwe s equations. We do not need a oca reguar decomposition here, but rather combine the more powerfu interpoant recenty defined in [17], the Scott-Zhang interpoant, a goba reguar decomposition, and some ideas reated to discrete reguar decompositions as in [24] in order to prove our oca upper bound. Remark 2 Proof of a posteriori bounds for harmonic forms is sighty simper than for genera probems of Hodge-Lapace type. Our technique for proof of ocaized a posteriori bounds does however carry over to the more genera case. Proof of such upper bounds requires testing various terms with v T V T and then subtracting π v T via Gaerkin orthogonaity. Empoying a goba reguar decomposition yieds v T = dϕ + z, and subsequenty v T = π T v T = dπ T ϕ + π T z. Subtracting off π v T via Gaerkin orthogonaity yieds v T π v T = d[(i π )π T ϕ]+(i π )π T z. Proceeding as above using a combination of standard residua estimation toos and the Scott-Zhang interpoant wi genericay ead to ocaized upper bounds. It is crucia that each of the terms d[(i π )π T ϕ] and (I π )π T z is individuay ocay supported. The Fak- Winther interpoant pays a critica roe as a of its major properties are empoyed in the proof. 5 Contraction We empoy a standard adaptive finite eement method of the form sove estimate mark refine. (5.1) Our contraction proof foows the standard outine [7] in that it combines an a posteriori error estimate, an estimator reduction property reying on properties of the marking scheme, and an orthogonaity resut in order to estabish stepwise contraction of a propery defined error notion.

16 16 A. DEMLOW In the mark step we use a buk (Dörfer) marking. That is, we fix 0 < θ 1 and choose a minima set M T such that η (M ) θη (T ). (5.2) The foowing is an easy consequence of Theorem 1 and P 1 P0 1. Proposition 4 Let 0 < θ 1, and et M T satisfy (5.2). Let aso θ = θ( P P 1 ) 2 and θ = θ P 1 2. Then for 0 and 0, µ (M ) 2 θ µ (T ) 2 θ µ (T ) 2. (5.3) Remark 3 Note that P P 1 = 1 when β = 1, and for β > 1 the deviation of P P 1 from 1 is dependent on the isotropy of P. Thus if P q i P q j for 1 i, j β, then P P 1 1 even if H and H do not overap strongy. If β = 1 or P is isotropic, the theoretica and practica AFEM s wi therefore mark the same eements for refinement. We next estabish continuity of the theoretica indicators (aso known as an estimator reduction property). The proof is standard and is thus omitted. Lemma 4 Given 0 < θ 1, et M T satisfy (5.2). Assume aso that each T M is bisected at east once in passing from T to T +1. Then there are constants C 2 and 1 > λ > 0 independent of essentia quantities such that for any α > 0, any 0, and any 0, µ +1 (T +1 ) 2 (1 + α)(1 λ θ )µ (T ) 2 + C 2 (1 + 1 α ) (P P +1 )q j 2 (1 + α)(1 λ θ )µ (T ) 2 + C 2 (1 + 1 α ) (P P +1 )q j 2. Here θ and θ are as defined in Proposition 4. (5.4) Athough H H +1, the Hodge decomposition and compex-conforming structure of the finite eement spaces nonetheess yieds the foowing essentia orthogonaity resut. Theorem 2 For q H, q P +1 q 2 = q P q 2 (P P +1 )q 2. (5.5) Proof It suffices to prove (q P +1 q, (P P +1 )q) = 0. This is a consequence of (P P +1 )q Z +1, which hods due to the nestedness Z Z +1 and the fact that P : H H aso acts as the L 2 projection from H to Z. Assembing the above estimates yieds the foowing contraction resut. The proof is standard, except we must track dependence of constants on the mesh eve.

17 AFEM for harmonic forms 17 Theorem 3 For each > 0, there exist 0 < ρ < 1 and γ > 0 such that for, E γ µ +1 (T +1 ) 2 ρ ( E 2 + γ µ (T ) 2). (5.6) Here 1 > ρ 0 ρ 1... ρ := im ρ > 0. ρ depends on P 1 but not on other essentia quantities, and ρ is independent of essentia quantities. Finay, 0 < γ < C2 1 with C 2 as in (5.4). 1 Proof Given 0 and α as in (5.4), et γ = C 2(1+α 1 ). (Here we suppress the dependence of α on.) Then 0 < γ < C2 1, as asserted. Appying (5.5) to q j (j = 1,..., β) and summing the resuting equaities, then adding the resut to (5.4) mutipied through by γ, yieds E γ µ 2 +1 E 2 + γ (1 + α)(1 λ θ )µ 2. (5.7) Let now 0 < ρ < 1. From (4.6) we have (1 ρ )E 2 C 1(1 ρ ) P 1 2 µ 2, so that [ E γ µ 2 +1 ρ E 2 + γ (1 + α)(1 λ θ ) + γ 1 C 1 (1 ρ ) P 1 2] µ 2. (5.8) We next set ρ = (1+α)(1 λ θ )+γ 1 C 1 (1 ρ ) P 1 2 and sove for ρ. Before doing so, we introduce the shorthand K = 1 λ θ and M = C 1 C 2 P 1 2. Then ρ = α2 K + α(k + M ) + M α(1 + M ) + M. (5.9) Recaing that α > 0 is arbitrary, we minimize the above expression with respect to α to obtain ρ = 2 M K (1 + M K ) + M 2 + M + K K M (1 + M ) 2. (5.10) We now anayze the dependence of ρ on. Reca that M = C 1 C 2 P 1 2 and K = 1 λθ P 1 2 are decreasing in. Tedious but eementary cacuations aso show that for 0 < K < 1 and M > 0, ρ is increasing in both M and K. Thus (5.6) hods for a 0 with ρ = ρ 0. We in turn see that P, P 1 1 as, so M C 1 C 2 := M. In addition, K 1 λθ := K as. Thus ρ ρ 0 and ρ decreases to ρ = 2 C 1 C 1 (1 λθ)(c 1 C 2 + λθ) + C 2 1C C 1 C 2 + (1 λθ)(1 C 1 C 2 ) (1 + C 1 C 2 ) 2 (5.11) as. This competes the proof.

18 18 A. DEMLOW Remark 4 In Remark 3 we estabished that the theoretica and practica AFEM mark the same eements for refinement when P is isotropic, and in particuar when β = 1. We coud sharpen the proof of Theorem 3 to take advantage of this fact by redefining K = 1 λ θ = 1 λθ( P 1 P ) 2. However, doing so woud compromise monotonicity of the sequence {ρ }, and M depends on and not on the product P 1 P in any case. P 1 Remark 5 Dependence of ρ on P 1 seems unavoidabe in our proofs. We prove a contraction by combining the orthogonaity reationship (5.5), the estimator reduction inequaity (5.4), and the a posteriori estimate (4.6) in a canonica way [7]. The orthogonaity reationship (5.5) indicates that the error E 2 is reduced by β (P +1 P )q j 2 at each step of the AFEM agorithm. This quantity is directy reated to the theoretica indicators µ in the estimator reduction inequaity (5.4). Combining these reationships eads to an error reduction on the order of µ (T ). On the other hand, E is uniformy equivaent to the practica estimator η. Because the theoretica and practica estimators are reated by P 1 1 η µ P η, reducing E by µ (T ) is 1 equivaent to a reduction ying between E P 1 and P E. Remark 6 For eiptic source probems, a contraction is obtained from the initia mesh with contraction factor independent of essentia quantities [7]. On the other hand, for eiptic eigenvaue probems such a contraction resut hods ony if the initia mesh is sufficienty fine [19], [6]. The situation for harmonic forms is intermediate between those encountered in source probems and eigenvaue probems. No initia mesh fineness assumption is needed to guarantee a contraction, but we ony show that the contraction constant is independent of essentia quantities on sufficienty fine meshes. In the case of eigenvaue probems a simiar transition state ikey exists in which AFEM can be proved to be contractive, but with contraction constant improving as resoution of the target invariant space improves. 6 Quasi-Optimaity Our proof of quasi-optimaity foows a more or ess standard outine, simpified somewhat by ack of data osciation but made more compicated by improvement in the contraction factor as the mesh is refined. 6.1 Approximation casses Given rate s > 0 and r [H] β, we et r As = sup r j P T r j 2 inf N s N>0 #T #T 0 N 1/2. (6.1)

19 AFEM for harmonic forms 19 A s is then the cass of a r [H] β such that r As <. If we appied A s to arbitrary r [HΛ(Ω)] β, it woud be natura to repace the projection P T onto H T used in (6.1) by the L 2 projection onto the fu discrete space V T. We show in Proposition 5 that best approximation over the fu finite eement space is equivaent up to a constant to best approximation over H T ony. Thus our definition of A s makes use of the fu approximation strength of the finite eement space V T, even though at first gance it may not seem that this is the case. Proposition 5 Given T T and q H, q P T q inf q q T. (6.2) q T V T Proof From (27) of [3], we have q P HT q (I π T )q with π T a commuting cochain projection. Thus for q H, q T V T, we may use (2.4) to obtain q P T q (q q T ) π CW (q q T ) (1 + C) q q T, (6.3) where C is the L 2 stabiity constant for π CW. 6.2 Rate optimaity We first state and prove two emmas which are more or ess standard in this context. It is important, however, that the constants in these two emmas are entirey independent of essentia quantities. Lemma 5 Let C 1 and C 2 be the constants from (4.3) and (4.8), respectivey, and assume that θ < 1 C 1C 2. Then for T T with there hods that q P T q [1 θc 1 C 2 ] q P q, (6.4) η ( ˆR T T ) θη (T ). (6.5) Proof Empoying in turn (4.3), (6.4), the triange inequaity, and (4.8) yieds θc 2 η (T ) θc 1 C 2 q P q q P q q P T q P q P T q C 2 η ( ˆR T T ). (6.6) Dividing through by C 2 competes the proof. Lemma 6 The coection of marked eements M T defined by the marking strategy (5.2) satisfies #M q 1/s A s E 1/s. (6.7)

20 20 A. DEMLOW Proof By definition of A s there exists a partition T T such that and #T #T 0 q 1/s A s [(1 θc 1 C 2 )E ] 1/s (6.8) q P T q (1 θc 1 C 2 )E. The smaest common refinement T of T and T is in T with #T #T #T #T 0 (cf. [32] ast ines of the proof of Lemma 5.2). Since V T V T, (6.2) and the ast equation above yied q P VT q q P VT q (1 θc 1 C 2 )E. Thus η( ˆR T T ) θη by Lemma 5. Since M is the smaest subset of T with η(m ) θη, we concude that #M # ˆR T T #T #T #T #T 0. We finay state our optimaity resut. Theorem 4 Assume as in Lemma 5 that θ < 1 C 1C 2, and assume that q A s. Given 0, there exists a constant C depending on P 1 and the constant C ref, from (2.3) but independent of other essentia quantities such that E C (#T #T ) 1/s q As,. (6.9) Proof We first compute using Theorem 1, (4.3), and the fact that γ C 1 2 that for k, Thus E 2 k + γ µ k (T k ) 2 E 2 k + γ η k (T k ) 2 (1 + γ )E 2 k E 2 k. (6.10) We then use (5.6) to obtain E 1/s k From (2.3), it then foows that k= E 1/s k (E 2 k + γ µ k (T k ) 2 ) 1/2s. (6.11) ρ ( k)/2s (E 2 + γ µ (T ) 2 ) 1/2s. (6.12) 1 1 #T #T = #R Tk T k+1 C ref, #M k C ref, q 1/s A s C ref, q 1/s A s 1 k= k= ρ ( k)/2s (E 2 + γ µ (T ) 2 ) 1/2s 1 k= E 1/s k ( ) 1 C ref, 1 ρ 1/2s 1/s q A s (E 2 + γ µ (T ) 2 ) 1/2s. (6.13) ( ) 1 Setting C := CC ref, 1 ρ 1/2s and rearranging the above expression competes the proof.

21 AFEM for harmonic forms 21 Remark 7 As is standard in AFEM optimaity resuts, θ is required to be sufficienty sma in order to ensure optimaity. θ must however ony be sufficienty 1 sma with respect to C 1C 2, which is entirey independent of the dimension β of H, P 1, and other essentia quantities. In contrast to eiptic eigenvaue probems [19] [6], we do not require an initia fineness assumption on T 0 in order to guarantee that the threshod vaue for θ is independent of essentia quantities. The constant C does however depend on P 1, and it is not cear that this constant wi improve (decrease) as the mesh increases even though P 1 1. The factor (1 ρ 1/2s ) 1 is nonincreasing, but the factor C ref, arising from (2.3) is not uniformy bounded in. According to [33, Theorem 6.1] it may in essence depend on the degree of quasi-uniformity of the mesh T and thus may degenerate as the mesh is refined. In order to guarantee a uniform constant, we appy Theorem 4 with = 0 and obtain a constant C 0 which depends on P0 1 in addition to T 0. 7 Extensions In this section we briefy discuss possibe extensions of our work. 7.1 Essentia boundary conditions. Many of our resuts extend directy to the case of essentia boundary conditions in which the requirement tr q = 0 in (2.1) is repaced by tr q = 0, with the atter condition imposed directy on the finite eement spaces. The major hurde in obtaining an immediate extension is the avaiabiity of quasi-interpoants which possess the necessary properties. The Christiansen-Winther interpoant in Section 2.3 has been defined and anayzed aso for essentia boundary conditions, whie the Fak-Winther interpoant was fuy anayzed in [17] ony for natura boundary conditions. We used the properties of the Fak-Winther interpoant ony to obtain the ocaized a posteriori upper bound (Lemma 3), which is necessary to obtain a quasi-optimaity resut but not a contraction. The contraction resut given in Theorem 3 thus extends immediatey to the case of essentia boundary conditions. There is indication given in [17] that the properties of the Fak-Winther interpant transfer naturay to essentia boundary conditions, in particuar by simpy setting boundary degrees of freedom to 0. Assuming this extension our quasi-optimaity resuts aso hod for homogeneous essentia boundary conditions. A suitabe interpoant is aso defined and anayzed in the paper [36] for the practicay important case d = 3, k = 1.

22 22 A. DEMLOW 7.2 Harmonic forms with coefficients In eectromagnetic appications the magnetic permeabiity A is a symmetric, uniformy positive definite matrix having entries in L (Ω). If A is nonconstant, then the space H A (Ω) = {v L 2 (Ω) 3 : cur v = 0, div Av = 0, Av n = 0 on Ω} (7.1) is the natura space of harmonic forms, but differs from that in (1.1). It is simiary possibe to modify the more genera definition (2.1) to incude coefficients. As is pointed out in [3, Section 6.1], the finite eement exterior cacuus framework appies essentiay verbatim to this situation once the inner products used in a of the reevant definitions are modified to incude coefficients. Our a posteriori estimates and the contraction resut of Section 5 simiary appy with minima modification. Extension of the optimaity resuts of Section 6 is possibe but compicated by the presence of osciation of the coefficient A in the anaysis; cf. [6], [7], [12]. 7.3 Approximation of cohomoogous forms In some appications it is of interest to compute P H f for a given (non-harmonic) form f. This is for exampe the case in the finite eement exterior cacuus framework for soving probems of Hodge-Lapace type. There f is the righthand-side data. Because the Hodge-Lapace probem is ony sovabe for data orthogona to H, it is necessary to compute P H f and sove the resuting system with data f P H f; cf. [25] for other appications. A possibe adaptive approach to this probem is to adaptivey reduce the defect between H and H as we do above and then project f onto H. However, this approach requires computation of a mutidimensiona space, whie the origina probem ony requires computation of a one-dimensiona space (that spanned by P H f). An aternate method woud be to approximate the fu Hodge decomposition. More precisey, one coud compute P B f and P Z f by soving two constrained eiptic probems, but expense is an obvious disadvantage of this method aso. It may be desirabe to instead adaptivey compute P H f directy. It might for exampe be the case that some members of H have singuarities which are not shared by P H f (such situations arise in eigenfunction computations). The task of constructing an AFEM for approximating ony P H f appears difficut, however. Assume for the sake of argument the extreme case where P H f 0, but f H. The indirect approach of first controing the defect between H and H and then projecting f woud continue to function with no probems in this case. On the other hand, it is not cear how to directy construct an a posteriori estimate for P H f P f which woud be nonzero when H f. In particuar, it is not difficut to construct an estimator for P f P H P f (cf. [15]), but such an estimator woud be 0 and thus not reiabe in this case.

23 AFEM for harmonic forms Aternate methods for computing harmonic forms: Cutting surfaces We have argey ignored the actua method for obtaining H by simpy assuming that we in some fashion produced an orthonorma basis for this space. This viewpoint is consistent with the finite eement exterior cacuus framework that we have argey foowed. It aso fits we with eigenvaue or SVD-based methods for computing H, which are genera with respect to space dimension and form degree and which may be easiy impemented using standard inear agebra ibraries. Discussion of methods for producing such a basis consistent with the FEEC framework may be found in [25]. However, the process of producing H is in and of itsef not entirey straightforward, and the method for producing it may affect the structure and properties of the resuting adaptive agorithm. Different methods with potentiay advantageous properties have been expored especiay in three space dimensions [16], [28]. The method of cutting surfaces is such an exampe. Let Ω R 3, and assume that there exist β reguar and nonintersecting cuts (two-dimensiona hypersurfaces) σ j such that Ω 0 = Ω \ β σ j is simpy connected. The assumption that such cuts exist is nontrivia with respect to domain topoogy and excudes for exampe the compement of a trefoi knot in a box [4]. The methodoogy we discuss here thus does not appy in a situations where our theory above appies. A domain for which such a set of cutting surfaces exist is caed a Hemhotz domain. Determination of cutting surfaces on finite eement meshes is aso a nontrivia and possiby computationay expensive probem [16], athough in an adaptive setting one coud potentiay compute the cutting surfaces at ow cost on a coarse mesh and transfer them to subsequent refinements. Assuming that Ω is Hemhotz, et ϕ j sove ϕ j = 0 in Ω 0, ϕ j n = 0 on Ω, ϕ j σi = δ ij and ϕ j n i σi = 0, 1 i β. (7.2) The set { ϕ j } β then serves as a basis for the space H1 of vector fieds defined in (1.1). That is, the harmonic fieds may be recaimed from potentias consisting of H 1 functions. Next assume that the cuts σ j each consist of unions of faces in a simpicia mesh T. Let now V 0 be piecewise inear Lagrange eements on Ω, and et Vj 0 be the set of functions which are continuous and piecewise inear in Ω \ σ j and which satisfy v h σj = 1. The canonica finite eement approximation to ϕ j is to find ϕ j T V j 0 such that Ω\σ j ϕ j T v = 0 for a v V 0. ϕ j T is ony unique up to a constant, but since we are ony interested in ϕ j T this has itte effect on our discussion. One can aso verify that ϕ j T is a discrete harmonic fied ying in the owest-degree Nédéec edge space. Thus as in the continuous case, the discrete harmonic fieds can be recovered from potentias. The same procedure may be appied with other compex-conforming pairs of spaces. An anaogous procedure aso exists for essentia boundary conditions.

24 24 A. DEMLOW An obvious adaptive procedure for approximating H 1 is to adaptivey compute ϕ j T, j = 1,..., J, using standard AFEM for scaar Lapace probems. Convergence and optimaity foows from standard resuts such as those found in [7] with sight modification to account for boundary conditions. The basis { ϕ j } β that we thus approximate is not orthonorma, but has the advantage of being fixed using criterion that are passed on to the discrete approximations. Aso note that here the approximation map ϕ j ϕ j T is ceary inear. If we however orthonormaized the vectors { ϕ j T } in order to produce {qj T } as in our previous assumptions, then the approximation { ϕ j } {q j T } woud be noninear. This discussion indicates that whie producing an orthonorma basis of forms is a noninear procedure, the noninearity is mid and not intrinsic to the task of finding some basis for the space of harmonic forms. 7.5 Aternate methods for computing harmonic forms: Adaptive correction of a cohomoogy basis We next describe another possibe option for computing the space H 1 of forms given in (1.1). Given a coarse initia mesh T 0, one may first compute a basis {q j 0 }β for the space H1 0 of harmonic forms on T 0. (More broady, one coud use as a starting point any basis for the first cohomoogy group on T 0, that is, a ineary independent set { q j 0 }β of cur-free finite eement vector fieds that are not gradients of scaar potentias.) Assuming that #T 0 is sma, computation of H 1 0 can be achieved reativey cheapy using a brute-force inear agebra technique such as an svd sover. Let then φ j H 1 (Ω) sove φ j, τ = q j 0, τ, τ H1 (Ω). (7.3) Because H 1 0 Z 1, we have P H q j 0 = qj 0 P Bq j 0 = qj 0 φj. In addition, Lemma 3 guarantees that {q j 0 φj } β is a basis for H1. This suggests an aternative and potentiay cheaper method for adaptive computation of H 1, namey to first find a basis for the discrete harmonic forms on a coarse mesh, then adaptivey compute discrete approximations {φ j }β to the β source probems in (7.3). The set {q j 0 φj }β is then a basis for the discrete harmonic forms H 1. Adaptive finite eement approximation of the corrections φ j is nonstandard because the right-hand-side data for the probem (7.3) is ony in H 1 (Ω). Convergence anaysis of such AFEM is thus more difficut than for L 2 data, but was carried out under broad assumptions in [10] for appropriatey defined error estimators and AFEM. Besides the potentia use of a sighty nonstandard AFEM that is ess ikey to be avaiabe in standard codes, there are two main possibe disadvantages of this approach. The first is that a straightforward appication woud empoy β meshes, as in the case of cutting surfaces. Depending on the appication, using mutipe meshes coud be costy and inconvenient if β > 1. This can easiy be remedied by summing the eementwise error indicators from the j source probems (7.3) as we do in (4.2), but a direct appication