ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 10, October 2014, Pages S (2014) Article electronically publised on February 26, 2014 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL DOUGLAS N. ARNOLD, RICHARD S. FALK, JOHNNY GUZMÁN, AND GANTUMUR TSOGTGEREL Abstract. In 1976, Dodziuk and Patodi employed Witney forms to define a combinatorial codifferential operator on cocains, and tey raised te question weter it is consistent in te sense tat for a smoot enoug differential form te combinatorial codifferential of te associated cocain converges to te exterior codifferential of te form as te triangulation is refined. In 1991, Smits proved tis to be te case for te combinatorial codifferential applied to 1-forms in two dimensions under te additional assumption tat te initial triangulation is refined in a completely regular fasion, by dividing eac triangle into four similar triangles. In tis paper we extend te result of Smits to arbitrary dimensions, sowing tat te combinatorial codifferential on 1-forms is consistent if te triangulations are uniform or piecewise uniform in a certain precise sense. We also sow tat tis restriction on te triangulations is needed, giving a counterexample in wic a different regular refinement procedure, namely Witney s standard subdivision, is used. Furter, we sow by numerical example tat for 2-forms in tree dimensions, te combinatorial codifferential is not consistent, even for te most regular subdivision process. 1. Introduction Let M be an n-dimensional polytope in R n, triangulated by a simplicial complex T of maximal simplex diameter, wic we orient by fixing an order for te vertices. (Altoug we restrict ourselves to polytopes for simplicity, several of te results below can easily be extended to triangulated Riemannian manifolds.) We denote by Λ k =Λ k (M) te space of smoot differential k-forms on M. Te Euclidean inner product restricted to M determines te Hodge star operator Λ k Λ n k, and te inner product on Λ k given by u, v = u v. Te space L 2 Λ k is te completion of Λ k wit respect to tis norm, i.e., te space of differential k-forms wit coefficients in L 2. We ten define HΛ k to be te space of forms u in L 2 Λ k wose exterior derivative du, wic may be understood in te sense of distributions, belongs to L 2 Λ k+1. Tese spaces combine to form te L 2 de Ram complex 0 HΛ 0 d HΛ 1 d d HΛ n 0. Received by te editors December 18, Matematics Subject Classification. Primary 58A10, 65N30; Secondary 39A12, 57Q55, 58A14. Key words and prases. Consistency, combinatorial codifferential, Witney form, finite element. Te work of te first autor was supported by NSF grant DMS Te work of te second autor was supported by NSF grant DMS Te work of te fourt autor was supported by an NSERC Discovery Grant and an FQRNT Nouveaux Cerceurs Grant c 2014 American Matematical Society Reverts to public domain 28 years from publication

2 5488 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL Viewing te exterior derivative d as an unbounded operator L 2 Λ k to L 2 Λ k+1 wit domain HΛ k, we may define its adjoint d. Tus a differential k-form u belongs to te domain of d if te operator v u, dv L2 Λ k is bounded on L2 Λ k 1,andten d u, v L 2 Λ k 1 = u, dv L 2 Λ k, v HΛk 1. In particular, every u wic is smoot and supported in te interior of M belongs to te domain of d and d u =( 1) k(n k+1) d u. Let Δ k (T )denotetesetofk-dimensional simplices of T.WedenotebyC k (T ) te space of formal linear combinations of elements of Δ k (T ) wit real coefficients, te space of k-cains, and by C k (T )=C k (T ) te space of k-cocains. Te coboundary maps d c : C k (T ) C k+1 (T ) ten determine te cocain complex. Te de Ram map R maps Λ k onto C k (T ) taking a differential k-form u to te cocain (1.1) R u : C k (T ) R, c u. Te canonical basis for C k (T ) consists of te cocains a τ, τ Δ k (T ), were a τ takes te value 1 on τ and zero on te oter elements of Δ k (T ). Te associated Witney form is given by k W a τ = k! ( 1) i λ i dλ 0 dλ i dλ k, i=0 were λ 0,...,λ k are te piecewise linear basis functions associated to te vertices of te simplex listed, i.e., λ i is te continuous piecewise linear function equal to 1 at te it vertex of τ and vanising at all te oter vertices of te triangulation. Te span of W a τ, τ Δ k (T ), defines te space of Λ k of Witney k-forms. Its elements are piecewise affine differential k-forms wic belong to HΛ k and satisfy dλ k Λk+1. Tus te Witney forms comprise a finite-dimensional subcomplex of te L 2 de Ram complex called te Witney complex: 0 Λ 0 d Λ 1 d d Λ n 0. Te Witney map W maps C k (T ) isomorpically onto Λ k and satisfies (1.2) W d c c = dw c, c C k (T ), i.e., is a cocain isomorpism of te cocain complex onto te Witney complex. Altoug Witney k-forms need not be continuous, eac as a well-defined trace on te simplices in Δ k (T ), so te de Ram map (1.1) is defined for u Λ k. Te Witney map is a one-sided inverse of te de Ram map: R W c = c for c C k (T ). Te reverse composition π = W R :Λ k Λ k defines te canonical projection into Λ k. In [3] and [4], Dodziuk and Patodi defined an inner product on cocains by declaring te Witney map to be an isometry: (1.3) a, b = W a, W b L2 Λ k, a,b Ck (T ). Tey ten used tis inner product to define te adjoint δ c of te coboundary: (1.4) δ c a, b = a, d c b, a,b C k (T ). Since te coboundary operator d c may be viewed as a combinatorial version of te differential operator of te de Ram complex, its adjoint δ c may be viewed as a c

3 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL 5489 combinatorial codifferential, and togeter tey define te combinatorial Laplacian on cocains given by Δ c = d c δ c + δ c d c : C k (T ) C k (T ). Te work of Dodziuk and Patodi concerned te relation between te eigenvalues of tis combinatorial Laplacian and tose of te Hodge Laplacian. Dodziuk and Patodi asked weter te combinatorial codifferential δ c is a consistent approximation of d in te sense tat if we ave a sequence of triangulations T wit maximum simplex diameter tending to zero and satisfying some regularity restrictions, ten (1.5) lim W δ c R u d u =0, for sufficiently smoot u Λ k belonging to te domain of d. Here and encefort te norm denotes te L 2 norm. Since C k (T )andλ k are isometric, we may state tis question in terms of Witney forms, witout invoking cocains. Define te Witney codifferential d :Λk by Λ k 1 (1.6) d u, v L2 Λ = u, dv k 1 L 2 Λ k, u Λk,v Λ k 1. Combining (1.2), (1.3), and (1.4), we see tat d = W δ c W 1. Terefore, W δ c R = d π, and te question of consistency becomes weter (1.7) lim d π u d u =0, for smoot u in te domain of d. In Appendix II of [4], te autors suggest a counterexample to (1.7) for 1-forms (i.e., k = 1) on a two-dimensional manifold, but, as pointed out by Smits [7], te example is not valid, and te question as remained open. Smits imself considered te question, remaining in te specific case of 1-forms on a two-dimensional manifold, and restricting imself to a sequence of triangulations obtained by regular standard subdivision, meaning tat te triangulation is refined by dividing eac triangle into four similar triangles by connecting te midpoints of te edges, resulting in a piecewise uniform sequence of triangluations. See Figure 5 for an example. In tis case, Smits proved tat (1.5) or, equivalently, (1.7) olds. Smits s result leaves open various questions. Does te consistency of te 1-form codifferential on regular meses in two dimensions extend to Mes sequences wic are not obtained by regular standard subdivision? More tan two dimensions? Te combinatorial codifferential on k-forms wit k>1? In tis paper we sow tat te answer to te second question is affirmative, but te answers to te first and tird are negative. More precisely, in Section 2 we present a simple counterexample to consistency for a quadratic 1-form on te sequence of triangulations sown in Figure 1. Wile tese meses are not obtained by regular standard subdivision, tey may be obtained by anoter systematic subdivision process, standard subdivision, as defined by Witney in [8, Appendix II, 4]. Next, in Section 3, we recall a definition of uniform triangulations in n-dimensions wic was formulated in te study of superconvergence of finite element metods, and we use te superconvergence teory to extend Smits s result on te consistency of te combinatorial codifferential on 1-forms to n-dimensions, for triangulations tat are

4 5490 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL uniform or piecewise uniform. In Section 4, we provide computational confirmation of tese results, bot positive and negative. Finally, in Section 5, we numerically explore te case of 2-forms in tree dimensions and find tat te combinatorial codifferential is inconsistent, even for completely uniform mes sequences. 2. A counterexample to consistency We take as our domain M tesquare( 1, 1) ( 1, 1) R 2, and as initial triangulation te division into four triangles obtained via drawing te two diagonals. We refine a triangulation by subdividing eac triangle into four using standard subdivision. In tis way we obtain te sequence of crisscross triangulations sown in Figure 1, wit te mt triangulation consisting of 4 m isoceles rigt triangles. We index te triangulation by te diameter of its elements, so we denote te mt triangulation by T were =4/2 m. Using tis triangulation, te autors of [5] sowed tat superconvergence does not old for piecewise linear Lagrange elements. Figure 1. T 2, T 1, T 1/2, T 1/4, te first four crisscross triangulations. Define p : M R by p(x, y) =x x 3 /3 and let u = dp =(1 x 2 )dx Λ 1 (M). Now for q HΛ 0 (M) (i.e., te Sobolev space H 1 (M)), we ave ( ) q q dq = dx + x y dy = q q dy x y dx, so u, dq L 2 Λ 1 = M u dq = M (1 x 2 ) q dx dy = 2xq dx dy = 2x, q L x 2 Λ 0. M Tus u belongs to te domain of d and d u =2x. As an alternative verification, we may identify 1-forms and vector fields. Ten u corresponds to te vector field (1 x 2, 0) wic as vanising normal component on M, and so belongs to te domain of d = div and d u = div(1 x 2, 0) = 2x.

5 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL 5491 Set w = d π u. Now w Λ 0, i.e., it is a continuous piecewise linear function. Te projections π into te Witney forms form a cocain map, so π u = π dp = dπ p =gradπ p,wereπ p is piecewise linear interpolant of p. Tus w Λ 0 is determined by te equations (2.1) w qdxdy= grad π p grad qdxdy, q Λ 0. M M It turns out tat we can give te solution to tis problem explicitly. Since w is a continuous piecewise linear function, it is determined by its values at te vertices of te triangulation T. Te coordinates of te vertices are integer multiples of /2. In fact te value of w at a vertex (x, y) depends only on x and for 1isgiven by, x = 1, 0, 1 <x<1, xa multiple of,, x =1, w (x, y) = 6+2, x = 1+/2, 6x, 1+/2 <x<1 /2, xan odd multiple of /2, 6 2, x =1 /2. A plot of te piecewise linear function w is sown in Figure 3 for =1/2. To verify te formula it suffices to ceck (2.1) for all piecewise linear functions q tat vanis on all vertices except one. Tere are several cases depending on ow close te vertex is to te boundary, and te computation is tedious, but elementary. Here we only give te details wen te vertex is (x, y) wit 1 +/2 <x<1 /2 and x is an odd multiple of /2. To tis end, let q be te piecewise linear function tat is one on vertex (x, y) and vanises on all te remaining vertices. In tis case, te support of q is te union of te four triangles T 1,T 2,T 3,T 4 tat ave (x, y) as a vertex (see Figure 2). According to te formula, in te support of q, one as w =6xq. A simple calculation ten sows tat te left-and side of (2.1) is 4 w qdxdy=6x q 2 dx dy =4xm, M T i were m = 2 /4= T i for any i. To calculate te rigt-and side of (2.1) for tis q, we calculate tat (1, 0), on T 1, grad q = 2 (0, 1), on T 2, ( 1, 0), on T 3, (0, 1), on T 4, i=1 and (p(x) p(x 2 grad π p = 2 ), 0), on T 1, ( 1 2 [p(x + 2 ) p(x 2 )],p(x) 1 2 [p(x + 2 )+p(x 2 )]), on T 2, (p(x + 2 ) p(x), 0), on T 3, ( 1 2 [p(x + 2 ) p(x 2 )], 1 2 [p(x + 2 )+p(x 2 )] p(x)), on T 4.

6 5492 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL Hence, M grad π p grad qdxdy= 4 i=1 T i π p grad qdxdy = 16 2 (p(x) 1 2 [p(x 2 )+p(x + )])m =4xm. 2 Tis verifies (2.1) for tis piecewise linear function q. (x + /2,y+ /2) T 4 T 1 (x, y) T 3 T 2 (x /2,y /2) Figure 2. Te support of te piecewise linear function q. Figure 3. Te spiked surface is te grap of te piecewise linear function w = d π f for =1/4. Te plane is te grap of te linear function d u. (Color available online.) Finally, we note tat, since w essentially oscillates between 6x and 0, it does not converge in L 2 to d u (or to anyting else) as tends to zero.

7 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL Consistency for 1-forms on piecewise uniform meses We continue to consider a sequence of triangulations T indexed by a positive parameter tending to 0. We take to be equivalent to te maximal simplex diameter c max diam T C, T Δ n (T ) for some positive constants C, c independent of (trougout we denote by C and c generic constants, not necessarily te same in different occurrences). We also assume tat te sequence of triangulations is sape regular in te sense tat tere exists c>0 suc tat ρ(t ) c diam T, for all T T and all, wereρ(t ) is te diameter of te ball inscribed in T. We begin wit some estimates for te approximation of a k-form by an element of Λ k. For tis we need to introduce te spaces of differential forms wit coefficients in a Sobolev space. Let m be a non-negative integer and u a k-form defined on a domain M R n,wicwemayexpandas (3.1) u = u i1 i k dx i 1 dx i k. 1 i 1 < <i k n Using multi-index notation for partial derivatives of te coefficients u i1 i k, we define te mt Sobolev norm and seminorm by u 2 H m Λ = k D α u i1 i k 2 L 2 (M), 1 i 1 < <i k n α m u 2 H m Λ = k D α u i1 i k 2 L 2 (M), 1 i 1 < <i k n α =m and define te space H m Λ k (M) to consist of all k-forms in M for wic te Sobolev norm u Hm Λ k is finite. Wit tis notation, we can state te basic approximation result tat for any sape regular sequence of triangulations tere is a constant C suc tat (3.2) inf v Λ k u v C u H 1 Λ k, u H1 Λ k (M). For a proof, see [1, Teorem 5.8]. Since H 1 Λ k is dense in L 2 Λ k, tis implies tat (3.3) dist(f,λ k ):= inf f v 0as 0, v Λ k f L 2 Λ k (M). In addition to te best approximation estimate (3.2), we also need an O() estimate on te projection error u π u. For tis we require more regularity of u, sinceπ u is defined in terms of traces of u on k-dimensional faces, wic need not be defined on H 1 Λ k. Lemma 3.1. Let {T } be a sape regular sequence of triangulations of M R n and k an integer between 0 and n. Let l be te smallest integer so tat l>(n k)/2. Ten tere exists a constant C, depending only on n and te sape regularity constant, suc tat l (3.4) π u u L2 Λ k C m u Hm Λ k, u Hl Λ k (M). m=1

8 5494 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL Proof. First we note tat te canonical projection is defined simplex by simplex, as (π u) T = π T (u T ), were, for v a k-form on T, π T v is its interpolant into te space of Witney forms on te single simplex T. Terefore, it is enoug to prove tat l (3.5) u π T u L2 Λ k (T ) C m u Hm Λ k (T ), u H l Λ k (T ), m=1 wit te constant C depending on T only troug its sape constant. We prove tis first for te unit rigt simplex in R n, ˆT, wit vertices at te origin and te n points (1, 0,...,0), (0, 1, 0,...),... Since l>(n k)/2, we obtain, by te Sobolev embedding teorem, tat π ˆT u L 2 Λ k ( ˆT ) C u H l Λ k ( ˆT ), and so, by te triangle inequality, u π ˆT u L 2 Λ k ( ˆT ) C u H l Λ k ( ˆT ). Now let ū = n! ˆT u,aconstantk-form on ˆT equal to te average of u. Ten ū =ū, so π ˆT u π ˆT u L 2 Λ k ( ˆT ) = (u ū) π ˆT (u ū) L 2 Λ k ( ˆT ) C u ū Hl Λ k ( ˆT ) C( u ū L 2 Λ k ( ˆT ) + l u Hm Λ k ( ˆT ) ), wereweaveusedtefacttatū is a constant form, so its mt Sobolev seminorm vanises for m 1. Now we invoke Poincaré s inequality u ū L2 Λ k ( ˆT ) C u H 1 Λ k ( ˆT ). Putting tings togeter, and writing û instead of u, weavesowntat l (3.6) û π ˆT û L 2 Λ k ( ˆT ) C û Hm Λ k ( ˆT ), û Hl Λ k ( ˆT ). m=1 Tis is te desired result (3.5) in te case T = ˆT. To obtain te result for a general simplex, we scale via an affine diffeomorpism F : ˆT T.Ifuis te k-form on T given by (3.1), ten (3.7) F u = n {1 i 1 < <i k n} j 1,...,j k =1 (u i1 i k F ) Fi 1 ˆx j 1 Fik ˆx j k m=1 dˆx j 1 dˆx j k. Eac of te partial derivatives F i p / ˆx j q is a constant bounded by. Using te cain rule and cange of variables in te integration, we find tat (3.8) c F u Hm Λ k ( ˆT ) (vol T ) 1/2 m+k u Hm Λ k (T ) C F u Hm Λ k ( ˆT ), were te constants c and C depend only on m and n and te sape regularity constant of T. Combining (3.6) and (3.8) we get u π T u L2 Λ k (T ) C(vol T ) 1/2 k û π ˆT û L 2 Λ k ( ˆT ) l C(vol T ) 1/2 k û Hm Λ k ( ˆT ) C wic establises (3.5). m=1 l m u Hm Λ k (T ), m=1

9 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL 5495 Our approac to bounding te norm of te consistency error is to relate it to anoter quantity wic as been studied in te finite element literature, namely (3.9) A (u) := sup v Λ k 1 u π u, dv. v Teorem 3.2. Assume te approximation property (3.3). Ten, for any smoot u L 2 Λ k belonging to te domain of d we ave lim d u d π u =0 lim A (u) =0. Tis follows immediately from Lemma 3.3. Lemma 3.3. Let 1 k n, andletu L 2 Λ k be smoot and in te domain of d. Ten (3.10) A (u) d u d π u dist(d u, Λ k 1 )+A (u). Proof. Te first inequality is straigtforward. For any v Λ k 1, u π u, dv = d u d π u, v d u d v v π u. For te second inequality, we introduce te L 2 -ortogonal projection P : L 2 Λ k 1 and invoke te triangle inequality to get Λ k 1 (3.11) d u d π u d u P d u + P d u d π u =dist(d u, Λ k 1 )+ w, were w = P d u d π u Λ k.now (3.12) w 2 = P d u d π u, w = u π u, dw, and ence (3.13) w = u π u, dw w wic completes te proof. u π u, dv sup = A (u), v Λ v k 1 Tus we wis to bound u π u, dv / v for smoot u in te domain of d and v Λ k. An obvious approac is to apply te Caucy Scwarz inequality and ten use te approximation estimate (3.4) to obtain (3.14) u π u, dv u π u dv C u Hl Λ k dv. To continue, we need to bound dv / v for v an arbitrary non-zero element of Λ k. Because Λk consists of piecewise polynomials, it is possible to bound its derivative in terms of its value using a Bernstein type inequality or inverse estimate. Tis gives tat (3.15) dv C 1 v, v Λ k, were =min T Δn (T ) diam T. Unfortunately, even if we assume tat our triangulations are quasiuniform, i.e., tat c for some fixed c>0, tis just leads to te bound A (u) C u Hl Λ k, wic does not tend to zero wit. In fact, we cannot ope to get a bound wic tends to zero witout furter ypoteses, since, as we ave seen, even for te nice mes sequence and form u considered in te previous section, d is not consistent, and so A (u) does not tend to zero.

10 5496 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL Noneteless, for very special mes sequences it is possible to improve te bound (3.14) from first to second order in. Tis was establised by Brandts and Křížek in teir work on gradient superconvergence [2]. Te mes condition is embodied by te following concept. Definition 3.4 ([2]). A triangulation T on M is called uniform if tere exist n linearly independent vectors e 1,...,e n, suc tat (1) Every simplex in T contains an edge parallel to eac e j. (2) If an edge e is parallel to one of te e j and is not contained in M, ten te union P e of simplices containing e is invariant under reflection troug te midpoint m e of e, i.e., 2m e x P e for all x P e. Te crisscross triangulations sown in Figure 1 satisfy te first condition of te definition, but not te second, and so are not uniform. On te oter and, te mes sequence tat is obtained by starting from a single triangle, or from a division of a square into two triangles and applying regular standard subdivision, is uniform. See te first two rows of Figure 4. A uniform triangulation of te cube in n dimensions is obtained by subdividing it into m n subcubes, and dividing eac of tese into n! simplices saring a common diagonal, wit all te diagonals of te subcubes cosen to be parallel. Te 3D case is sown in Figure 4. We refer to [2] for more details. Figure 4. Uniform triangulations. Teorem 3.4 of [2] claims tat if {T } is a sape regular family of uniform triangulations of M, and if u is a smoot 1-form, ten tere exists a constant C>0suc tat π u u, dv C 2 u H 2 Λ 1 dv, for all v Λ 0 H 1 (M) and>0. Here H 1 (M) denotes te space of H 1 (M) functions wit vanising trace on M. However, teir proof uses te inequality (cf.

11 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL 5497 Figure 5. A piecewise uniform sequence of triangulations. (1.5) of [2]) π u u L 2 Λ 1 C u H 1 Λ 1, were C is a constant independent of u. Tis would imply tat π can be continuously extended to H 1 Λ 1, wic is impossible for n 3. Fortunately, te proof in [2] works verbatim if te above inequality is replaced by (3.4). Hence, te following result is essentially proved in [2]. Teorem 3.5. Let {T } be a sape regular family of uniform triangulations of M, and let u be a smoot 1-form. Furtermore, let l be te smallest integer so tat l>(n 1)/2. Ten tere exists a constant C>0 suc tat (3.16) π u u, dv C 2 u Hl Λ 1 dv, for all v Λ 0 H 1 (M) and 1 >>0. Next we consider piecewise uniform sequences of triangulations. Definition 3.6. A family T of triangulations of te polytope M is called piecewise uniform if tere is a triangulation T of M suc tat for eac, T is a refinement of T and for eac T T and eac, te restriction of T to T Δ n (T ) is uniform. If, as in [7], we start wit an arbitrary triangulation of a polygon and refine it by standard regular subdivision, te resulting sequence of triangulations is piecewise uniform. Tis is illustrated in Figure 5. Te following teorem sows tat d is consistent for 1-forms on piecewise uniform meses, tus generalizing te main result of [7] from 2 to n dimensions. Teorem 3.7. Assume tat te family of triangulations {T } is a sape regular, quasiuniform, and piecewise uniform. Let u H l Λ 1 (M) be a 1-form in te domain of d,werel is te smallest integer satisfying l>(n 1)/2. Ten we ave (3.17) lim d u d 0 π u =0. Proof. Let T denote te triangulation of M wit respect to wic te triangulations T are uniform. We will apply Teorem 3.5 to te uniform mes sequences obtained

12 5498 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL by restricting T to eac T T. To tis end, let K = T T of T,andset Σ = { T T T K }. We can decompose an arbitrary function v Λ 0 as T denote te skeleton (3.18) v = w + v T, T T were w Λ 0 is supported in Σ and v T Λ0 is supported in T. Indeed, we just take w to coincide wit v at te vertices of te triangulation contained in K and to vanis at te oter vertices, wile v T = v at te vertices in te interior of T and vanises at te oter vertices. Because te mes family is sape regular and quasiuniform, tere exist positive constants C, c suc tat c v 2 v(x) 2 C v 2, v Λ 0, x Δ 0 (T ) from wic we obtain te stability bound (3.19) w + v T C v. T T Using te decomposition (3.18) of v we get π u u, dv π u u, dw + π u u, dv T T T (3.20) C u H l Λ 1 (Σ ) dw + C 2 u H l Λ (M) dv 1 T T T C u Hl Λ 1 (Σ ) w L2 (Σ ) + C u Hl Λ (M) v 1 T T T C ( u H l Λ 1 (Σ ) + u H l Λ 1 (M)) v, were we ave used te Caucy Scwarz inequality, te projection error estimate (3.4), te second order estimate (3.16) (wic olds on te uniform meses on eac T ), te inverse estimate of (3.15), and te L 2 -stability bound (3.19). Since te volume of Σ goes to 0 as 0, so does u Hl Λ 1 (Σ ).TusA (u) vanises wit, and te desired result is a consequence of Teorem 3.2. Remark 3.8. Te preceding proof sows tat as long as te triangulation is mostly uniform, in te sense tat te volume of te defective region goes to 0 as 0, we obtain consistency. One can also extract information on te convergence rate. For instance, using te fact tat Σ is O(), we obtain u Hl Λ 1 (Σ ) C u Cl Λ 1 for u C l Λ 1 (M). 4. Computational experiments for 1-forms In tis section, we present numerical computations confirming te consistency of d for 1-forms on uniform and piecewise uniform meses in 2 and 3 dimensions, and oter computations confirming its inconsistency on more general meses. Te four tables in tis section display te results of computations wit various mes sequences. In eac case we sow te maximal simplex diameter, tenumberof simplices in te mes, te consistency error d π f d f, and te apparent order

13 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL 5499 inferred from te ratio of consecutive errors. All computations were performed using te FEniCS finite element software library [6]. Te first two tables concern te problem on te square described in Section 2, i.e., te approximation of d u were u =(1 x 2 )dx. Table 1 sows te results wen te piecewise uniform mes sequence sown in Figure 5 is used for te discretization. Notice tat te consistency error clearly tends to zero as O(). Table 1. Wen computed using te 2-dimensional piecewise uniform mes sequence of Figure 5, te consistency error tends to 0. n triangles error order e e e e e e e 02 1, e e 02 5, e e 02 20, e By contrast, Table 2 sows te counterexample described analytically in Section 2, using te mes sequence of Figure 1, obtained by standard subdivision. In tis case, te consistency error does not converge to zero, as is clear from te computations. Table 2. Wit te mes sequence of Figure 1, te consistency error does not tend to 0. n triangles error order e e e e 02 1, e 02 4, e 02 16, Similar results old in 3 dimensions. We computed te error in d u on te cube ( 1, 1) 3, were again u is given by (1 x 2 )dx. We calculated wit two mes sequences, bot starting from a partition of te cube into six congruent tetraedra, all saring a common edge along te diagonal from ( 1, 1, 1) to (1, 1, 1). We constructed te first mes sequence by regular subdivision, yielding te meses sown in Figure 6. Tese are uniform meses, and te numerical results given in Table 3 clearly demonstrate consistency. For te second mes sequence we applied standard subdivision, obtaining te sequence of structured but non-uniform triangulations sown in Figure 7. In tis case d is inconsistent. See Table Inconsistency for 2-forms in 3 dimensions We ave seen tat for 1-forms, d is consistent if computed using piecewise uniform mes sequences, but not wit general mes sequences. It is also easy to see tat consistency olds for n-forms in n-dimensions for any mes sequence.

14 5500 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL Figure 6. Uniform mes sequence in 3D, obtained by regular subdivision. Figure 7. As in 2D, te mes sequence in 3D obtained by standard subdivision is not uniform. Tis is because te canonical projection π onto te Witney n-forms (wic are just te piecewise constant forms) is te L 2 ortogonal projection. Now if v is a Witney (n 1)-form, ten dv is a Witney n-form, so te inner product u π u, dv =0. TusA (u), defined in (3.9), vanises identically, and so d is consistent by Teorem 3.2. Having understood te situation for 1-forms and n- forms, tis leaves open te question of weter consistency olds for k-forms wit k strictly between 1 and n. In tis section we study 2-forms in 3 dimensions and give numerical results indicating tat d is not consistent, even for uniform meses. Let u =(1 x 2 )(1 y 2 )dx dy, a 2-form on te cube M =( 1, 1) 3. Te corresponding vector field is (0, 0, (1 x 2 )(1 y 2 )) wic as vanising tangential

15 ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL 5501 Table 3. Te consistency error for d on 1-forms in 3D tends to zero wen using te uniform mes sequence of Figure 6. n tetraedra error order e e e e e 01 3, e e 01 24, e e , e e 02 1,572, e Table 4. Te consistency error for d on 1-forms in 3D, using te non-uniform mes sequence of Figure 7, does not tend to zero. n tetraedra error order e e e e e 01 3, e e 01 24, e e , e components on M. Terefore u belongs to te domain of d and d u is te 1- form corresponding to curl u, i.e., d u = 2(1 x 2 )ydx +2x(1 y 2 )dy. Table 5 sows te consistency error d π u d u L 2 Λ 1 computed using te sequences of uniform meses displayed in Figure 6. Tis mes sequence yields a consistent approximation of d for 1-forms, but te experiments clearly indicate tat tis is not so for 2-forms. Table 5. Te consistency error does not tend to zero for 2-forms, even on a uniform mes sequence. n triangles error order e e e e e e e e e e References [1] Douglas N. Arnold, Ricard S. Falk, and Ragnar Winter, Finite element exterior calculus: from Hodge teory to numerical stability, Bull. Amer. Mat. Soc. (N.S.) 47 (2010), no. 2, , DOI /S MR (2011f:58005) [2] Jan Brandts and Mical Křížek, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J. Numer. Anal. 23 (2003), no. 3, , DOI /imanum/ MR (2004i:65105) [3] Jozef Dodziuk, Finite-difference approac to te Hodge teory of armonic forms, Amer.J. Mat. 98 (1976), no. 1, MR (53 #11642) [4] J.DodziukandV.K.Patodi,Riemannian structures and triangulations of manifolds,j.indian Mat.Soc.(N.S.)40 (1976), no. 1-4, 1 52 (1977). MR (58 #7742)

16 5502 D. N. ARNOLD, R. S. FALK, J. GUZMÁN, AND G. TSOGTGEREL [5] Ricardo Durán,María Amelia Muscietti, and Rodolfo Rodríguez,On te asymptotic exactness of error estimators for linear triangular finite elements, Numer. Mat. 59 (1991), no. 2, , DOI /BF MR (92b:65086) [6] Anders Logg, Kent-Andre Mardal, Gart N. Wells, et al., Automated solution of differential equations by te finite element metod, Springer, [7] Lieven Smits, Combinatorial approximation to te divergence of one-forms on surfaces, Israel J. Mat. 75 (1991), no. 2-3, , DOI /BF MR (93d:57052) [8] Hassler Witney, Geometric integration teory, Princeton University Press, Princeton, N. J., MR (19,309c) Scool of Matematics, University of Minnesota, Minneapolis, Minnesota address: Department of Matematics, Rutgers University, Piscataway, New Jersey address: Division of Applied Matematics, Brown University, Providence, Rode Island address: jonny Department of Matematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 0B9 address: