Double Complexes and Local Cochain Projections

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1 Double Complexes and Local Cochain Projections Richard S. Falk, 1 Ragnar Winther 2 1 Department o Mathematics, Rutgers University, Piscataway, New Jersey Centre o Mathematics or Applications and Department o Mathematics, University o Oslo, 0316 Oslo, Norway Received 3 December 2013; accepted 13 August 2014 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI /num he construction o projection operators, which commute with the exterior derivative and at the same time are bounded in the proper Sobolev spaces, represents a key tool in the recent stability analysis o inite element exterior calculus. hese so-called bounded cochain projections have been constructed by combining a smoothing operator and the unbounded canonical projections deined by the degrees o reedom. However, an undesired property o these bounded projections is that, in contrast to the canonical projections, they are nonlocal. he purpose o this article is to discuss a recent alternative construction o bounded cochain projections, which also are local. A key tool or the new construction is the structure o a double complex, resembling the Čech-de Rham double complex o algebraic topology Wiley Periodicals, Inc. Numer Methods Partial Dierential Eq 000: , 2014 Keywords: cochain projections; inite element exterior calculus; stability analysis I. INRODUCION he construction o projection operators onto inite element spaces which commute with the governing dierential operators has always been a central eature o the analysis o mixed inite element methods, c. Res. [1, 2]. However, the act that the canonical projections, deined rom the degrees o reedom, are usually not bounded in the natural Sobolev norms has resulted in additional complexity o the stability arguments or such methods. In act, or a long time, it was not known how to construct commuting projections which also are bounded in spaces like H(curl) and H(div), without making extra regularity assumptions. However, during the last decade rather general constructions o such projections have been given, leading to elegant and compact stability proos. Correspondence to: Ragnar Winther, Centre o Mathematics or Applications and Department o Mathematics, University o Oslo, 0316 Oslo, Norway ( ragnar.winther@cma.uio.no) his is an open access article under the terms o the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modiications or adaptations are made. Contract grant sponsor: European Research Council [European Union s Seventh Framework Programme (FP7/ )/ERC]; contract grant number: he Authors Numerical Methods or Partial Dierential Equations Published by Wiley Periodicals, Inc.

2 2 FALK AND WINHER he irst successul construction was given by Schöberl in [3], where a smoothing operator, constructed by perturbing the mesh, is combined with the canonical projection to obtain a commuting projection operator which also is bounded. In a related paper, Christiansen [4] proposed to use a more standard smoothing operator deined by a molliier unction. In the setting o inite element exterior calculus, variants o such constructions, all based on a proper combination o smoothing and canonical projections, are analyzed in [5, Section 5], [6, Section 5], and [7]. hese operators are bounded in the appropriate norms and commute with the exterior derivative. However, these projections lack another key property o the canonical projections; they are not locally deined. In act, it has not been clear i it is possible to construct bounded and commuting projections which are locally deined. In recent work, such projections were constructed in [8]. he construction is inspired by the well-known Clément operator [9], which is based on local projection operators deined on macroelements. In its original orm, where the inite element space consists o continuous piecewise polynomial subspaces o the Sobolev space H 1, the Clément operator is not a projection. However, by modiying the Clément operator, such that the local projections are deined with respect to piecewise polynomial spaces, a projection is easily obtained. he more challenging part, successully addressed in Re. [8], is to obtain commuting projections. he construction proposed in Re. [8] is discussed in the setting o exterior calculus and the de Rham complex on bounded domains in R n, and covers all orders o the basic inite element spaces described in Res. [5, 6]. However, the lowest order case, corresponding to the Whitney elements [10], represents in many ways the most challenging part o the construction. A key diiculty is to relate the projection onto the space o Whitney k-orms, constructed rom local projections on macroelements deined rom subsimplexes o the mesh o dimension k, to the corresponding projection onto Whitney (k + 1)-orms, constructed by local projections on the corresponding macroelements associated to (k + 1)-dimensional subsimplexes. In Re. [8], a double complex structure, which resembles the Čech-de Rham double complex, c. Re. [11], is introduced as a tool to handle this diiculty. Such structures were apparently irst introduced by Weil in his proo o de Rham s theorem, c. Re. [12]. As ar as we know, the construction given in Re. [8] represents the irst time a double complex has been utilized in numerical analysis. he purpose o the present article is to explain the construction given in [8] in the simplest possible setting. In particular, we want to motivate why double complexes seems to be a natural tool or generalizing Clément type operators, based on local projections on macroelements, to obtain bounded commuting projections. In the discussion below, we will thereore restrict the discussion to the lowest-order case o Whitney elements and to the case o two space dimensions. II. HE DE RHAM COMPLEX AND IS DISCREIZAION For a given polygon R 2, we consider the associated de Rham complex o the orm H 1 ( ) grad rot H (rot, ) L 2 ( ). (2.1) Here the operator rot is given by rot v := x2 v 1 x1 v 2, mapping a vector ield v into scalar ields. he space H (rot, ) is the space o all vector ields v L 2 ( ) 2 with rot v L 2 ( ), while H 1 ( ) is the corresponding Sobolev space consisting o all u L 2 ( ) with grad u L 2 ( ) 2. he sequence (2.1) is a complex since rot grad = 0. o enhance readability, we will continue to use bold typeace or vector-valued quantities and or operators returning vector ields. We will consider the simplest inite element subsimplex o (2.1). Hence, or a given triangulation h o, weletw( h ) H 1 ( ) be the corresponding space o continuous piecewise linear Numerical Methods or Partial Dierential Equations DOI /num

3 DOUBLE COMPLEXES AND LOCAL COCHAIN PROJECIONS 3 unctions and P 0 ( h ) L 2 ( ) the space o piecewise constants. he space N ( h ) H (rot, ) is requently reerred to as the rotated Raviart homas space in the numerical analysis literature. It consists o all piecewise rigid motions with continuous tangential components on all edges o h. Alternatively, the spaces W( h ), N ( h ), and P 0 ( h ) correspond to the space o Whitney orms o order 0, 1, and 2, respectively. It is a simple act that W( h ) grad rot N ( h ) P 0 ( h ) (2.2) is a subcomplex o (2.1). In particular, grad(w ( h )) N ( h ) and rot(n ( h )) P 0 ( h ). Our goal is to deine projection operators πh 0, π 1 h, and π h 2 such that the diagram grad rot H 1 ( ) H (rot, ) L 2 ( ) πh 0 π 1 h πh 2 grad W( h ) N ( h ) rot P 0 ( h ) commutes. In addition, the projections πh i H 1 ( ), H (rot, ), and L 2 ( ), respectively. will be local and bounded in the norms o III. HE PROJECION π 0 h We will irst deine the projection π 0 h. We let j( h ) be the set o subsimplexes o h o dimension j. In other words, 0 ( h ) is the set o vertices, 1 ( h ) the set o edges, and 2 ( h ) the set o triangles, while ( h ) is the union o the three sets. For each subset ( h ), the associated macroelement is given by := { h, ( ) }. A vertex macroelement and an edge macroelement are shown in Fig. 1. I u H 1 ( ), then the piecewise linear unction πh 0 u is determined by its values at each vertex, since π 0 h u = π 0 h u(y)λ y, y 0 ( h ) where λ y is the piecewise linear hat unction associated to the vertex y, that is, λ y (y) = 1 and λ y 0 on the complement o the macroelement y. Hence, to determine πh 0 u, it is enough to FIG. 1. (a) Vertex macroelement and (b) edge macroelement. Numerical Methods or Partial Dierential Equations DOI /num

4 4 FALK AND WINHER determine πh 0u(y) or all y 0( h ). However, since unctions u H 1 ( ) in general do not have point values, we cannot take πh 0 u(y) to be u(y). Instead, the idea o the Clément operator is to compute πh 0u(y), or all y 0( h ), rom a local projection P y applied to u. he local projections P y are deined by solving discrete Laplace Neumann problems on the macroelements y. We let W( y,h ) be the restriction o the space W( h ) to y, that is, W( y,h ) is the space o piecewise linear unctions on y. he solution operator or the discrete Laplace Neumann problem can naturally be separated into two parts, a mean value contribution and a second part which only depends on the gradient o the unction u. More precisely, we deine P y u by P y u = y 1 u dx + Q 0 y u, where y = dx. he operator y Q0 y maps H 1 ( y ) into W( y,h ) such that the mean value o Q 0 y v over y is zero, and grad(q 0 yv v) grad w dx = 0, w W( y,h). y his deines Q 0 y v uniquely. he operator π h 0 is then deined by the condition π h 0u(y) = P yu(y) or each y 0 ( h ). Alternatively, π 0 h u = P y u(y)λ y. y y 0 ( h ) he operator π 0 h is bounded in H 1 ( ). Furthermore, it is straightorward to check that it is a projection, that is, π 0 h u = u i u W( h), since we obviously have P y u(y) = u(y) in this case. IV. HE PROJECION π 1 h In this section, we will construct the projection π 1 h. In addition to the macroelements, we will also need extended macroelements given by = y 0 ( ) y. FIG. 2. he extended macroelement or =[y 0, y 1 ]. Numerical Methods or Partial Dierential Equations DOI /num

5 DOUBLE COMPLEXES AND LOCAL COCHAIN PROJECIONS 5 Alternatively, is the union o all triangles o h which intersect. An example o an extended macroelement is shown in Fig. 2. In the special case that dim = 0, that is, is a vertex, then =. In general, i, g ( h ) with g ( ) then g and e g e. We let e,h be the restriction o h to. We will assume throughout that all the macroelements are simply connected. On these macroelements, we utilize discrete complexes o the orm W( e,h ) curl R ( e,h ) div P 0 ( e,h). (4.1) Here curl denotes the two-dimensional curl-operator, that is, the operator which maps a scalar ield u to the vector ield curlu = ( x2 u, x1 u). he complex property ollows since div curl = 0. he spaces W( e,h ) and P 0 ( e,h) consist o piecewise linear and piecewise constant unctions on, and restricted to vanishing boundary values in the piecewise linear case, and to vanishing integral over in the piecewise constant case. Finally, the space R ( e,h) is the lowest order Raviart homas space on with vanishing normal components on the boundary. Since the macroelements are simply connected, it ollows that the complex given in (4.1) is exact. his means that any element in P 0 ( e,h) can be expressed as div v, where v R ( e and any divergence ree element o R ( e,h) is equal to curl w or a unique w W( Recall that the projection π 1 h is required to satisy the commuting property π 1 h grad u = grad π 0 h u, u H 1 ( ). (4.2),h ), e,h ). Since the let hand side o this identity only depends on grad u, so must the right hand side. Furthermore, since we want π 1 h to be local, we need to see that grad π h 0 u depends locally on grad u. We introduce the mean value operator, Mh 0 : L2 ( ) W( h ),by M 0 h u = ( ) uz 0 y dx λ y, where z 0 y = y 1. hen we have that y 0 ( h ) π 0 h u = M0 h u + y y 0 ( h ) Q 0 y u(y)λ y. (4.3) From the deinition o the operator Q 0 y, we see that the second term here already depends on grad u. Furthermore, we observe that grad M 0 h u = ( ) uz 0 y dx grad λ y. y 0 ( h ) Let =[y 0, y 1 ] 1 ( h ) be a ixed edge with vertices y 0 and y 1, and consider the tangential component o grad Mh 0 u on.wehave ( ) grad M 0 h u (y 1 y 0 ) ds = uz 0 y 1 dx uz 0 y 0 dx = u(δz 0 ) dx, y 1 y 0 y Numerical Methods or Partial Dierential Equations DOI /num

6 6 FALK AND WINHER where (δz 0 ) = zy 0 0 zy 0 1 and is the length o. Here, we assume that the unctions zy 0 are extended by zero outside y, such that (δz 0 ) is a piecewise constant unction on the extended macroelement, and with integral equal to zero. In other words, (δz0 ) belongs to the space P 0 ( e,h), and by the exactness o the complex (4.1), it ollows that there is a unique unction z 1 R ( e ) such that,h div z 1 = (δz0 ), and z 1 curl w dx = 0, w W( e,h ). Hence, rom integration by parts we obtain grad M 0 h u (y 1 y 0 ) ds = u div z 1 dx = grad u z 1 dx. Recall that unctions in N ( h ) are uniquely determined by the integrals o the tangential components over all edges. From the calculations above, we can thereore conclude that grad M 0 h u = 1 ( h ) ( grad u z 1 dx ) φ, (4.4) where φ N ( h ) is the Whitney 1-orm associated to the edge =[y 0, y 1 ], scaled such that φ (y 1 y 0 ) ds =. In other words, φ = λ y0 grad λ y1 λ y1 grad λ y0, and any v N ( h ) admits a representation o the orm v = =[y 0,y 1 ] 1 ( h ) 1 For any v L 2 ( ) 2, we now deine M 1 h v N ( h) by M 1 h v = 1 ( h ) ( v (y 1 y 0 ) ds φ. v z 1 dx ) φ. he identity grad Mh 0u = M1 h grad u ollows rom (4.4). For each y 0( h ), we introduce the operator Q 1 y, : L2 ( y ) 2 W( y,h ) deined by (grad (Q 1 y, v) v) grad w dx = 0, w W( y,h), y with the mean value o Q 1 y, v set to zero. Hence, by construction, we have Q 1 y, grad u = Q0 yu. (4.5) Numerical Methods or Partial Dierential Equations DOI /num

7 DOUBLE COMPLEXES AND LOCAL COCHAIN PROJECIONS 7 hereore, i we deine an operator S 1 h : L2 ( ) 2 N ( h ) by S 1 h v = M1 h v + y 0 ( h ) Q 1 y, v(y) grad λ y, then the desired commuting relation grad πh 0u = S1 h grad u ollows, c. (4.3) and (4.5). However, the operator S 1 h will in general not be a projection onto the space N ( h). hereore, the operator π 1 h will instead be o the orm π 1 h v = S1 h v + ) ( 1 (I S 1 h )Q1 v (y 1 y 0 ) ds φ, =[y 0,y 1 ] 1 ( h ) where Q 1 is a local projection deined with respect to the extended macroelement e. he operator Q 1 : H (rot, e ) N ( e,h) is deined by (Q 1 e v v) grad w dx = 0, w W(,h ), rot(q 1 v v)rot ψ dx = 0, ψ N ( e,h). (4.6) hese conditions determine Q 1 v N ( e,h) uniquely as a consequence o the exactness o the complex (2.2) restricted to the domain. Furthermore, the operator Q1 is a local projection. In act, the operator Q 1 admits the decomposition Q 1 v = grad Q0 v + Q2 rot v, (4.7) where the operators Q 0 and Q2 are deined by subsystems o (4.6). More precisely, Q0 v W( e,h ) has integral zero over e and satisies (grad Q 0 v v) grad w dx = 0, w W( e,h ), while Q 2 u N ( e,h) is determined by Q 2 u grad w dx = 0, w (rot Q 2 u u) rot ψ dx = 0, ψ W( e,h ), N ( e,h ), or any u L 2 ( ). he operator π 1 e e h is a projection onto N (,h), since or any v N (,h ),wehave π 1 h v = ) ( 1 Q 1 v (y 1 y 0 ) ds φ = v. =[y 0,y 1 ] 1 ( h ) Numerical Methods or Partial Dierential Equations DOI /num

8 8 FALK AND WINHER On the other hand, it ollows rom (4.7) that or w H 1 ( ), Q1 grad w = grad Q0 grad w, and this implies that on, S 1 h Q1 grad w = S1 h grad Q0 grad w = grad π 0 h Q0 grad w = grad Q 0 grad w = Q1 grad w. hereore, it ollows that or any =[y 0, y 1 ] 1 ( h ) and w H 1 ( ), (I S 1 h )Q1 grad w (y 1 y 0 ) = 0 on, (4.8) and as a consequence, π 1 h grad w = S1 h grad w = grad π 0 h w. We have thereore seen that π 1 h is a projection operator, which is local and satisies the desired commuting relation (4.2). V. HE DOUBLE COMPLEX In the construction o the projection π 1 h above, we have already implicitly used a double complex structure. o see this more clearly, consider the direct sum over all y 0 ( h ) o the complexes o the orm (4.1). his gives the exact complex y 0 ( h ) W( e,h ) curl y 0 ( h ) R ( e,h ) div y 0 ( h ) P 0 ( e,h ). Here the dierential operators curl and div are applied to each component in the sum. he operator δ = δ 0, introduced in the construction o the operator π 1 h above, represents another operator naturally acting on these spaces. he operator δ 0 is o the orm δ 0 : y 0 ( h ) V 0 (y) 1 ( h ) V 1 ( ), (δ 0 u) = u y0 u y1 i =[y 0, y 1 ]. Here the space V m ( ) is a substitute or any o the local spaces W( e,h ), R ( e,h ),or P 0 ( e,h ) or m ( h ). Furthermore, it is implicitly assumed that the unctions u y are extended by zero outside y. In act, i u P y 0 ( h ) 0 ( y,h ) and all the components o u = { u y y 0 ( h ) } have the same mean value with respect to y, the unction δ 0 u will be in P 1 ( h ) 0 ( e,h ). his was exactly what we utilized above to deine z 1 = { } z 1 1 ( h ) R ( e,h) such that div z 1 = δz 0. We similarly deine an operator δ = δ 1 : g 1 ( h ) V 1(g) 2 ( h ) V 2( ) by (δ 1 u) = u [y1,y 2 ] u [y0,y 2 ] + u [y0,y 1 ] i =[y 0, y 1, y 2 ], Numerical Methods or Partial Dierential Equations DOI /num

9 DOUBLE COMPLEXES AND LOCAL COCHAIN PROJECIONS 9 where the notation [,..., ] is used to denote convex combination. We obtain a double complex o the orm 0 ( h ) 1 ( h ) W( e,h ) curl 0 ( h ) R ( e,h ) div 0 ( h ) δ 0 δ 0 δ 0 W( e,h ) curl R ( e,h ) div 1 ( h ) δ 1 δ 1 δ 1 2 ( h ) W( e,h ) curl 2 ( h ) R ( e,h ) div 1 ( h ) 2 ( h ) P 0 ( e,h ) P 0 ( e,h ) P 0 ( e,h ) his is a double complex in the sense that each row and each column is a complex. In particular, δ 1 δ 0 = 0. Furthermore, the operators δ 0 and δ 1 commute with the dierential operators curl and div, that is, δ i curl = curl δ i and δ i div = div δ i. In particular, consider the unction δ 1 z 1 2 ( h ) R ( e,h ), where z1 = { } z 1 the unction introduced above to deine the operator M 1 h. his unction is divergence ree, since divδ 1 z 1 = δ 1 div z 1 = δ 1 δ 0 z 0 = 0. 1 ( h ) is As a consequence o the exactness o the last row o the double complex above, we conclude that there is a unique z 2 W( 0 ( h ) e,h ) such that curl z2 = δ 1 z 1. he components o the unction z 2 will be utilized in the construction o the operator πh 2 below. VI. HE PROJECION πh 2 It remains to construct the locally deined projection πh 2 onto P 0( h ) such that π 2 h rot v = rot π 1 h v. We start by computing rot S 1 h v = rot M1 h v.wehave rot M 1 h v = ( ) v z 1 dx rot φ. 1 ( h ) Let =[y 0, y 1, y 2 ] 2 ( h ), where the vertices are ordered counter clockwise. he unction rot M 1 h is a constant on and the only nonzero contributions in the sum above on arise rom the three edges [y 1, y 2 ], [y 0, y 2 ], and [y 0, y 1 ]. From the edge =[y 1, y 2 ] we have rot φ [y1,y 2 ] dx = 1 φ [y1,y 2 ] (y 2 y 1 )ds = 1. [y 1,y 2 ] Similar calculations show rot φ [y0,y 2 ] dx = 1, and rot φ [y0,y 1 ] dx = 1. Numerical Methods or Partial Dierential Equations DOI /num

10 10 FALK AND WINHER We thereore obtain that rot M 1 h v dx = e v (δ 1 z 1 ) dx = For any u L 2 ( ), we now deine M 2 h u by M 2 h u = 2 ( h ) e e v curl z 2 dx = uz 2 dx. e (rot v)z 2 dx. he identity Mh 2 rot v = rot M1 h v = rot S1 hv is a consequence o the calculations above. From the deinition o the operator π 1 h, we now obtain rot π 1 h v = M2 h rot v + ) ( 1 (I S 1 h )Q1 v (y 1 y 0 ) ds rot φ =[y 0,y 1 ] 1 ( h ) = M 2 h rot v + ) ( 1 (I S 1 h )Q2 rot v (y 1 y 0 ) ds rotφ, =[y 0,y 1 ] 1 ( h ) where the last identity ollows by combining (4.7) and (4.8). Hence, i we deine Sh 2 : L2 ( ) P 0 ( h ) by S 2 h u = M2 h u + ( 1 (I S 1 h )Q2 u (y 1 y 0 ) ds) rot φ, =[y 0,y 1 ] 1 ( h ) then the identity Sh 2 rot v = rot π 1 hv ollows by construction. However, in the present case, the operator Sh 2 is also a projection. o see this, let u P 0( h ). It is enough to show that S 2 h u dx = u dx, 2 ( h ). (6.1) However, due to the exactness o the complex (2.2), restricted to e e, there is a v N (,h ) such that rot v = u on e. hereore, by the projection property o the operator π 1 h we obtain S 2 h u dx = S 2 h rot v dx = rot π 1 h v dx = rot v dx = u dx. We thereore deine π h 2 to be the operator S2 h. VII. CONCLUSIONS We have constructed Clément-type projection operators or discretization o the de Rham complex in two space dimensions. Only the lowest-order inite element spaces, that is, the Whitney orms, are considered. he projections are locally deined, they commute with the dierential operators o the de Rham complex, and they are bounded in the natural Sobolev norms. A discussion in the general case, covering higher-order piecewise polynomial spaces and arbitrary space dimensions, can be ound in the recent paper [8]. Numerical Methods or Partial Dierential Equations DOI /num

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