A UNIFYING THEORY OF A POSTERIORI ERROR CONTROL FOR NONCONFORMING FINITE ELEMENT METHODS

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1 A UNIFYING THORY OF A POSTRIORI RROR CONTROL FOR NONCONFORMING FINIT LMNT MTHODS C. CARSTNSN AND JUN HU Abstract. Residual-based a posteriori error estimates were derived witin one unifying framework for lowest-order conforming, nonconforming, and mixed finite element scemes in [C. Carstensen, Numerisce Matematik 100 (2005) ]. Terein, te key assumption is tat te conforming first-order finite element space V c annulates te linear and bounded residual l written V c kerl. Tat excludes particular nonconforming finite element metods (NCFMs) on parallelograms in tat V c kerl. Te present paper generalises te aforementioned teory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin metods. Te key assumption is te existence of some bounded linear operator Π : V c V nc wit some elementary properties. It is conjectured tat te more general ypotesis (H1)-(H3) can be establised for all known NCFMs. Applications on various nonstandard finite element scemes for te Laplace, Stokes, and Navier-Lamé equations illustrate te presented unifying teory of a posteriori error control for nonconforming finite element metods. 1. Unified Mixed Approac to rror Control Suppose tat te primal variable u V (e.g., te displacement field) is accompanied by a dual variable p L (e.g., te flux or stress field). Typically L is some Lebesgue and V is some Sobolev space; suppose trougout tis paper tat L and V are Hilbert spaces and X := L V. Given bounded bilinear forms (1.1) a : L L R and b : L V R and well establised conditions on a and b [16, 20], te linear and bounded operator A : X X, defined by (1.2) (A(p, u))(q, v) := a(p, q) + b(p, v) + b(q, u), is bijective. Ten, given rigt-and sides f L and g V, tere exists some unique (p, u) X wit (1.3) (1.4) a(p, q) + b(q, u) = f(q) for all q L, b(p, v) = g(v) for all v V. Date: October 31, Matematics Subject Classification. 65N10, 65N15, 35J25. Key words and prases. a posteriori error analysis, finite element metod, nonconforming finite element metod, error estimates reliability. Te first autor CC was supported by DFG Researc Center MATHON Matematics for key tecnologies in Berlin and te German Indian Project DST-DAAD (PPP-05). Te second autor JH was partially supported by National Science Foundation of Cina under Grant No

2 2 C. CARSTNSN AND JUN HU Suppose (p, ũ ) L V is some approximation to (p, u) and define (1.5) (1.6) Res L (q) := f(q) a(p, q) b(q, ũ ) for all q L, Res V (v) := g(v) b(p, v) for all v V. Here and trougout, ũ is some continuous and not necessarily discrete function establised as te key ingredient in [23]; owever, te subindex in ũ refers to te fact tat ũ migt be closely related (or designed wit some post-processing) to some discrete function u and ence tat ũ is on our disposal. Since A : X X is an isomorpism, tere olds (1.7) p p L + u ũ V Res L L + Res V V. Here and trougout, an inequality a b replaces a C b wit some multiplicative mes-size independent constant C > 0 tat depends only on te domain and te sape (e.g., troug te aspect ratio) of elements (C > 0 is also independent of crucial parameters as te Lamè parameter λ below). Finally, a b abbreviates a b a. Remark 1.1. Note tat (1.3)-(1.4) are a primal mixed formulation wit L := L 2 () m n for te Laplace, Stokes, and Navier-Lamé equations under consideration. Trougout tis paper, te discrete component p is derived from u, e.g, p = T u in case p = u for te Laplace equation; wile u is solved from te discrete problem in te displacement-oriented formulation (Sections 4-6 below). Te examples in [23] include conforming, nonconforming and mixed finite element scemes for te Laplace, Stokes, and Navier-Lamé equations. Tis paper will consider suc applications in Section 4, 5, and 6 below for wit focus on NCFMs displayed in Table 4.1, 5.1, 6.1, and 6.2. Te applications of te present teory to mortar and discontinuous Galerkin metods are also condidered in Section 4 for te Poisson problem. Terein, te norms of Res L and Res V are estimated under te general ypotesis tat eac of tose as te form (1.8) Res(v) := g v dx + g v ds for v V. Here and below, V belongs to some Sobolev space V = H0 1()m and g L 2 () m, wile g L 2 ( ) m wit some domain R n and te union of edges (if n = 2) or faces (if n = 3) related to a regular triangulation of. Some required key property in [23] on bot Res = Res L and Res = Res V reads (1.9) V c ker Res V. In tis situation, a typical result of an explicit residual-based error estimation reads (1.10) Res 2 V T g 2 L 2 () + g 2 L 2 () =: η2. Here and trougout, T and denote local mes-sizes in te underlying triangulation, i.e., T T = diam(t) for any T T, and = diam() for any.

3 UNIFYING THORY OF A POSTRIORI RROR CONTROL 3 V c includes te first-order finite element functions to ensure (1.10). Details on te notation and te concrete examples will be given below. Te terms in (1.8) often result from some discretisation of te equilibration condition (1.4),e.g., via an integration by parts, and ence te term Res V is referred to as te equilibration residual. Te first aim of tis paper is te generalisation of (1.10) for Res = Res V in Teorem 2.1 of Section 2 to allow te control of certain nonstandard finite element scemes witout te condition (1.9) in Sections 4-6. Here, one key teory is to replace (1.9) by assumptions (H1)-(H3) on some Clement-type operator J and some linear bounded operator Π between te conforming and nonconforming finite element spaces. For te Laplace, Stokes, and Navier-Lamé equations considered erein, one can observe from te definitions of a(, ) and b(, ) in Sections 4-6 below tat te consistency residuum Res L from (1.5) can also be written in te form (1.8). Wit some bounded linear operator A : L := L 2 () m n L, te norm of Res L allows te form (1.11) min ũ V Res L L min ũ V A(p ) Dũ L 2 (). Terein Dũ denotes te functional matrix of all first-order partial derivatives (e.g., te gradient and possibly also te Green strain of linear elasticity) of te Sobolev function ũ in Sections 4-6. Remark 1.2. Tis observation can also be found in [23, Teorem 2.2] for te Laplace equation wit A = id te identity operator. For te Stokes and Navier-Lamé equations, te operators A are dev and µ C 1 wit te operators dev (and µ) and C 1 of Sections 5 and 6. Since A(p ) Dũ L 2 () D T u Dũ L 2 () for te aforementioned problems, te second aim of tis paper reads min D T u Dũ 2 L ũ V 2 () γ τ ([D T (ψ z u )]) 2 L 2 () =: µ2 z K() for te jumps [D T (u ψ z )] of a discrete nonconforming finite element function u times a weigt-function ψ z across some side wit vertex z; details on te notation can be found in Section 3. Te second main result (Teorem 3.1) olds for all piecewise gradients and employs a localisation argument wit te (modified) at functions (ψ z : z K) of te free nodes K. Ten, a summary of tese two aims (See, Teorem 2.1 and Teorem 3.1) and (1.7) concludes te main result of tis paper (1.12) p p L η + µ + osc(g) for te unified a posterior error estimate of te nonconforming finite element metods wit (H1)-(H3) of Section 2 and for all aforemented problems. Tis conclusion will be exibited for eac problem in Sections 4-6, wat is left is to ceck te well-posdeness of (1.3)-(1.4) (or (1.2)) for eac problem and (H1)-(H3) for eac nonconforming finite element sceme; see Sections 4-6 for furter details.

4 4 C. CARSTNSN AND JUN HU Te rest of tis paper is organized as follows. Wile Section 2-3 treat general assertions on (1.10)-(1.11) were condition (1.9) is substituted by (H1)-(H3), Sections 4-6 conclude tis paper wit particular model examples in 2D (and some in 3D) wit first reliability proofs for many nonstandard finite element error estimates. Trougout tis paper, V c nc and V denote conforming and nonconforming finite element spaces based on a regular triangulation T of ; ν denotes te normal unit vector along te boundary ; τ denotes te tangent vector along te boundary for 2D. Colon : denotes te scalar product in R m n, i.e., A : B := m j=1 n k=1 A jkb jk. 2. Reliability control of te equilibrium residual Tis section establises an explicit residual-based error estimate (1.8) for a class of nonstandard finite element scemes. Let V = H 1 0 ()m and L = L 2 (; R m n ) denote standard Sobolev and Lebesgue spaces on some bounded Lipscitz domain in R n wit a piecewise flat boundary Γ. Suppose tat te closure is covered exactly by a regular triangulation T of into (closed) triangles or parallelograms in 2D, tetraedrons or parallelepipeds in 3D (or oter unions of simplices). It is assumed, tat (2.1) = T and T 1 T 2 = 0 for T 1, T 2 T wit T 1 T 2, were denotes te volume (as well as te modulus of a vector etc. were tere is no real risk of confusion). Te remaining assumptions on te sape regularity of T are idden in te following abstract conditions. (H1) Tere exists a Clement-type operator J : V V c into some (conforming) V of T -piecewise smoot functions suc tat, for all v V and subspace V c T T (2.2) 1 T v Jv L 2 (T) + 1/2 T v Jv L 2 ( T) + D(v Jv) L 2 (T) Dv L 2 (ω T ), wit some neigbourood ω T of T suc tat (ω T : T T ) as finite overlap (2.3) max x card{t T : x ω T } 1. (H2) Tere exists a nonconforming space V nc L 2 () m of T -piecewise smoot and, in general, discontinuous functions V nc H 1 (T ) m V. Given distinct T 1, T 2 T, teir intersection T 1 T 2 as zero volume measure by (2.1) but possibly a positive surface measure. Te set of all interior (edges or faces etc.) T 1 T 2 = is denoted by. For any v V nc, te jump (2.4) [v ] (x) := (v T2 )(x) (v T1 )(x) for x. across wit = T 1 T 2 is fixed up to te sign wic results from te orientation of te unit vector ν on (e.g. ν points outward of T 2 ). Te sape regularity of T and is described by te assumption (2.5) T diam(ω T ) for all, T T wit T.

5 UNIFYING THORY OF A POSTRIORI RROR CONTROL 5 Remark 2.1. Te trace inequality yields, for v V and T T [44, 18], (2.6) v L 2 ( T) 1/2 T v L 2 (T) + 1/2 T Dv L 2 (T). Hence te trace term wit L 2 ( T) in (2.2) is estimated by te oter two L 2 (T) norms. More over, if (T) denotes te set of all wit T, te sape regularity (2.5) sows tat (2.7) (T) 1 v Jv 2 L 2 () Dv L 2 (ω T ). Remark 2.2. Te conforming functions are given as tose wit vanising jumps, i.e., v V c implies [v ] = 0 for all. Te aforementioned standard assumptions are typical in finite element simulations. Te innovative condition on te nonstandard finite element space V nc and te conforming counterpart V c of (H1)-(H2) is te following. (H3) Tere exists some operator Π : V c V nc suc tat, for all v V c and all T T, tere olds (2.8) (Πv ) L 2 (T) v L 2 (ω T ) and v dx = Πv dx. Moreover, for some given discrete approximation p L 2 (; R m n ) and te T - piecewise gradient D T, tere olds (2.9) p : D T v dx = p : D T (Πv ) dx. A direct consequence of (2.8) is (2.10) 1 T v Πv L 2 (T) Dv L 2 (ω T ) for all T T. Given g L 2 () m and p as above, te residual Res V V is, for v V + V nc L 2 (, R m n ) defined by (2.11) Res V (v) := g v dx p : D T v dx. Te residual is supposed to stem from a nonstandard finite element sceme wit and ence V nc (2.12) Res V (v ) = 0 for all v V nc. Wit te abbreviation g T := T 1 T g(x)dx Rm, te data oscillation reads T T (2.13) osc(g) := ( T T 2 T g g T 2 L 2 (T) )1/2. Under te assumptions of (H1)-(H3), te residual-based error estimator (2.14) η := ( 2 T g + div p 2 L 2 (T) )1/2 + ( [p ] ν 2 L 2 () )1/2 T T is reliable in te following sense. Teorem 2.1. Tere olds Res V V η + osc(g).

6 6 C. CARSTNSN AND JUN HU Proof. Given any v V wit ΠJv V nc, (2.12) leads to Res V (v) = g (v ΠJv) dx p : D T (v ΠJv) dx. An elementwise integration by parts and a careful re-arrangement of boundary pieces leads to p : D(v Jv) dx = (div T p ) (v Jv) dx + [p ] ν (v Jv) ds. Te combination of te two identities wit (2.9), i.e., p : D T (Jv ΠJv) dx = 0, were v is replaced by Jv V c, reads Res V (v) = (g + div T p ) (v Jv) dx + g (Jv ΠJv) dx [p ] ν (v Jv) ds =: I 1 + I 2 + I 3. Te first integral I 1 on te rigt-and side is controlled wit (2.2)-(2.3), Hölder inequality and Caucy inequalities. Tis leads to I 1 ( T T 2 T g + div p 2 L 2 (T) )1/2 Dv L 2 (). Te second term I 2 requires (2.8), (2.10) and (2.13). Tis yields I 2 = g (Jv ΠJv) dx T T T = (g g T ) (Jv ΠJv) dx T T T T T T g g T L 2 (T) 1 T Jv ΠJv L 2 (T) osc(g)( Dv 2 L 2 (ω T ) )1/2 T T osc(g) Dv L 2 (). Standard arguments wit (2.1) (2.3) and (2.7) control te last term I 3 1/2 [p ] ν L 2 () 1/2 v Jv L 2 () ( [p ] ν 2 L 2 () )1/2 Dv L 2 (). Altogeter, tere follows te assertion Res V (v) = I 1 + I 2 + I 3 (η + osc(g)) Dv L 2 ().

7 UNIFYING THORY OF A POSTRIORI RROR CONTROL 7 3. Reliability control of te consistency residual Tis section establises a general control of te consistency residual (1.11). Given u V nc wit D T u L 2 (; R m n ) and te conforming finite element space V c from (H1)-(H3), let (ψ z : z K) denote a Lipscitz continuous partition of unity, (3.1) ψ z = 1 in. z K Moreover, for any z K, suppose tat, ψ z vanises outside an open and connected set z (3.2) supp ψ z z and max card{z K : x z } 1. x Given z K, let (z) := { : ψ z 0} denote te set of edges, were ψ z is nonvanising. For any edge let K() denote te set of all z K wit (z). Te tangential component of a vector v R n is defined as { v τ if n = 2, (3.3) γ τ (v) := v ν if n = 3. Te general estimator ( (3.4) µ := z K() is reliable in te following sense. γ τ ([D T (ψ z u )]) 2 L 2 () ) 1/2 Teorem 3.1. For n = 2, 3, tere olds min ũ V D T u Dũ L 2 () µ. Remark 3.1. In te examples below, 0 ψ z 1 is a finite sum of at functions and continuous suc tat γ τ ([D T (ψ z u )]) = γ τ (Dψ z )[u ] + ψ z γ τ ([D T u ]). Moreover, te polynomial [u ] as some zero on and allows an estimate (3.5) [u ] L 2 () γ τ ([D T u ]) L 2 (). Wit Dψ z L 1, one deduces (3.6) µ ( γ τ ([D T u ]) 2 L 2 () )1/2. Tis estimator is te frequently found version of te consistency error control [36, 37, 31, 27]. Remark 3.2. Teorem 3.1 generalizes [31]. To control te nonconformity, it was assumed terein tat (3.7) [v ] ds = 0 for and v ds = 0 for on for all v V nc. Te condition (3.7) is removed in Teorem 3.1 of te present paper. Proof of Teorem 3.1. Given z K let a z and b z denote te functions of te Helmoltz decomposition of D T (ψ z u ), i.e., D T (ψ z u ) = Da z + curl b z L,

8 8 C. CARSTNSN AND JUN HU Here a z H 1 0 ( z), b z H 1 ( z ) k wit z b z (x) dx = 0, and k = 1 for n = 2 wile k = 3 for n = 3. Since z curl b z : Da dx = 0 for any a H 1 0( z ), curlb z 2 L 2 ( z) = min Da D T (ψ z u ) 2 a H0 1(z) L 2 ( z) = (curl b z ) : D T (ψ z u ) dx. z An elementwise integration by parts followed by curl T D T 0 yields (curl b z ) : D T (ψ z u ) dx = ± b S z γ τ ([D T (ψ z u )]) ds z (z) γ τ ([D T (ψ z u )]) L 2 ( S (z)) b z L 2 ( S (z)), were (z) := { : ψ z 0}. Te well-known trace teorem on eac element domain K, namely leads to te estimate A Poincaré inequality gives b z L 2 ( K) 1/2 K b z L 2 (K) + 1/2 K Db z L 2 (K), b z L 2 ( S (z)) z 1/2 b z L 2 ( z) + 1/2 z Db z L 2 ( z). b z L 2 ( z) z Db z L 2 ( z) z curl b z L 2 ( z). Te latter inequality results from te stability of te Helmoltz decomposition [27, 44] wit an z -independent constant; it reads Db z L 2 ( z) = curl b z L 2 ( z) in 2D. Te combination of te proceeding tree inequalities leads to b z L 2 ( S (z)) 1/2 z curl b z L 2 ( z). Since T z z := diam( z ), z K, te aforementioned arguments imply Da z D T (ψ z u ) L 2 ( z) 1/2 γ τ ([D T (ψ z u )]) L 2 ( S (z)). Since ψ z 1 and ũ := a z H0 1 (), tis estimate plus te finite overlap of z K z K all z and (z) prove te assertion. In fact, D T u Dũ 2 L = z K(D T (ψ z u ) Da z ) 2 L z K Da z D T (ψ z u ) 2 L 2 ( z) z K ( 1/2 γ τ ([D T (ψ z u )]) 2 L 2 ( S (z)) z K() γ τ ([D T (ψ z u )]) 2 L 2 () ).

9 UNIFYING THORY OF A POSTRIORI RROR CONTROL 9 4. Application to Laplace quation Tis section is devoted to te Poisson problem and its residual-based a posteriori finite element error control. Subsection 4.1 introduces te model problem and Subsection 4.2 some required notations. Subsection 4.3 presents a list of examples. Subsections present te applications of te teory to te mortar and dg finite element metods. Subsection concerns te extension of te present teory to te ig-order nonconforming finite element metod Model Problem. Te Lebesgue and Sobolev spaces L 2 () and H 1 () are defined as usual and (4.1) L := L 2 () n and V := H 1 0 () := {w H1 () : w = 0 on }. Te gradient operator maps V into L. Given g L 2 () let u V denote te solution to te Poisson Problem (4.2) u + g = 0 in and u = 0 on. Ten, te flux p := u L and u V satisfy (4.3) (A(p, u))(q, v) := a(p, q) + b(p, v) + b(q, u)! = gv dx for all (q, v) X = L V. Trougout tis section, (1.1)-(1.7) old for (4.4) a(p, q) := p q dx and b(p, v) := p v dx. Te operator A : X X is bounded, linear, and bijective [23] Nonconforming finite element metods and unified a posteriori error estimators. Let P k (T) and Q k (T) denote te space of algebraic polynomials of total and partial degree k, respectively, and set P k (T) = P k (T) and P k (T) = Q k (T) for a triangle (or tetraedron) and parallelogram (or parallelepiped), respectively. Define P k (T ) := {v L 2 () : T T, v T P k (T)} for k = 0, 1; (4.5) S 1 (T ) := P 1 (T ) C() and V c := S 1 0(T ) := S 1 (T ) V. Let N denote te set of nodes (i.e., vertices of elements in T )u. T and denote T - and -piecewise constant functions on and = defined by T T := T := diam(t) and := := diam() for T T and. For a given quadrilateral or parallelepiped element T T, F T : ˆT = [ 1, 1] n T denotes te canonical bilinear transformation. Let V nc denote some nonconforming finite element space specified in Table 4.1. For te moment solely suppose tat T v L for any v V nc, were T denote te T -piecewise action of te gradient operator. Te finite element solution u V nc is te unique solution to (4.6) T u T v dx = gv dx for all v V nc.

10 10 C. CARSTNSN AND JUN HU picture name reference space Crouzeix-Raviart [35] V CR Wilson [73, 64] V Wil Han [45] V Han NR (midpoint) [60] V RT,P NR (average) [60] V RT,A CNR [50] V CRT DSSY [40] V DSSY Table 4.1. Nonconforming lements for te Laplace quation (4.2) wit (H1)-(H3) and te rror stimate (4.8). Te aim is to estimate te flux error p p for te discrete flux p := T u L = L 2 () n. For any ũ V tere olds (1.7) for Res L L and Res V V defined, for all q L and v V, by Res L (q) := q ( ũ p ) dx and (4.7) Res V (v) := gv dx + p v dx xamples. Tis subsection presents a list of 2D and 3D nonconforming finite element spaces V nc of Table 4.1 wit (H1)-(H3), so tat (4.8) p p L 2 () η + µ + osc(g) wit η from (2.14), µ from (3.4), and osc(g) from (2.13). Tis list below is not compreensive. In fact, we conjecture tat all known NCFMs could be analyzed in te present framework. Only te triangular Crouzeix-Raviart element as already been analyzed in [23]. Te present unifying teory leads to new error control (4.8) for all nonconforming finite elements of Subsubsections Te triangular Crouzeix-Raviart element. Based on te regular triangulation T into simplices, te set of midpoints M of edges (or faces), te non-conforming Crouzeix-Raviart finite element space reads (in 2D and 3D) (4.9) V CR Since V C := {v P 1 (T ) : v continuous at M and v = 0 at M }. V CR, ten tere olds (H1)-(H3) wit Π = id; cf. Section 4 of [31] for proofs. Similar arguments verify (H1)-(H3) in 3D as well; we terefore omit te details.

11 UNIFYING THORY OF A POSTRIORI RROR CONTROL Te Quadrilateral Wilson element. Let B denote one of te nonconforming quadratic bubble function spaces on te reference element ˆT = [ 1, 1] n, i.e., { span{1 (ξ 2 + η 2 )/2} or span{1 ξ 2, 1 η 2 } for n = 2, B := span{1 ξ 2, 1 η 2, 1 ζ 2 } for n = 3. Te nonconforming quadrilateral Wilson finite element space V Wil (4.10) V Wil = S B wit te factors S := {v H 1 0 () : T T, ˆv = v F T Q 1 ( ˆT)}, B := {v L 2 () : T T, ˆv = v F T B}. [73, 64] reads Tis element is excluded from te analysis of [31, 23] since (3.7) is violated. However, tere olds (H1)-(H3) wit Π = id, te proof is immediate since V c V Wil Te parallelogram nonconforming Han element. Consider te functional (4.11) F (v) = 1 v ds for all (T) and T T. Te parametric formulation of rectangular and parallelogram elements of Han [45] is introduced by (4.12) Q nc H := span{ 1, ξ, η, ξ2 5 3 ξ4, η η4 }. Te nonconforming Han finite element space ten reads (wit [ ] := along ) (4.13) V Han := { v L 2 () : T T, v T F T Q nc H and, F ([v]) = 0 }. Ten tere olds (H1)-(H3) wit te associated interpolation operator Π for V Han [45], te proof follows from ΠV c = V CRT V Han [50] wit V CRT from Subsubsection below. Furter details for te properties of Π can be found in Section 4 of [31], Remark 2.5 and Lemma 3.1 of [50] Te parallelogram nonconforming rotated Q 1 elements. Rannacer and Turek introduce two types of parallelogram nonconforming elements [60], called te NR elements. Te first element RTA uses te average of te function over te edge ( or face) as te local degree of freedom, and te second one RTP uses te value at te midside point ( or midpoint ) of te edge ( or face ) instead. Define { (4.14) Q nc R := span{ 1, ξ, η, ξ 2 η 2 } for n = 2, span{ 1, ξ, η, ζ, ξ 2 η 2, ξ 2 ζ 2 } for n = 3 ten nonconforming space V RT,A V RT,P is defined in (4.13) wit Q nc H replaced by Qnc R, and is defined in (4.9) wit P 1 (T ) replaced by Q nc R. For 2D, following a similar argument for te Han element, one proves tat te average version element satisfies (H1)-(H3) wit te canonical interpolation operator Π for V RT,A ; [31] contains furter details.

12 12 C. CARSTNSN AND JUN HU Te midside point version element is not included in [31] since te condition (3.7) is violated by tis element. However, tere olds equally (H1)-(H3) for it wit te canonical interpolation operator Π of V RT,P. In fact, we ave (4.15) ΠV c = V CRT and V CRT V RT,P, contains te linear part of V c, and only te nonlinear part is excluded [50]. Wit tis fact, (H3) follows from straigt forward investigations. For 3D, define te local interpolation operator Π T : H 1 (T) Q nc R F 1 T by (4.16) F (Π T v) = F (v) for (T) for all v H 1 (T). Since FÊ(v) = 0 for v = ξη, ξζ, ηζ, ξηζ wit Ê ( ˆT), we conclude for any v = a 0 + a 1 ξ + a 2 η + a 3 ζ + a 4 ξη + a 4 ξζ + a 6 ηζ + a 7 ξηζ tat (4.17) Π T v = a 0 + a 1 ξ + a 2 η + a 3 ζ, wit some interpolation constants a 0,...,a 7. Te global interpolation operator Π is defined by Π T = Π T for any T T. Ten (H1)-(H3) eventually follows from (4.17). Remark 4.1. Te analysis does not cover te non-parametric variant of tis element except on parallelogram meses Te parallelogram constrained nonconforming rotated Q 1 elements. Te constrained rotated nonconforming finite element (referred to as CNR element) introduced in [50] is obtained by enforcing a constraint on te NR element on eac element for 2D. Te space of te CNR element reads (4.18) V CRT := {v V RT,A : T T, 1 v ds + 3 v ds = 2 v ds + wit { 1,, 4 } = (T) numbered counterclockwise}. 4 v ds For rectangular and parallelogram meses, te element is equivalent to te P 1 - quadrilateral element of [59]. Ten tere olds (H1)-(H3) wit te interpolation operator Π of V CRT. Te proof follows from te argument for te NR element wit te midside point version. We refer to Section 4 of [31] for more details. Te goal-oriented error control of tis element is given in [43] Te parallelogram DSSY elements. Te DSSY element is obtained by introducing on te reference element [40] wit θ 1 (t) = t t4 and θ 2 (t) = t t t6 and (4.19) Q nc D := { span{1, ξ, η, θ l (ξ) θ l (η)} for l = 1, 2 for n = 2, span{1, ξ, η, ζ, (ξ ξ4 ) (η η4 ), (ξ ξ4 ) (ζ ζ4 )} for n = 3. Te nonconforming finite element spaces V DSSY are defined as in (4.13) wit Q nc H replaced by Q nc DSSY D. Tere olds (H1)-(H3) wit te interpolation operator Π of V, cf. te proof in Section 4 of [31] for 2D. Arguments similar to tose of Subsection verify (H1)-(H3) for 3D.

13 UNIFYING THORY OF A POSTRIORI RROR CONTROL 13 Remark 4.2. Te parallelogram nonconforming element of [57] can also be analyzed by tis unifying teory Comments on mortar finite element metods. Anoter class of nonconforming FM is known as mortar FM [13, 14] were te continuity of u over te common side of two subdomains K and K + in some locally quasi-uniform regular decomposition T H of into triangles is enforced by Lagrange multipliers. Te a posteriori error estimates wit te saturation assumptions are presented in [15, 74]. A more general one is analyzed in [12]. For te ease of te discussion, suppose tat n = 2 and tat te partition T is obtained from T H by refining some of te triangles in T H by some finite number k of successive red-refinements (i.e., cutting a triangle into 4 congruent subtriangles by connecting its edges midpoints) so tat te ratio of te diameters of two neigbouring triangles wit adjusted edges is bounded by 2 k. Notice tat (2.5) olds for all edges of T wile te equivalence wit ω T depends on k. Let V nc be te mortar finite element space wit respect to T as in [12]. Wit V c := V P 1(T H ) one can prove (H1) by along te lines of [24]. Since V c V nc, (H3) olds for Π = id. Ten, Teorem 2.1 reads Res 2 V T T H 2 T g + div p 2 L 2 (T) + H [p ] ν 2 L 2 () for H T := max{diam(k) : T K T H } and H := max{diam(k) : K, K T H }. Moreover, Teorem 3.1 yields ( wit T = T etc.) min Res L 2 L H ũ V γ τ ([D T ((ψ z u )]) 2 L 2 (). z K() Terein, ψ z is te partition of unity based wit respect to T H and H Dψ z L 1. Tis reliability error estimate is essential Teorem 3.4 in [12]. In fact, since (in 2D) ( s [ψ zu ] = z) s ψ [u ] + ψ z [ u / s], tere olds (wit an inverse estimate [u ]/ s L 2 () H 1 [u ] L 2 ()) tat H s [ψ zu ] 2 L 2 () H H 2 T [u ] 2 L 2 () + H [ u / s] 2 L 2 () (H H 2 T + H 1 ) [u ] 2 L 2 () H 1 [u ] 2 L 2 (). Altogeter, te upper bounds for (1.7) wit p := D T u and p = u reads p p L 2 () HT 2 g+divp 2 L 2 (T) + H [p ] ν 2 L 2 () + [u ] 2 L 2 (). T T H 1 Terein, denote te set of edges on te boundary. Notice [u ] = 0 on edges interior to T T H. In comparison to [12, Teorem 3.4], te factor 2 k terein is idden erein te mes-sizes H T, H.

14 14 C. CARSTNSN AND JUN HU 4.5. Comments on discontinuous Galerkin metods. Te feature for te discontinous Galerkin(abbreviated dg ereafter) metods [4, 38, 71, 5, 7, 8, 52] lies in tat te trial and test spaces consist of piecewise discontinuous polynomials. A posteriori error estimates for dg type metods are considered in [53, 62, 61, 21, 9, 48] for second order elliptic problems, in [46] for te Stokes problem, and in [47, 72] for plane elasticity. Tis subsection comments on te extension of te unifying teory to dg FM. For any v P k (T ), te average across = T 1 T 2 reads < v > (x) := 1/2((v T1 )(x) + (v T2 )(x)) for x. Wit some appropriately cosen constant γ, te modified bilinear form is defined as a γ (u, v ) := u v dx + γ 1 [u ] [v ] ds T T T (< u > ν [v ] + < v > ν [u ] ) ds u ν v + v ν u ) ds + γ ( 1 u v ds for any u, v P k (T )+H 1 0(). Tis is te symmetric dg metod from [7, 8, 52, 53]. Te discontinuous Galerkin solution u P k (T ) is caracterized by (4.20) a γ (u, v ) = (g, v ) L 2 () for any v P k (T ). From V c P k(t ), tere olds (H3) wit Π = id. Teorem 3.1 yields min Res L 2 L ũ V γ τ ([D T ((ψ z u )]) 2 L 2 (). z K() To bound Res V V, let v V and deduce Res V (v) = gv dx + T u v v dx = gv dx + T u v v dx < u > ν [v] ds + γ < u > ν [v ] ds + γ 1 [u ] [v] dx It follows from Jv V P k (T ) tat Res V V η + osc(g) + ( 1 [u ] 2 L 2 () + 1 u v ds. 1 u 2 L 2 ()) 1/2. Remark 4.3. A combination of te above estimates for Res V V and min ũ V Res L L wit (1.7) recovers te estimate p p L 2 () Res V V + min ũ V Res L L,

15 UNIFYING THORY OF A POSTRIORI RROR CONTROL 15 wic appeared in Teorem 3.1 of [9] and Teorem 3.1 from [53] witout te assumption u H 2 (). Were p = u and p = T u. Remark 4.4. For brevity, we only consider te a posteriori error estimate of te symmetric dg metods for te Poisson equation, te analysis wit corresponding modifications can equally apply to te Stokes problem in Section 5, and te elasticity in Section 6. In particular, tis yields te a posteriori error control from Teorem 4.1 of [72] for te plane elasticity, and Teorem 3.1 of [46] for te Stokes problem. Moreover, te unifying teory can be generalized to oter dg metods reviewed in [5] Comments on ig-order nonconforming scemes. In tis paper, we focus on te first-order nonconforming finite element metod. Te present unifying teory can be extended to ig-order nonconforming finite element metods wit te corresponding modifications in (H1)-(H3). In fact, Teorem 3.1 olds equally for all nonstandard finite element metods. We only need to modify te te conforming space V c in (H1) and (H3) and its associated Clemént interpolation operator. For instance, (H3) reads Πv qdx = v qdx for any q P k 1 (T) and for any v V c. T T 5. Applications to te Stokes Problem 5.1. Te Stokes Problem. Te unsymmetric formulation of te Stokes problem reads: Given g L 2 () n seek (u, p) H0 1()n L 2 0 (), suc tat for all (v, q) H0 1()n L 2 0 (), (5.1) µ Du : Dv dx p div v dx q div u dx = g v dx. Here, L 2 0 () := {q L2 () : q dx = 0} L2 ()/R fixes a global additive constant in te pressure p (note tat p is not te flux from te previous section). Te unique existence of solution to (5.1) is well known. Set (5.2) 1 a(σ, τ) := µ dev σ : dev τ dx for all σ, τ L := {τ L2 (, R n n ), trτ dx = 0}. Te deviatoric-part operator dev is defined as (5.3) dev F = F (tr(f)/n) id for any F R n n. wit tr(f) = F F nn. It is known tat te operator A : X = L V X, defined for (σ, u) X by (5.4) (A(σ, u))(τ, v) := a(σ, τ) (σ, Dv) L 2 () (τ, Du) L 2 () is a linear, bounded and bijective, cf. e.g., [23].

16 16 C. CARSTNSN AND JUN HU 5.2. Nonconforming finite element metods and unified a posteriori error estimators. Given some nonconforming finite element space V nc for V := H0 1()n and Q L 2 0(), te finite element solution (u, p ) V nc Q to (5.1) satisfies, for all (v, q ) V nc Q, (5.5) µ D T u : D T v dx p div T v dx + div T u q dx = g v dx. Given te unique discrete solution u H 1 (T ) n and p L 2 0 (), set (5.6) σ := µd T u p id L and define te linear functional Res V : V := H0 1()n R by (5.7) Res V (v) = (g v σ : Dv) dx for v V := H0 1 ()n. Te teory of Section 3 sows tat te norm of te residual Res L reads (5.8) Res L L D(ũ ) dev D T (u ) L 2 (). Given any ũ V wit σ := µdu p id, te unifying teory in te form of (1.7) and (5.4) prove (5.9) σ σ L + u ũ V D(ũ ) D T (u ) L 2 () + div T u L 2 () + Res V (v) V xamples. Tis subsection lists some examples of nonconforming finite element scemes wit (H1)-(H3) from te literature displayed in Table 5.1. Ten, it follows from (5.9), te definitions of σ and σ wit a straigtforward investigation, Teorem 2.1, and Teorem 3.1, tat (5.10) Du D T u L 2 () + p p L 2 () min D(ũ ) D T (u ) L ũ V 2 () + div T u L 2 () + Res V (v) V η + µ + div T u L 2 () + osc(g). Tis recovers te result from [36, 33] for te Crouzeix-Raviart element, and is new for five parallelogram elements of Subsubsection Te Crouzeix-Raviart element. Tis is a triangular element wit te velocity space for te space V CR V nc Q L 2 0(). Since V c V c V CR := V CR V CR from Subsection 4.3.1, and te piecewise constant pressure space V CR, tere olds (H1)-(H3) wit Π = id Four parallelogram elements. Tere are four parallelogram elements in te literature including te parallelogram Han element, te parallelogram nonconforming rotated (NR) element of Rannacer and Turek [60], te parallelogram CJY element [22], and te parallelogram constrained nonconforming rotated element of Hu, Man and Si [49]. Tese elements employ te piecewise constant pressure space

17 UNIFYING THORY OF A POSTRIORI RROR CONTROL 17 picture name reference space Crouzeix-Raviart [35] V CR Han [45] V Han NR [60] V RT,A Hu-Man-Si [49] V CRT CJY [22] V DSSY V CR V Han V RT,A V CRT V DSSY Kouia-Stenberg [54] V c V CR Table 5.1. Nonconforming lements for te Stokes Problem (5.1) wit (H1)-(H3) and te rror stimate (5.10). Q L 2 0 (). Te velocity spaces for tese metods are cosen from te following list. V nc :=V Han V CRT V Han V CRT, V RT,A, V DSSY V RT,A, V DSSY. Herein V Han, V RT,A, V CRT and V DSSY denote te nonconforming finite element spaces from te respective Subsubsections Ten tere olds (H1)-(H3) wit te canonical interpolation operators Π for tese nonconforming finite element spaces. Te proof follows wit te results of Section 4; furter details are omitted. Remark 5.1. Te parallelogram nonconforming finite elements from [33] can also be analyzed in te present framework to recover te a posteriori error estimation on for te isotropic mes terein Te Kouia-Stenberg element. Te Stokes problem in its form (5.1) is equivalent to te symmetric form wit ε(u) := sym(du) := 1/2(Du+Du T ) replacing Du in (5.1). Te velocity space [54] reads V nc := V c V CR. Since V c V c V c V CR, tere olds (H1)-(H3) wit Π = id, cf. [23]. 6. Linear elasticity Tis section is devoted to te Navier-Lamé equation and its locking-free nonconforming finite element approximation. Te presented unifying teory leads to a posteriori error estimates wic are robust wit respect to te Lamé parameter λ. Subsection 6.1 displays te model problem and Subsection 6.2 NCFMs and teir unifying error control. Subsection 6.3 presents some examples. Subsection 6.4 discusses te unsymmetric formulation for linear elasticity and te examples for tis case are given in Subsection 6.5

18 18 C. CARSTNSN AND JUN HU 6.1. Model Problem. Adopt te notation of te previous sections and te following linear stress-strain relation, for λ, µ > 0, (6.1) CF := λ tr(f) id +2µ F and C 1 F := 1 2µ F λ 2µ(nλ + 2µ) tr(f) id, for F Rn n. Te weak form of te linear elasticity problem reads: Given g L 2 () n find u V := H0 1()n wit (6.2) ε(v) : σ dx = g v dx and σ = Cε(u) for all v V. Define te operator A : X = L V X for any (σ, u) X by (6.3) (A(σ, u))(τ, v) := (C 1 σ, τ) L 2 () (σ, ε(v)) L 2 () (τ, ε(u)) L 2 (). Here, L := {σ L 2 (, Rsym n n ), tr σ dx = 0}. Te operator A is linear, bounded, and bijective wit λ-independent operator norms of A and A 1 [17, 28] Nonconforming finite element metods and unified a posteriori error estimators. Wit te nonconforming finite element approximation u V nc to u and te discrete Green strain ε T (v) := (D T v + D T v T ))/2 L 2 (; Rsym n n ), set (6.4) σ = 2µε T (u ) + λπ 2 div T u id. Trougout tis section, Π 2 : L 2 () L 2 () denotes some reduction operators in te context of te locking penomena, and te discrete stress σ is supposed to satisfy (6.5) σ : ε T (v ) dx = g v dx for all v V nc. We define te continuous and discrete pressures as (6.6) p = λ div u and p = λπ 2 div T u. Teorem 6.1. For any ũ V tere olds ε(u) ε T (u ) L 2 () + p p L 2 () + ε(u ũ ) L 2 () (6.7) ε T (u ) ε(ũ ) L 2 () + Res V V + div T u Π 2 div T u L 2 (). Proof. Te unifying teory wit (1.7) and (6.3) reads in te present notations (6.8) σ σ L + ε(u ũ ) L 2 () C 1 σ ε(ũ ) L 2 () + Res V V. Ten te assertion follows from te definitions of σ, C 1, p, and p xamples. Tis subsection analyzes finite element metods depicted in Table 6.1 for te planar elasticity problem. Tese scemes satisfy (H1)-(H3). Ten, te estimate (6.7) wit Teorem 2.1 and Teorem 3.1 leads to (6.9) ε(u) ε T (u ) L 2 () + p p L 2 () min ε T (u ) ε(ũ ) L ũ V 2 () + Res V V + div T u Π 2 div T u L 2 () µ + η + div T u Π 2 div T u L 2 () + osc(g).

19 UNIFYING THORY OF A POSTRIORI RROR CONTROL 19 picture name reference space Kouia-Stenberg [54] V c V CR Zang [75] V Wil V Wil Ming [56] V c V RT,A Table 6.1. Nonconforming lements for te Linear lasticity Problem (6.2) wit (H1)-(H3) and te rror stimate (6.9). Te error control for te Kouia-Stenberg element as already been analyzed in [23]. Te a posteriori error estimator (6.9) for te Falk elements, te Zang element, and te Ming element is new Te Falk elements. Two nonconforming triangular finite element metods are proposed in [41] for te linear elasticity equation for k = 2, 3 wit Π 2 = id and (6.10) := {v L 2 () 2 : T T, v T P k (T) 2 and v is continuous (res. vanises) at te V nc Since V c V c V nc tere olds (H1)-(H3) wit Π = id. k Gauss points on eac interior (resp. boundary) edge} Te Kouia-Stenberg element. Tis triangular element for te symmetric formulation (6.1) and Π 2 = id [54] is defined by te nonconforming finite element space (6.11) V nc := V c V CR. Since V c V c V c V CR tere olds (H1)-(H3) wit Π = id, cf. also [23] Te Zang element. Tis element is proposed in [75] based on te nonconforming quadrilateral Wilson element [73, 64] wit Π 2 = id. In tis element, (6.12) V nc := V Wil V Wil. Since V c V c V Wil V Wil tere olds (H1)-(H3) wit Π = id Te Ming element. In Ming s dissertation [56], a parallelogram nonconforming element is proposed based on te nonconforming rotated Q 1 space from [60] for planar elasticity. Te nonconforming finite element space reads (6.13) V nc := V c V RT,A were Π 2 = Π 0 : L 2 () Q 0 denotes te piecewise constant projection operator wit Q 0 te piecewise constant space. Following te arguments in Subsubection and [31], one proves (H1)-(H3) for te associated interpolation operator Π.

20 20 C. CARSTNSN AND JUN HU picture name reference space Brenner-Sung [19] V CR Lee-Lee-Seen [55] V RT,A Hu-Man-Si [49] V CRT V CR V RT,A V CRT Table 6.2. Nonconforming lements for te Linear lasticity Problem in Unsymmetric Formulation wit (H1)-(H3) and te rror stimate (6.19) Te unsymmetric formulation. For te pure Diriclet boundary condition under consideration, one can use te equivalent unsymmetric formulation and ten define te following formal stress-strain relation, for F R n n, (6.14) CF := (λ+µ) tr(f) id+µf and C 1 F := 1 µ F λ + µ tr(f) id. µ(nλ + (n + 1)µ) Given some nonconforming finite element space V nc, te finite element solution u V nc satisfies (6.15) σ : D T v dx = g v dx for all v V nc. Given te unique discrete solution u V nc, set (6.16) σ = µd T u + (λ + µ)π 2 div T u id, Te continuous and discrete pressures reads (6.17) p = (λ + µ) div u and p = (λ + µ)π 2 div T u. Define te operator A : X = L V := {τ L 2 (, R n n ), trτ dx = 0} H1 0() n X for any (σ, u) X as (A(σ, u))(τ, v) := (C 1 σ, τ) L 2 () (σ, Dv) L 2 () (τ, Du) L 2 (). Te arguments for te symmetric case in [17] sow tat te operator A is linear, bounded, and bijective wit λ-independent operator norms of A and A 1. Following te argument for te symmetric case, one proves Teorem 6.2. For any ũ V tere olds tat (6.18) Du D T u L 2 () + p p L 2 () + D(u ũ ) L 2 () D T u Dũ L 2 () + Res V V + div T u Π 2 div T u L 2 () xamples. Tree nonconforming finite elements are listed below as examples wit te unsymmetric formulation and are summarized in Table 6.2. Tere olds tat (6.19) Du D T u L 2 () + p p L 2 () µ + η + div T u Π 2 div T u L 2 () + osc(g). Tis a posteriori error estimator is brand new for tese elements.

21 UNIFYING THORY OF A POSTRIORI RROR CONTROL Te Brenner-Sung element. Tis triangular element is proposed in [19] wit Π 2 = id, and (6.20) V nc := V CR V CR. Since V c V c V CR V CR tere olds (H1)-(H3) wit Π = id Te Lee-Lee-Seen element. In tis parallelogram element [55], bot components of te displacement are approximated by te nonconforming rotated Q 1 space from [60], namely (6.21) V nc := V RT,A V RT,A. Te reduction integration operator is te same as in te Ming elements. (H1)-(H3) is satisfied by tis element wit te canonical interpolation operator Π for V nc. It follows te arguments for te nonconforming rotated Q 1 element in Subsubsection Te Hu-Man-Si element. Tis parallelogram element is designed in [49] witout reduction integration. Te nonconforming finite element space is te constrained nonconforming rotated Q 1 from [50]. Tere also olds (H1)-(H3) wit te canonical interpolation operator Π. Te proof can be found in Subsubsection Remark 6.1. Our conditions and terefore analysis in tis paper can be extended to oter nonstandard finite element metods for te elasticity, for instance, te Wang-Qi element from [70] and te enanced strain finite element from [63, 17]. Acknowledgments Te second autor JH tankfully acknowledges te Alexander von Humboldt Fellowsip during is stay at te Department of Matematics of Humboldt-Universität zu Berlin, Germany. References [1] M. Ainswort and J.T. Oden, A posteriori error estimation in finite element analysis, Wiley- Interscience [Jon Wiley & Sons], New York, [2] M. Ainswort, Robust a posteriori error estimation for nonconforming finite element approximation, Preprint available at ttp:// aas98107/papers.tml. [3] M. Ainswort, A posteriori error estimation for non-conforming quadrilateral finite elements, Int. J. of Numerical Analysis and Modeling, 2 (2005), [4] Douglas N. Arnold, An interior penalty finite element metod wit discontinuous elements, IAM J. Numer. Anal. 19 (1982), [5] Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and Donatella Marini, Discontinuous Galerkin metods for elliptic problems, Discontinuous Galerkin Metods: Teory, Computation and Applications, B. Cockburn, G. Karniadakis, and C. W. Su, eds., ecture Notes in Computational Science and ngineering 11, Springer-Verlag, New York, 2000, [6] I. Babuška and T.Strouboulis, Te Finite lement Metod and its Reliability, Te Clarendon Press Oxford University Press, [7] G.Baker, Finite element metods for elliptic equations using nonconforming elements, Mat.Comp.,31(1977), [8] G.A. Baker, W.N. Jureidini, and O.A. Karakasian, Piecewise solenoidal vector fields and te Stokes problem, SIAM J. Numer. Anal. 27 (1990),

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