Nonconforming FEM for the obstacle problem

Transcription

1 IMA Journal of Numerical Analysis (2017) 37, doi: /imanum/drw005 Advance Access publication on May 10, 2016 Nonconforming FM for the obstacle problem C. Carstensen and K. Köhler Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D Berlin, Germany Corresponding author: [Received on 6 March 2015; revised on 13 November 2015] he paper is dedicated to Professor Peter Wriggers on the occasion of his 65 birthday. he main motivation for the application of the Crouzeix Raviart nonconforming finite element method (NCFM) to the obstacle problem in this paper is that it allows for fully computable guaranteed lower bounds of the energy and so for simple a posteriori error control. A further fully computable and guaranteed upper error bound follows from Braess work, extended to the Crouzeix Raviart NCFM. his error bound competes with the error control from the lower energy bounds. Both a posteriori estimates are efficient with respect to the total error. he paper circumvents variational crimes through a medius analysis and the design of conforming companions. his leads to an improved a priori error analysis for the NCFMs under minimal regularity assumptions on polyhedral domains. Numerical evidence supports the a priori convergence analysis and confirms guaranteed error control with moderate efficiency indices for uniform and adaptive mesh refinement. Keywords: obstacle problem; variational inequality; nonconforming; finite elements; medius analysis; a priori error analysis; a posteriori error analysis; guaranteed upper error bounds; lower energy bounds; adaptive mesh refinement. 1. Introduction he obstacle problem is the simplest mathematical model of a variational inequality, with countless applications and related models in free boundary value problems. he trial functions are restricted to some convex set K and any discretization replaces this set by a discrete approximation K h.ifk h K the discretization is called conforming and it is called nonconforming otherwise. he particular motivation for the application of the Crouzeix Raviart nonconforming finite element method (NCFM) in Wang (2003) remains less clear. herein, compared with conforming finite element methods (CFMs), higher regularity assumptions are made to prove linear convergence for convex domains R 2 with smooth boundary. he refined a priori error analysis of this paper shows, under the minimal regularity assumption (i.e., Δu L 2 ()), that the NCFM converges with optimal convergence rates for arbitrary polyhedral domains R d (d = 2, 3) and hence the Crouzeix Raviart NCFM becomes competitive with the CFM. his paper also explores the a posteriori error control from two different points of view. In the first place, the Crouzeix Raviart NCFM allows for the computation of guaranteed lower bounds for the energy. Some simple postprocessing leads to a computable estimate for the energy difference and hence also for the error in the energy norm. In the second place, the results in Braess (2005) for the CFM are adapted to the Crouzeix Raviart NCFM. he two error estimates for NCFM are comparable (up to unknown multiplicative constants). he authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 NONCONFORMING FM FOR H OBSACL PROBLM 65 Given a bounded polyhedral Lipschitz domain R d (d = 2, 3) with boundary, the energy product a : H 1 () H 1 () R on the Hilbert space H 1 () reads a(u, v) := u vdx, for all u, v H 1 () and induces the energy seminorm := a(, ) 1/2, which is a norm on the vector space V := H0 1 () := {v H 1 () v = 0on }, and the corresponding local variant ω := L 2 (ω) for ω.given some source term f L 2 (), define F L 2 () by F(v) := fv dx, for all v L 2 (). We assume that the obstacle χ H 1 () and the Dirichlet boundary value u D C( ) H 1/2 ( ) satisfy χ u D a.e. along in order to ensure that the closed and convex subset K := {v A χ v a.e.} of A := {v H 1 () v = u D along } is nonempty. he well-established weak formulation of the obstacle problem leads to a unique u K, see Kinderlehrer & Stampacchia (1980, Chapter 2, heorem 2.1), with F(v u) a(u, v u), for all v K. (1.1) he obstacle problem is also characterized by the minimization of the energy functional (v) := 1 a(v, v) F(v) over all v K. (1.2) 2 hroughout this paper, the exact solution u K and the Lagrange multiplier Λ := F a(u, ) V are approximated by the discrete solution u NC in some discrete analogue K NC of K and a certain novel discrete Lagrange multiplier Λ NC in the discrete space CR 1 ( ) (cf. (2.1)). he first main result of this paper establishes an a priori error estimate for the Crouzeix Raviart NCFM under the known regularity property Δu L 2 () for Δχ L 2 () (cf. Rodrigues, 1987, Proposition 5:2.2) so that λ := f + Δu L 2 (). he second main result yields two guaranteed lower bounds for the minimal energy (u) and so allows for an a posteriori control of the error u u NC NC in the discrete (i.e., piecewise) energy norm NC := NC L 2 (). he third main result is an explicit residual-based a posteriori error analysis with reliable and efficient control over the error u u NC NC + Λ Λ NC with the dual norm in H 1 () up to data oscillation terms. his extends the a posteriori error analysis of Bartels & Carstensen (2004), Braess (2005) for the Courant FM to the Crouzeix Raviart NCFM; cf. also the recent work Braess et al. (2008) on the a posteriori error analysis for mixed FMs and Gudi & Porwal (2014) for dg FMs. Numerical experiments confirm guaranteed error control with moderate over-estimation by a factor typically in the range and support adaptive over uniform mesh refinement. An empirical comparison of the two a posteriori error estimates is included and shows that both converge with the same convergence rate but the residual-based a posteriori error is better by a small multiplicative constant. he remaining parts of this paper are organized as follows. Section 2 introduces the discretization of the obstacle problem and discusses the approximation of the nonhomogeneous Dirichlet data and the

3 66 C. CARSNSN AND K. KÖHLR design of a specific conforming companion to the discrete solution. Section 3 presents a new a priori error analysis under the reduced regularity assumption. Section 4 derives two guaranteed lower bounds for the minimal energy and two a posteriori error estimates, followed by the discussion of efficiency in Section 5. he paper concludes with three computational benchmark examples in Section 6 on uniform and adapted triangulations. he paper applies standard notation for Lebesgue and Sobolev spaces and their norms L 2 (), := L 2 (), as well as their local variants L 2 (ω) and ω for ω. he integral mean is denoted by. Moreover A B abbreviates A CB for some generic constant C (which solely depends on the shape regularity of the underlying triangulation in Section 2.1) and A B abbreviates A B A. he analysis in this paper is essentially carried out explicitly for the two-dimensional case but the generalization to three dimensions is straightforward (with additional explanations stated whenever necessary). 2. Preliminaries his section introduces the discretization of the obstacle problem and discusses the approximation of the nonhomogeneous Dirichlet data and of some conforming companion to a nonconforming Crouzeix Raviart function. 2.1 Discretization Let R 2 be a bounded polygonal Lipschitz domain partitioned in a shape-regular triangulation into triangles in the sense of Ciarlet (1978), with nodes N, interior nodes N () and nodes on the boundary N ( ). he set of edges is denoted by, with interior edges () := { }, and edges ( )along the boundary. Given any node z N, let (z) denote the set of all triangles with z N () the set of the three vertices of a triangle, let (z) 1 denote the number of triangles in (z) and let ω z := (z) denote the node patch around z; ω := z N () ω z denotes a patch around each triangle. Any edge has length, midpoint mid() and unit normal ν ; mid( ) := {mid() } denotes the set of the midpoints of all edges. For any k N 0, set P k () :={v k : R v k is a polynomial of degree k}; P k ( ) :={v k L (), v k P k ()}; CR 1 ( ) :={v NC P 1 ( ) v NC continuous at mid( )}; V NC :={v NC CR 1 ( ) ( ), v NC (mid()) = 0}; V 1 ( ) :=P 1 ( ) C 0 (); (2.1) V 2 ( ) :=P 2 ( ) C 0 (); A NC :={v NC CR 1 ( ) ( ), v NC (mid()) = u D ds}; K NC :={v NC A NC (), χ ds v NC(mid())}. he triangulation is regular in the sense that any two distinct triangles in with nonempty intersection are either identical or share exactly one common node or one common edge. he triangulation is shape regular in the sense that any interior angle of any triangle is bounded from below by some universal positive constant γ 0 and all the generic constants hidden in the notation (or ) solely depend on γ 0 > 0. Given

4 NONCONFORMING FM FOR H OBSACL PROBLM 67 the triangulation, define the (local) mesh size h P 0 ( ), the L 2 projection Π 0 : L 2 () P 0 ( ) and the oscillation osc(f, ) of f by h := h := diam(), Π 0 f := f dx := f dx/ for all and f L 2 () (as well as for vector-valued functions in L 2 (; R 2 ), etc.) and osc 2 (f, ) := h (f Π 0 f ) 2 L 2. () For any subsets 1, 2, set dx := 1 dx 1 and L2 ( 1 2) := L 2 ( ( 1 2 )). With the piecewise gradient NC v NC P 0 ( ; R 2 ) of any discrete function v NC CR 1 ( ), the discrete energy product a NC :CR 1 ( ) CR 1 ( ) R reads a NC (u NC, v NC ) := NC u NC NC v NC dx, for all u NC, v NC CR 1 ( ) and induces the discrete energy seminorm NC := a NC (, ) 1/2 in CR 1 ( ). With the discrete Friedrichs inequality v NC L 2 () v NC NC, for all v NC V NC (see Brenner, 2003), this is a norm in V NC. he local variant on ω of this discrete energy norm reads NC(ω) := NC L 2 (ω). he discrete analogue to the variational inequality (1.1) seeks u NC K NC with F(v NC u NC ) a NC (u NC, v NC u NC ), for all v NC K NC. (2.2) As in the continuous case (1.2), the discrete solution u NC K NC is also the minimizer of the analogous discrete energy functional NC (v NC ) := 1 2 a NC(v NC, v NC ) F(v NC ) over all v NC K NC. (2.3) ach edge () is associated with its edge-oriented basis function ψ CR 1 ( ), which satisfies ψ 1 along and ψ (mid(f)) = 0 for any other edge F \{}, and has support ω := { ()}. Lemma 2.1 For each edge (), the solution u NC to the discrete variational inequality (2.2) satisfies the discrete consistency condition ( abbreviates orthogonality in R, i.e., a b means ab = 0 for a, b R) 0 u NC (mid()) χ ds F(ψ ) a NC (u NC, ψ ) 0. (2.4) Proof. he discrete consistency condition follows from direct considerations with the degrees of freedom in (2.2).

5 68 C. CARSNSN AND K. KÖHLR he discrete consistency conditions are the discrete analogue of the well-known (continuous) consistency condition (Kinderlehrer & Stampacchia, 1980) that the solution u Hloc 2 () to (1.1) satisfies 0 u χ f + Δu 0 a.e. in, (2.5) where abbreviates pointwise orthogonality. 2.2 wo interpolation operators he conforming and nonconforming interpolation operators read I C : C() P 1 ( ) C(), v z N v(z)ϕ z, I NC : H 1 () CR 1 ( ), v ( ) v ds ψ. Here and throughout this paper, ϕ z denotes the (conforming) nodal basis function associated with the node z N and ψ is the edge-oriented basis function of CR 1 ( ) associated with the edge. Known interpolation error estimates in two dimensions involve the constants 1/4 + 2/j1,1 2 κ C (γ ) := 1 cos(γ ) and κ NC := 1/48 + 1/j1, (2.6) for the maximal interior angle γ in the triangle K and the smallest positive root j 1, of the Bessel function of the first kind. Lemma 2.2 (properties of the interpolation operators) Any v H 1 () and its interpolation I NC v satisfy a b h 1 (v I NCv) L 2 () κ NC v I NC v NC ; NC I NC v = Π 0 v. Any v H 2 (K) on a triangle K with diameter h K and κ C (γ ) from (2.6) satisfies c h 1 K (v I Cv) L 2 (K) κ C(γ ) D 2 v L 2 (K). Remark 2.3 he assertions constants (Ciarlet, 1978). a c hold also in the three-dimensional case with different universal Proof. he proof of a follows as in Carstensen et al. (2012) with the improved constant in Carstensen & Gallistl (2014, heorem 4).

6 NONCONFORMING FM FOR H OBSACL PROBLM 69 he proof of b follows from an integration by parts on each triangle and the integral mean property of I NC along each edge. Assertion c is contained in Carstensen et al. (2012, heorem 3.1). 2.3 Nonhomogeneous dirichlet data he nonhomogeneous Dirichlet data u D leads to affine spaces, where u D needs to be adapted to the corresponding discretization. Let the edge := conv{a, B} and define the corresponding bubble function b := 6ϕ A ϕ B with the nodal basis functions ϕ A, ϕ B for the nodes A, B, which satisfies supp(b ) = supp(ϕ A ) supp(ϕ B ) and b ds = 1. Given the Dirichlet data u D C( ), the functions u D1 P 1 ( ) C() and u D2 P 2 ( ) C() approximate the nonhomogeneous Dirichlet data. he function u D1 is defined by the nodal values and the function u D2 is given by u D2 := u D1 + u D1 (z) := ( ) Given u D1 and u D2, define the affine spaces { u D (z) for z N ( ), 0 for z N () (u D u D1 ) dsb P 2 ( ) C(). A 1 := u D1 + V 1 ( ), A 2 := u D2 + V 2 ( ), and recall A NC := I NC u D + V NC. Lemma 2.4 Let u D C( ) H 2 ( ( )). hen the quadratic approximation u D2 P 2 ( ) C() satisfies u D ds = u D2 ds for any edge ( ), u D2 (z) = u D (z) at any node z N ( ) and u D2 I NC u D2 NC h3/2 2 u D s 2 L 2 ( ). Proof. he first two properties follow from the definition of u D2. Since the function u D2 satisfies u D2 P 1 () for any interior edge (), it follows that u D2 I NC u D2 = 0 only on triangles with () ( ) =. Let be such a triangle and assume first that =. hen it holds that u D2 = u D1 + (u D u D1 ) dsb. he properties of the interpolation operator I NC lead to (u D2 I NC u D2 ) = (u D u D1 ) ds(b I NC b ).

7 70 C. CARSNSN AND K. KÖHLR he bubble function b and its nonconforming interpolation I NC b satisfy b I NC b NC() 1. It remains to estimate (u D u D1 ) ds. Let the edge := conv{a, B} of length h = = A B have the vertices A and B. Without loss of generality suppose A = (0, 0) and B = (0, h) and write u D (s) for u D (s,0). he definition of u D1 shows I := (u D u D1 ) ds = h 0 ( u D (s) u ) D(0) + u D (h) ds/h. 2 he function ζ(s, t) := { 1 for t < s, 1 for s < t satisfies h h ζ(x, y) dy dx = 0. For the constant c := h u D(t)/ t dt/h R, the fundamental theorem of calculus along (0, h) leads to I = 1 2h = 1 2h = 1 2h h ( s 0 0 h h 0 0 h h 0 0 u D (t) h dt t s ζ(s, t) u D(t) dt ds t ( ud (t) ζ(s, t) t ) u d (t) dt ds t ) c dt ds. Since u D H 2 (), the Cauchy inequality in L 2 (0, h) and ζ 2 = 1, followed by the Poincaré inequality, lead to h I u D 2 s c L 2 (0, h) h3/2 2 π 2 u D s 2 L 2 (0, h). his proves u D2 I NC u D2 NC() h 3/2 2 u D s 2 L 2 (). In the second case where two edges of belong to ( ), the triangle inequality is used to obtain the general result (with some hidden extra factor 2). he sum of all these estimates concludes the proof. Remark 2.5 In three dimensions, the bubble functions are defined analogously and the above proof employs an interpolation error estimate for the nodal interpolation.

8 NONCONFORMING FM FOR H OBSACL PROBLM 71 Lemma 2.6 For any v D C( ) H 2 ( ( )) with v D = 0atN ( ) there exists w D H 1 () with w D = v D, w D = 0 for (), I NC w D = 0, integral mean Π 0 w D = 0 and w D h3/2 2 v D s 2 L 2 ( ). Proof. he proof follows from Bartels et al. (2004, heorem 4.2). herein a function w D is defined by harmonic extension. Denote this function by w D. It can be modified to w D to achieve the property Π 0 w D = 0. o this end, define, for each with := conv{a, B, C}, the cubic bubble function b := 60ϕ A ϕ B ϕ C, which satisfies supp(b ) = and b dx = 1. hen the function w D is given by w D := w D w D dxb, for all. 2.4 Conforming companion he design of two conforming companions to any v NC V NC in V 1 ( ) and V 2 ( ) starts with the map J 1 : V NC V 1 ( ) defined, for v NC V NC,by J 1 (v NC ) := z N () ( (z) ) v NC (z) ϕ z. (z) Recall that ϕ z P 1 ( ) C()denotes the P 1 nodal basis function associated with the node z in N (). Define J 2 : V NC V 2 ( ) for v NC V NC by J 2 (v NC ) := J 1 (v NC ) + ( ( ) ) v NC J 1 (v NC ) ds b (2.7) with the bubble function b associated with the edge. Lemma 2.7 (Carstensen et al., 2014, Proposition 2.3) Given v NC V NC, the function v 2 := J 2 (v NC ) V 2 ( ) satisfies a v NC v 2 NC() min v v NC NC(ω ), for any ; v V b v NC v 2 NC min v v NC NC. v V Any w NC A NC satisfies w NC = I NC u D2 + (w NC I NC u D2 ) and w NC I NC u D2 V NC. A conforming companion of w NC is designed with the boundary approximation u D2 of u D and the aforementioned operator J 2, namely w 2 := u D2 + J 2 (w NC I NC u D2 ) A 2. (2.8)

9 72 C. CARSNSN AND K. KÖHLR Fig. 1. Commutative diagram between the discrete and continuous spaces related to the admissible functions; w D H 1 () extends u D u D2 (cf. Lemma 2.6). Figure 1 illustrates the relation between the vector spaces V NC, V 2 ( ) and the affine spaces A NC, A 2. Lemma 2.8 (properties of the conforming companion) Let u D C( ) H 2 ( ( )). Given any w NC A NC, the conforming companion w 2 A 2 from (2.8) satisfies a I NC w 2 = w NC ; b w 2 I NC w 2 NC() min + h3/2 2 u D s 2 L2 ( ω ) v A c w 2 I NC w 2 NC min v A w NC v NC(ω ), for all ; w NC v NC + Proof. A direct integration of (2.8) along any edge shows his implies a. w 2 dx = = (u D2 + J 2 (w NC I NC u D2 )) ds u D2 ds + h3/2 2 u D s 2 L 2 ( ). J 1 (w NC I NC u D2 ) ds + ((w NC I NC u D2 ) J 1 (w NC I NC u D2 )) ds b ds = u D2 dx + (w NC I NC u D2 ) ds = I NC u D2 (mid()) + w NC (mid()) I NC u D2 (mid()) = w NC ds.

10 NONCONFORMING FM FOR H OBSACL PROBLM 73 he proof of b utilises Lemmas he triangle inequality yields w NC w 2 NC u D2 I NC u D2 NC + w NC I NC u D2 w 2 + u D2 NC. Lemma 2.4 shows u D2 I NC u D2 NC h 3/2 2 u D / s 2 L Recall v NC := w NC I NC u D2 V NC and ( ). 2 v 2 := J 2 (v NC ) = u D2 + w 2 V 2 ( ) from (2.7). Lemma 2.7 a implies w NC I NC u D2 w 2 + u D2 NC() min v V v w NC + I NC u D2 NC(ω ). With the approximations u D2 and I NC u D2 of the Dirichlet data u D, Lemma 2.6 shows the existence of some function w D H 1 () which satisfies w D = (u D u D2 ), I NC w D = 0 and w D h 3/2 2 u D / s 2 L he triangle inequality yields ( ). 2 min v w NC + I NC u D2 NC(ω ) min w w NC NC(ω ) + u D2 I NC u D2 NC(ω ) + w D NC(ω ). v V w A Lemmas 2.4 and 2.6 estimate the second and third terms to prove b. he proof of c follows from the summation of the squares of inequality b and the finite overlap of the element patches ω. 3. A priori error analysis his section proves an a priori error estimate for the error u u NC NC for the solutions u K and u NC K NC to the continuous and discrete obstacle problem (1.1) and (2.2). he result uses only the regularity property Δu L 2 () guaranteed for f L 2 () and Δχ L 2 () (see Rodrigues, 1987, Proposition 5:2.2 for a proof) and generalizes Wang (2003) to singular solutions (e.g., as in the example of Section 6.4). he a priori result employs four subsets of the triangulation : + :={ u >χa.e. in }, 0 :={ u = χ a.e. in }, M := \ ( + 0 ) and :={ > 0}. (3.1) In other words, + denotes the triangles without contact, 0 those with full contact, M contains the triangles at the interface and the triangles with at least one edge on the boundary ( > 0 means that has positive length). heorem 3.1 (A priori error estimate) Let χ H 2 (). hen the continuous and discrete solutions u K and u NC K NC to the obstacle problem satisfy u u NC NC u I NC u NC + h ( f + λ ) L 2 () + h D 2 χ h L2 ( 0 M) + 3/2 2 u D / s 2 L. 2 ( ) he theorem above shows that the nonconforming FM requires weaker regularity than the conforming Courant FM, as it requires only that Δu L 2 (). his is in contrast to the a priori error analysis

11 74 C. CARSNSN AND K. KÖHLR for the Courant FM presented in Falk (1974), where full H 2 regularity is assumed and extra work is required for reduced elliptic regularity with u H 1+s () for 0 s < 1. Proof of heorem 3.1. Step 1 of the proof utilises a NC (u NC, u) = a NC (u NC, I NC u) and u 2 := u D2 +J 2 (u NC I NC u D2 ) with I NC u 2 = u NC from Section 2.4. With the abbreviation Λ NC (v NC ) := F(v NC ) a NC (u NC, v NC ) for v NC V NC, (3.2) the discrete variational inequality (2.2) shows for I NC u K NC that 0 Λ NC (u NC I NC u). Since u NC = I NC u 2, Lemma 2.2. b shows Π 0 NC (u 2 u NC ) = Π 0 NC (u I NC u) = 0. his leads to u u NC 2 NC LHS2 := u u NC 2 NC + Λ NC(u NC I NC u) = a(u, u u 2 ) + a(u I NC u, u 2 u NC ) + F(u NC I NC u). With λ := f + Δu L 2 (), the consistency conditions (2.5) read 0 u χ λ 0( abbreviates pointwise orthogonality a.e.). his, an integration by parts and w D H 1 (), designed as in Lemma 2.6 with w D = (u D u D2 ), imply a(u, u u 2 ) = a(u, u u 2 w D ) + a(u, w D ) = ( λ)(u u 2 ) dx + F(u u 2 ) + w D Δu dx + a(u, w D ) = ( λ)(χ u NC ) dx + ( λ)(u NC u 2 ) dx + F(u u 2 ) + w D Δu dx + a(u, w D ). A Cauchy inequality for the term w DΔu dx followed by a Poincaré inequality (recall Π 0 w D = 0 and the Poincaré constant h /j 1,1 from Laugesen & Siudeja, 2010) prove w D Δu dx = (w D Π 0 w D )(Δu Π 0 Δu) dx osc(δu, )/j 1,1 w D. he design of w D yields w D ds = 0 for any edge and hence for any, w D dx = 0. his, an integration by parts, and a Cauchy inequality yield, for, that u w D dx = (u u NC ) w D dx u u NC NC() w D.

12 NONCONFORMING FM FOR H OBSACL PROBLM 75 he combination of the aforementioned estimates proves LHS 2 ( λ)(χ u NC ) dx + ( λ)(u NC u 2 ) dx + osc(δu, )/j 1,1 w D + u u NC NC w D + a NC (u I NC u, u 2 u NC ) + F(u u 2 + u NC I NC u). Since I NC u 2 = u NC, a Cauchy inequality, Lemma 2.2 a, and Lemma 2.8 c show for A := u u NC NC + h 3/2 2 u D / s 2 L that 2 ( ) ( λ)(u NC u 2 ) dx h λ L 2 () A, a NC (u I NC u, u 2 u NC ) u I NC u NC A, F(u u 2 + u NC I NC u) h f L 2 () A. With RHS 1 := u I NC u NC + h ( f + λ ) L 2 () + h 2 u D / s 2 L 2, the combination of the ( ) previous estimates and Lemma 2.6 results in ( LHS 2 ( λ)(χ u NC ) dx RHS 1 u u NC NC + RHS 1 ). (3.3) Step 2 is the analysis of I := ( λ)(χ u NC ) dx for any triangle. It resolves the subtle third case, where M is neither fully in the contact zone nor in the noncontact zone. Recall the three subsets +, 0 and M from (3.1). Case 1: For any + it holds that u >χa.e. on and the consistency condition (2.5) reveals λ = 0 and hence I = 0. Case 2: For any 0, u χ a.e. on. Since λ := (Δu + f ) 0, λ := λ dx 0. Since I NC χ u NC is affine on and (I NCχ u NC ) dx 0, for all edges (), it follows that (I NC χ u NC ) dx 0, whence 0 λ (I NC χ u NC ) dx. With e := u u NC = χ u NC on 0 and e := e dx, this leads to I = (λ λ)e dx λ (I NC χ u NC ) dx λ (χ I NC χ)dx (λ λ)(e e ) dx λ (χ I NC χ)dx.

13 76 C. CARSNSN AND K. KÖHLR Since Π 0 χ = NC I NC χ, a Poincaré inequality with the constant h /j 1,1 from Laugesen & Siudeja (2010), and Lemma 2.2 a (with κ NC ) in the previous estimate shows I osc(λ, )/j 1,1 e NC() + h 2 κ NC/j 1,1 λ L 2 () D 2 χ L 2. () Case 3: In the remaining case M, {x u(x) = χ(x)} < and so the compact subset C := {x u(x) = χ(x) u NC (x)} has a measure C <. (Note that u, χ, u NC are continuous on and so C is closed.) For x \ C it follows that either χ(x) <u(x) or u NC (x) <χ(x). he continuous consistency conditions (2.5) show that λ(x) = 0 for χ(x) <u(x) and since λ 0 a.e. in, it follows that for x with u NC (x) <χ(x), hese observations and a Cauchy inequality imply I C ( λ(x))(χ(x) u NC (x)) 0. ( λ)(χ u NC ) dx λ L 2 (C) C 1/2 χ u NC L (C). (3.4) For any edge, (χ u NC ) ds = (I NC χ u NC ) dx 0. (his follows for all interior edges () since u NC K NC, and from χ u D on, also for any edge on the boundary ( ).) Hence, the continuous function w := χ u NC satisfies w 0at least at one point on. Since w 0onC, the function w H 2 () has some zero x 0. herefore, for any x, let F := conv{x, x 0 }. A nondegenerate triangle K := conv{x 0, x, P} can be defined as follows (cf. Fig. 2): Any straight line s 0 through x and x 0 has parallels through the three vertices of. wo of them, s 1 and s 2, have a maximal distance d and the interior of the triangle lies between them. Hence d width() and elementary considerations prove width() = min{height of a vertex of }. he distance of the straight line s 0 to one of the lines s 1 or s 2 is d and the other is d d. Without loss of generality let d be the bigger distance with d d/2. herefore there exists a vertex P of such that the height d of P onto the line s 0, which includes F := conv{x, x 0 }, satisfies d d/2. hen, K := conv{x 0, x, P} satisfies K F width()/4. he triangle satisfies =h width()/2 = h width() 2 /(2width()). Since all angles γ in the triangle satisfy γ 0 γ (by shape regularity), the definition of the tangent shows h /2 tan γ 0 width(). his proves 1/2 tan 1/2 (γ 0 ) width().

14 NONCONFORMING FM FOR H OBSACL PROBLM 77 Fig. 2. Design of the triangle K in heorem 3.1. Since K it follows that F tan 1/2 (γ 0 ) K 1/2 4 F tan 1/2 (γ 0 ) 1/2 4 K. In other words, Suppose for the moment that w C 1 () with w(x 0 ) = 0. hen w(x) w(x 0 ) + 1 he trace identity (Carstensen et al., 2012, Lemma 2.1) yields F 0 F K 1/2 4cot1/2 (γ 0 ). (3.5) w(x 0 + t(x x 0 )) (x x 0 ) dt F w ds. F w ds = w dx + 1 D( w ) (x P) dx. K 2 K he combination of the previous two displayed formulas shows w(x) F K w dx + F D( w ) (x P) dx. 2 K

15 78 C. CARSNSN AND K. KÖHLR Notice that the derivative of the modulus function is bounded by the modulus of the derivative. he estimate (3.5) and a Cauchy inequality imply cot 1/2 (γ 0 ) w(x) w K + h D 2 w L 2 (K). Since x is arbitrary in and since D 2 w = D 2 χ it follows that χ u NC L () tan 1/2 (γ 0 )( χ u NC NC() + h D 2 χ L 2 () ). A triangle and a Poincaré inequality (with Π 0 χ = NC I NC χ) show χ u NC L () tan 1/2 (γ 0 )( I NC χ u NC NC() + h (1 + 1/j 1,1 ) D 2 χ L 2 ). (3.6) () stimate (3.6) holds for all w := χ u NC C 1 () H 2 (). A density argument reveals that it also holds for all χ H 2 (). Since (I NC χ u NC ) is constant on the triangle, it follows that I NC χ u NC NC() = 1/2 C 1/2 (I NC χ u NC ) L 2 (C) 1/2 C 1/2 ( (χ I NC χ) L 2 (C) + (χ u) L 2 (C) + (u u NC ) L 2 (C) A Poincaré inequality (with the Poincaré constant h /j 1,1 from Laugesen & Siudeja, 2010) yields (χ I NC χ) L 2 (C) (χ I NC χ) L 2 () h D 2 χ L 2 () /j 1,1. Since u χ = 0 on the compact set C, it holds that (u χ) L 2 (C) = 0. herefore ( I NC χ u NC NC() 1/2 C 1/2 h /j 1,1 D 2 χ L 2 + u u () NC NC() ). ). Combination with (3.4) and (3.6) results in I tan 1/2 (γ 0 ) h λ L 2 (C) ( u u NC NC() + h D 2 χ ) L 2 (1 + 2/j ( ) 1,1). his concludes Case 3. he summary of the three cases leads to ( λ)(χ u NC ) dx h λ L 2 () ( u u NC NC + h D 2 χ L2 ). (3.7) ( 0 M) he combination of (3.3) and (3.7) reads LHS 2 RHS 1 ( RHS 1 + h D 2 χ L2 ( 0 M) + u u NC NC ). A Young inequality and the absorption of u u NC NC LHS conclude the proof.

16 NONCONFORMING FM FOR H OBSACL PROBLM A posteriori error analysis his section provides guaranteed lower bounds for the exact energy (u) in (1.2) based on the discrete energy NC (u NC ) in (2.3), as well as reliable error estimators for the obstacle problem. Given the discrete Crouzeix Raviart solution u NC K NC of (2.2), define some discrete Lagrange multiplier λ NC := () ρ ψ / ψ 2 L 2 () with ρ := F(ψ ) a NC (u NC, ψ ) (4.1) with the edge-oriented basis function ψ V NC associated with the edge (). Recall Λ NC from (3.2) and notice that Λ NC (v NC ) = λ NC v NC dx, for all v NC V NC. In the sequel, Λ NC (v) will always denote the L 2 scalar product of any Lebesgue function v L 2 () with λ NC V NC. he residual-based a posteriori error estimate involves the continuous Lagrange multiplier Λ := F a(u, ) V with the L 2 representation λ = f + Δu. he dual norm Λ Λ NC reads Λ Λ NC := sup v(λ λ NC ) dx/ v. v V\{0} { } he subset := 0 < {x λ NC (x) >0} of is employed in heorems 4.1 and 4.4. he lower bounds for the exact energy (u) are given in the following theorem with the constant κ NC from Lemma 2.2 a. heorem 4.1 (Lower energy bounds) he discrete solution u NC and the continuous solution u to the obstacle problem satisfy a NC (u NC ) κ2 NC 2 h f 2 L 2 (u) Λ () NC(u NC I NC u) (u) ( 2/2 b NC (u NC ) κ NC h (f λ NC ) L 2 () + osc(λ NC, )) (χ u NC )Π 0 λ NC dx + (I NC χ χ)λ NC dx \ (u) (χ u)λ NC dx (χ u)π 0 λ NC dx (u). \ he lower energy bounds of heorem 4.1 are of separate interest, but may be used for error control of any v K based on the identity 1 2 u v 2 + Λ(u v) = (v) (u). (4.2)

17 80 C. CARSNSN AND K. KÖHLR Combination with heorem 4.1 b leads to the error bound in the following result where all summands on the left-hand side are non-negative and all summands on the right-hand side are computable. Corollary 4.2 Any v K satisfies 1 2 u v 2 + Λ(u v) + (χ u)λ NC dx + (χ u)π 0 λ NC dx \ (v) NC (u NC ) + 1 ( ) 2 κ NC h (f λ NC ) 2 L 2 () + osc(λ NC, ) + (χ u NC )Π 0 λ NC dx + (χ I NC χ)λ NC dx. \ Remark 4.3 In practice, v K can be chosen with the help of any conforming companion, for example J 2 from (2.7). hen v = max{χ, w D + u D2 + J 2 (u NC I NC u D2 )} K for w D H 1 () as in Lemma 2.6 with w D = u D u D2 is an admissible function. he following residual-based a posteriori error analysis generalizes Braess (2005). heorem 4.4 (Guaranteed upper error bound) Any v K satisfies a 1 2 u u NC 2 NC + Λ(u v) + (χ u)π 0 λ NC dx + (χ u)λ NC dx \ 1 2 v u NC 2 NC + (χ v)π 0 λ NC dx + (χ v)λ NC dx \ + 1 ( ) κnc h (f λ NC ) 2 L 2 () + osc(λ NC, 2 )/j 1,1 b Λ Λ NC u u NC NC + osc(f λ NC, )/j 1,1 + 1 Π0 (f λ NC )( mid( )) 2 L 2 + u () NC v NC + w D. Remark 4.5 he comparison of Corollary 4.2 with heorem 4.4 a is possible through the following formula (which follows from straightforward algebra): (v) NC (u NC ) = 1 2 v u NC 2 NC + Λ NC(u NC v) + (Λ NC F)(v I NC v). (4.3) heorem 4.1 plus (4.2) and (4.3) imply heorem 4.4 with different constants in the form u u NC 2 NC 3 u NC v NC + Λ NC (u NC v) + 3κ 2 NC h (f λ NC ) 2 L osc(λ () NC, ) (χ u NC )Π 0 λ NC dx + 2Λ NC (u NC v) + 2 (χ I NC χ)λ NC dx. \ he remaining parts of this section are devoted to the proofs of heorems 4.1 and 4.4. he analysis of the nonlinearity utilizes the subsequent lemma.

18 NONCONFORMING FM FOR H OBSACL PROBLM 81 Lemma 4.6 he continuous and discrete solutions u and u NC, and the discrete Lagrange multiplier Λ NC, satisfy Λ NC (u u NC ) = (χ u NC )Π 0 λ NC dx λ NC (I NC χ u) dx \ (χ u)π 0 λ NC dx + (1 Π 0 )λ NC (1 Π 0 )(u u NC ) dx. Proof. he discrete consistency conditions (2.4) show that the product λ NC (I NC χ u NC ) vanishes at the midpoint mid() of any edge. Since the three-point quadrature formula at the edge midpoints is exact for quadratic polynomials P 2 () on the triangle, it follows that λ NC (I NC χ u NC ) dx = 0 for any. Hence, with \ dx := \ dx, it holds that λ NC (u u NC ) dx = λ NC (u I NC χ)dx. \ \ It remains to perform the analysis for. Recall that the L 2 projection Π 0 onto P 0 ( ) is the piecewise integral mean operator with respect to the triangulation. he inequalities Π 0 λ NC 0 Π 0 (u NC I NC χ) a.e. and algebraic transformations motivate the split λ NC (u u NC ) dx = (u u NC )Π 0 λ NC dx + (1 Π 0 )λ NC (u u NC ) dx = (u χ)π 0 λ NC dx + (χ u NC )Π 0 λ NC dx + (1 Π 0 )λ NC (1 Π 0 )(u u NC ) dx. he combination of the aforementioned estimates concludes the proof. Proof of heorem 4.1. he proof is divided into seven steps. Step 1. he properties of the nonconforming interpolation operator and a direct calculation lead to 1 2 u I NCu 2 NC = (u) NC(I NC u) + F(u I NC u).

19 82 C. CARSNSN AND K. KÖHLR Step 2. Some algebra with the definition of Λ NC in (3.2) leads to Λ NC (u NC I NC u) I NCu u NC 2 NC = NC(I NC u) NC (u NC ). Step 3. Since NC(u I NC u) dx = 0 on each triangle, the Pythagorean theorem yields Step 4. he combination of Steps 1 3 leads to u u NC 2 NC = u I NCu 2 NC + I NCu u NC 2 NC. Λ NC (u NC I NC u) u u NC 2 NC = (u) NC(u NC ) + F(u I NC u). (4.4) Step 5 is the proof of assertion a. Lemma 2.2 a shows h 1 (u I NCu) L 2 κ () NC u I NC u NC κ NC u u NC NC. his, a Cauchy inequality and some Young inequality lead to F(u I NC u) κ2 NC 2 h f 2 L 2 () u u NC 2 NC. (4.5) he combination of (4.4) and (4.5) concludes the proof of a. Step 6. Identity (4.4) also reads Λ NC (u NC u) u u NC 2 NC = (u) NC(u NC ) + (F Λ NC )(u I NC u). Lemma 2.2. a leads to Λ NC (u NC u) u u NC 2 NC (u) NC(u NC ) + κ NC h (f λ NC ) L 2 () u u NC NC. (4.6) Step 7 concludes the proof of b. Lemma 4.6 shows Λ NC (u u NC ) = (χ u NC )Π 0 λ NC dx λ NC (I NC χ χ)dx \ λ NC (χ u) dx (χ u)π 0 λ NC dx \ + (1 Π 0 )λ NC (1 Π 0 )(u u NC ) dx. (4.7)

20 NONCONFORMING FM FOR H OBSACL PROBLM 83 he Poincaré inequality with constant h /j 1,1 from Laugesen & Siudeja (2010) for each triangle yields (1 Π 0 )λ NC (1 Π 0 )(u u NC ) dx osc(λ NC, )/j 1,1 u u NC NC( ). (4.8) he combination of this with (4.7) and (4.6) plus the absorption of u u NC NC conclude the proof of b in heorem 4.1. Proof of heorem 4.4. Given any v K, the definitions of Λ and Λ NC plus elementary algebra show for e := u u NC that Λ(u v) + a NC (e, u v) = F(1 I NC )(u v) + Λ NC (I NC (u v)) = (f λ NC )(1 I NC )(u v) dx + Λ NC (u v). he binomial theorem shows 2a NC (e, u v) = e 2 NC + u v 2 v u NC 2 NC. Lemma 2.2 and a Cauchy inequality yield (f λ NC )(1 I NC )(u v) dx κ NC h (f λ NC ) L 2 () u v. he combination of the above-displayed estimates proves 2Λ(u v) + e 2 NC + u v 2 2κ NC h (f λ NC ) L 2 () u v + v u NC 2 NC + 2Λ NC(u v). (4.9) Lemma 4.6 yields Λ NC (u v) = Λ NC (u NC v) + (χ u NC )Π 0 λ NC dx λ NC (χ u) dx \ (χ u)π 0 λ NC dx + (1 Π 0 )λ NC (1 Π 0 )(u u NC ) dx. (4.10) he properties of the integral, the Poincaré inequality with the constant h /j 1,1 from Laugesen & Siudeja (2010) on each triangle, and the Cauchy inequality prove (1 Π 0 )λ NC (1 Π 0 )(u u NC ) dx osc(λ NC, ) u v NC( )/j 1,1 + (1 Π 0 )λ NC (1 Π 0 )(v u NC ) dx.

21 84 C. CARSNSN AND K. KÖHLR It holds that Λ NC (u NC v) + (1 Π 0 )λ NC (1 Π 0 )(v u NC ) dx = (u NC v)π 0 λ NC dx + (u NC v)λ NC dx. (4.11) \ he first and second terms on the right-hand side of (4.11) satisfy (u NC v)π 0 λ NC dx + (χ u NC )Π 0 λ NC dx = (χ v)π 0 λ NC dx and (u NC v)λ NC dx \ λ NC (I NC χ u) dx = \ (u χ)λ NC dx + \ (χ v)λ NC dx. \ he combination with the above estimates and the absorption of u v conclude the proof of assertion a of heorem 4.4. he proof of b employs an auxiliary Laplace problem with right-hand side f λnc L 2 (). Following Braess (2005), let w denote the weak solution of the Laplace equation with Dirichlet boundary conditions u D and right-hand side f λ NC L 2 (), which corresponds to F Λ NC H 1 (). hat is, w H 1 () satisfies w = u D on and a(w, v) = (F Λ NC )(v), for all v V. (4.12) Since w is the solution to (4.12), any v V satisfies a(u w, v) = (Λ Λ NC )(v). he choice of v := u w V leads to u w Λ Λ NC. On the other hand, it holds that (Λ Λ NC )(v) u w v for any v V. Altogether it follows that he triangle inequality leads to Λ Λ NC = u w. Λ Λ NC u u NC NC + w u NC NC. By definition of λ NC, the discrete approximation u NC equals the nonconforming finite element solution to the Poisson model problem (4.12), namely a NC (u NC, v NC ) = (F Λ NC )(v NC ), for all v NC V NC with right-hand side f λ NC L 2 () and exact solution w. Hence, the error of the continuous and discrete solution w u NC NC to the Poisson model problem and its error control may follow from the a posteriori

22 NONCONFORMING FM FOR H OBSACL PROBLM 85 error analysis of variational equations. For instance, Carstensen & Merdon (2013, heorem 3.1) show that w u NC NC u NC v NC + osc(f λ NC, )/j 1,1 + 1 Π 0 (f λ NC )( mid( )) 2 L 2 + w () D. his concludes the proof of b of heorem fficiency his section discusses the efficiency of the global upper bound (GUB) from heorem 4.4 for piecewise affine obstacles χ V 1 ( ). For any v K, the a posteriori error estimate from heorem 4.4 leads to a computable global upper bound GUB(v) of the five non-negative error terms in LHS(v) GUB(v), LHS(v) := u u NC NC + Λ Λ NC + Λ(u v) 1/2 ( ) 1/2 ( ) 1/2 + λ NC Π 0 (χ u) dx + (χ u)λ NC dx, \ GUB(v) := v u NC NC + h (f λ NC ) L 2 () + osc(λ NC, ) + w D ( ) 1/2 ( ) 1/2 + λ NC Π 0 (χ v) dx + λ NC (χ v) dx. \ For χ V 1 ( ), the a posteriori error estimate LHS(v) RHS(v) is efficient in the sense that the converse inequality holds up to some generic factor and oscillation terms. Recall the assumption u D C( ) H 2 ( ( )) and suppose that v is postprocessed from u NC such that it holds that u v u u NC NC + h 3/2 2 u D s 2 L 2 ( ) heorem 5.1 (fficiency of GUB) Any function v K with (5.1) satisfies GUB(v) LHS(v) + osc(f, ) + osc(λ, ) + h 3/2 2 u D s 2. (5.1) L 2 ( ) Remark 5.2 Given any postprocessing v A with (5.1), the function max{v, χ} belongs to K and satisfies h 3/2 2 u D u max{v, χ} u u NC NC + + min{0, v χ}. s 2 L 2 ( ) he conforming companion u 2 from Section 2.4 satisfies this estimate..

23 86 C. CARSNSN AND K. KÖHLR Proof of heorem 5.1. Step 1 is the proof of h (f λ NC ) L 2 () osc(f, ) + Λ Λ NC + u u NC NC. (5.2) Proof. he triangle inequality and f := Π 0 f with f := f for imply h (f λ NC ) L 2 () osc(f, ) + h (f λ NC ) L 2 (). he efficiency of h (f λ NC ) L 2 () is shown as for boundary value problems in Verfürth (1996). Consider the cubic bubble function b := 60Π z N () ϕ z on the triangle and set w := (f λ NC )b which satisfies h (f λ NC ) 2 L 2 () h2 w (f λ NC ) dx. (5.3) he Cauchy and Friedrichs inequalities lead to w (f f ) dx w osc(f, ). An integration by parts yields (f λ NC )w dx = (Λ Λ NC )(w ) u w dx. Since w dx = 0, it follows that u w dx = ( u NC u NC ) w dx u u NC NC() w. he combination of the previous estimates shows ) h (f λ NC ) 2 L 2 () h2 w ( u u NC NC() + Λ Λ NC, + osc(f, ). (5.4) he triangle inequality implies w (f λ NC ) b L 2 () + b (f λ NC ) L 2 (). he properties of b (Verfürth, 1996, Lemma 1.3) result in (f λ NC ) b L 2 () h 1 b 1/2 (f λ NC ) An inverse estimate shows b (f λ NC ) L 2 () h 1 b 1/2 (f λ NC ) L 2 (). L 2 ().

24 NONCONFORMING FM FOR H OBSACL PROBLM 87 herefore, w h 1 b 1/2 (f λ NC ) (5.5) L (). 2 he combination of (5.4) and (5.5) implies h (f λ NC ) L 2 () osc(f, ) + Λ Λ NC, + u u NC NC(). his is already a local version of the assertion for some fixed triangle. o prove the global form, let w be defined as above on each triangle. hen w V and h 2 w V and (5.4) is replaced by h (f λ NC ) 2 L 2 () h 2 w ( osc(f, ) + λ λ NC, + u u NC NC() ). Since (5.5) holds, this leads to h (f λ NC ) L 2 () u u NC NC + Λ Λ NC + osc(f, ). his concludes the proof of (5.2). Step 2 verifies osc(λ NC, ) Λ Λ NC + osc(λ, ). (5.6) Proof of Step 2. Let b := 60Π z N () ϕ z be the cubic bubble function on the triangle and set w := b (λ NC Π 0 λ NC ). his gives h (λ NC Π 0 λ NC ) 2 L 2 () h2 w (λ NC Π 0 λ NC ) dx. Since w dx = 0, it follows that w (λ NC Π 0 λ NC ) dx = w (λ NC λ) dx + w (λ Π 0 λ) dx. With w := w on each triangle, the summation over all triangles results in h (λ NC Π 0 λ NC ) 2 L 2 h 2 w () (λ NC λ) dx + h 2 w (λ Π 0 λ) dx Λ Λ NC h 2 w + osc(λ, ) h w L 2 (). An inverse estimate as in Step 1 and a Friedrichs inequality prove h 2 w h w L 2 () osc(λ NC, ). he division by osc(λ NC, ) concludes the proof of (5.6).

25 88 C. CARSNSN AND K. KÖHLR Step 3 is the proof of ( ) 1/2 ( ) 1/2 (χ v)π 0 λ NC dx + (χ v)λ NC dx LHS(v). (5.7) \ Proof of Step 3. he proof starts with the split of the first term: (χ v)π 0 λ NC dx = (χ u)π 0 λ NC dx + λ(u v) dx + (λ λ NC )(v u) dx + (λ NC Π 0 λ NC )(1 Π 0 )(v u) dx. he Cauchy, Poincaré and Young inequalities and (5.1) show (λ NC Π 0 λ NC )(1 Π 0 )(v u) dx u u NC 2 NC + osc(λ NC, ) 2 + he second term satisfies (χ v)λ NC dx = \ h 3/2 2 u D s 2 L 2 ( ) (χ u)λ NC dx + (u v)(λ NC λ) dx + (u v)λ dx. \ \ \ he square roots of the terms λ NC Π 0 (χ u) dx, osc(λ NC, ) 2, u u NC 2 and \ (I NC χ u)λ NC dx are part of LHS(v). he combination of the remaining terms with (5.1) leads to λ(u v) dx + λ(u v) dx = Λ(u v) \. and (λ λ NC )(v u) dx + (λ λ NC )(v u) dx Λ Λ NC 2 \ + u u NC 2 NC. he square roots of these terms are part of LHS(v). his concludes the proof of (5.7). Step 4. he efficiency of (f λ NC) dx( mid( )) L 2 () + w D follows from Carstensen & Merdon (2013, heorem 3.1) and Lemma 2.6. he combination of Steps 1 4 and the triangle inequality conclude the proof of heorem Computational benchmarks his section is devoted to the performance of the a priori and a posteriori error estimates in three benchmark examples implemented as in Alberty et al. (1999) and Bahriawati & Carstensen (2005).

26 NONCONFORMING FM FOR H OBSACL PROBLM Numerical realization he output of the following algorithm are the two lower energy bounds, the value of two error estimators and the corresponding efficiency indices. he lower bounds μ 1, μ 2 of (u) from heorem 4.1 are given by μ 1 := NC (u NC ) κ 2 NC /2 h f 2 L 2, () μ 2 := NC (u NC ) 1 ( κ NC h (f λ NC ) 2 L 2 () + osc(λ NC, ) (χ u NC )Π 0 λ NC dx + (I NC χ χ)λ NC dx. \ With from Section 4 and v := max{χ, w D + u D2 + J 2 (u NC I NC u D2 )} K for w D H 1 () as in Lemma 2.6 with w D = u D u D2 and J 2 from (2.7), the estimator η 1 and η 2 are given by ( ) ( ) 2 η 1 := u NC v (v) NC (u NC ) + κ NC h (f λ NC ) L 2 () + osc(f, ) + 2 (χ u NC )Π 0 λ NC dx + 2 (χ I NC χ)λ NC dx, \ η 2 := v u NC 2 NC (κ + NC h (f λ NC ) L 2 () + osc(f, ) + 2 (χ v)π 0 λ NC dx + 2 (χ v)λ NC dx. \ Algorithm. INPU an initial triangulation 0. LOOP for all l = 0, 1, 2,...until termination repeating the three steps COMPU the discrete solution u NC on l with ndof unknowns. 2 SIMA the error u v for v := max{χ, w D + u D2 + J 2 (u NC I NC u D2 )} with η 1 and η 2 and compute the lower energy bounds μ 2 and μ 2. he related efficiency indices ff(η 1 ) and ff(η 2 ) read ff(η j ) := ηj 2 (v)/ u u NC NC for j = 1, 2. 3 RFIN l by red refinement of all triangles and compute l+1. OUPU efficiency indices ff(η 1 ),ff(η 2 ), and the lower energy bounds μ 1, μ 2 on each level l. 6.2 Square domain he function u(r, ϕ) := max{r , 0} 2 solves the obstacle problem from Nochetto et al. (2003) with the constant obstacle χ 0, the nonhomogeneous Dirichlet data u D = u and the right-hand side f (r, ϕ) which equals 16r for r > 0.7 and r 2 for r 0.7 on the square ) 2 ) 2

27 90 C. CARSNSN AND K. KÖHLR Fig. 3. Convergence history plot of the error u u NC NC and the two error estimators (left) and efficiency indices of the two error estimators (right) for the NCFM as functions of ndof in Section 6.2. Fig. 4. Upper and lower energy bounds in Sections 6.2 (left) and 6.3 (right). domain ( 1, 1) 2 in polar coordinates (r, ϕ) at the origin. Figure 3 displays the error estimators η 1 and η 2 of u u NC. On the left, the error estimator and the corresponding exact error converge with a convergence rate 0.5 with respect to ndof as anticipated by heorem 3.1 both for the uniform algorithm (described above) and an adaptive algorithm based on the error estimator η 2 as a refinement indicator with Dörfler marking and a bulk parameter Θ = 0.5. On the right, Fig. 3 shows that the a posteriori error estimators are efficient with efficiency indices between 2 and 3. Figure 4 shows the lower energy bounds μ 1 and μ 2 for adaptive and uniform mesh refinement and their convergence towards the exact energy on the left. Both lower bounds converge and they exhibit the same overall behaviour.

28 NONCONFORMING FM FOR H OBSACL PROBLM 91 Fig. 5. Convergence history plot of the error u u NC NC and the two error estimators (left) and efficiency indices of the two error estimators (right) for the NCFM as functions of ndof in Section Smooth obstacle he function u χ K from Gräser & Kornhuber (2009) solves the obstacle problem on the square domain with the smooth obstacle χ(x, y) := (x 2 1)(y 2 1), the homogeneous Dirichlet data u D := 0 and the source term f := Δχ. Figure 5 investigates the quality of the error estimators for u u NC on the left and confirms that the error estimator and the corresponding error converge with a convergence rate 0.5 as anticipated by heorem 3.1, for the adaptive and uniform mesh refinements. Figure 5 reveals on the right that all three error estimators are efficient with efficiency indices between 2 and 2.8. Figure 4 shows the lower energy bounds μ 1 and μ 2 on adaptive and uniform meshes and their convergence towards (u) on the right. 6.4 L-shaped domain he example from Bartels & Carstensen (2004) considers a zero obstacle and Dirichlet data u D χ 0 on the L-shaped domain := ( 2, 2) 2 \[0, 2] [ 2, 0] with the source term ( ) 7 f (r, ϕ) := r 2/3 g(r) sin(2ϕ/3) + 2 g(r) H(r 5/4), 3r r r 2 g(r) := max{0, min{1, 6s s 4 10s 3 + 1}} for s := 2(r 1/4) with the Heaviside function H. he exact solution u(r, ϕ) = r 2/3 g(r) sin(2ϕ/3) has a typical corner singularity at the reentrant corner and illustrates the superiority of an adaptive meshrefinement strategy that accompanies heorem 4.4. Figure 6 (left) displays a significantly improved convergence rate for the adaptive algorithm compared with uniform mesh refinement. he efficiency indices for guaranteed error control in Fig. 6 on the right range between 2 and 3 for uniform and adaptive

29 92 C. CARSNSN AND K. KÖHLR Fig. 6. Convergence history plot of the error u u NC NC and the two error estimators (left) and efficiency indices of the two error estimators (right) for the NCFM as functions of ndof in Section 6.4. mesh refinement. he errors in Fig. 6 on the left converge with the same rate as the estimators, which also follows from the efficiency indices displayed in Fig. 6 (right). 6.5 Comments All numerical experiments confirm the a priori convergence rates anticipated by heorem 3.1 even in Section 6.4 with a singular solution on a polygon; the theoretical result in Wang (2003) does not cover this situation. he guaranteed error estimates lead to upper error bounds confirmed in all numerical examples. Additional undisplayed numerical experiments with nonconforming and conforming finite element methods show comparable accuracies even in the presence of singular solutions. he lower energy bound μ 2 leads to a better approximation of the exact energy (u) on coarse grids. his behaviour also holds true for the experiment in Section 6.4 (undisplayed). On fine grids, the two lower energy bounds μ 1 and μ 2 lead to comparable bounds. Adaptive mesh refinement leads to optimal convergence rates in all considered experiments. In all numerical examples, the error estimator η 2 leads to slightly better efficiency indices with less over-estimation of the true error. Overall efficiency indices between 2 and 3.5 are obtained for the estimators η 1 and η 2. Funding his research is carried out in the framework of Matheon supported by instein Foundation Berlin. he author K. K. was supported by the Berlin Mathematical School. References Alberty, J., Carstensen, C. & Funken, S. (1999) Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms, 20, Bahriawati, C. & Carstensen, C. (2005) hree Matlab implementations of the lowest-order Raviart homas MFM with a posteriori error control. CMAM, 5, Bartels, S. & Carstensen, C. (2004) Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math., 99,

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