MIXED AND STABILIZED FINITE ELEMENT METHODS FOR THE OBSTACLE PROBLEM

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1 MIXED AND STABILIZED FINITE ELEMENT METHODS FOR THE OBSTACLE PROBLEM TOM GUSTAFSSON, ROLF STENBERG, AND JUHA VIDEMAN Abstract. We discretize te Lagrange multiplier formulation of te obstacle problem by mixed and stabilized finite element metods. A priori and a posteriori error estimates are derived and numerically verified. arxiv: v [mat.na] 11 Nov 017 Key words. Obstacle problem, mixed finite elements, stabilized finite elements AMS subject classifications. 65N30 1. Introduction. In its classical form, te obstacle problem is an arctype example of a variational inequality [35]. Te problem corresponds to finding te equilibrium position of an elastic membrane constrained to lie above a rigid obstacle. Oter examples of obstacle-type problems are found, e.g., in lubrication teory [40], in flows troug porous media [], in control teory [38] and in financial matematics [18]. Discretization of te primal variational formulation of te obstacle problem by te finite element metod as been extensively studied since te 1970 s. Error estimates for te membrane displacement in te H 1 -norm ave been obtained, e.g., by Falk [19], Mosco Strang [36] and Brezzi Hager Raviart [10]. For an overview of te early progress on te subject and te respective references, see te monograp of Glowinski [3]. Instead of focusing on te primal formulation, we study an alternative variational formulation based on te metod of Lagrange multipliers [, 9, 11, 7]. Te Lagrange multiplier formulation introduces an additional pysically relevant unknown, te reaction force between te membrane and te obstacle, wic in itself can be a useful tool especially in te context of contact mecanics, cf. Hlaváček et al. [31] or, more recently, Wolmut [49]. Furtermore, te alternative formulation leads naturally to an effective solution strategy based on te semismoot Newton metod [30, 46] and provides also a straigtforward justification for te related Nitsce-type metod tat follows from te local elimination of te Lagrange multiplier in te stabilized discrete problem [44]. A priori error analysis for finite element metods based on a Lagrange multiplier formulation of te obstacle problem as been performed by Haslinger et al. [8], Weiss Wolmut [48] and Scröder et al. [4, 3]. A posteriori error estimates were derived, e.g., in Bürg Scröder [13] or Banz Stepan [4] and, using a similar Lagrange multiplier formulation, in Veeser [47], Braess [6] and Gudi Porwal [6]. Te purpose of tis paper is to readdress te Lagrange multiplier formulation of te obstacle problem. We consider two metods. Te first is a mixed metod in wic Financial support from Tekes (Decisions nr. 4005/1 and 3305/31/015), Portuguese Science Foundation (FCOMP FEDER-09408) and Finnis Cultural Foundation is gratefully acknowledged. Department of Matematics and Systems Analysis, Aalto University, P.O. Box 11100, Aalto, Finland (tom.gustafsson@aalto.fitom.gustafsson@aalto.fi). Department of Matematics and Systems Analysis, Aalto University, P.O. Box 11100, Aalto, Finland (rolf.stenberg@aalto.firolf.stenberg@aalto.fi). CAMGSD/Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, Lisboa, Portugal (jvideman@mat.tecnico.ulisboa.ptjvideman@mat.tecnico.ulisboa.pt). 1

2 te stability is acieved by adding bubble degrees of freedom to te displacement. Te second approac is a residual-based stabilized metod similar to te metods used successfully for te Stokes problem [3, 0]. Te latter tecnique as been applied to variational inequalities arising from contact problems in Hild Renard [9]. Until our recent article on te Stokes problem [45], error estimates on stabilized metods were always built upon te assumption tat te exact solution is regular enoug, wit te additional regularity assumptions arising from te stabilizing terms. In [45], we realized tat tese extra regularity requirements can be dropped and only te loading term needs to be in L. In tis work, we extend tese ideas to variational inequalities. We empasize tat te improved error estimates can only be establised if te discrete stability bound is proven in correct norms, i.e. in te norms of te continuous problem and not in mes dependent norms as, for example, in Hild Renard [9] were te autors need to assume tat te solution to te Signorini problem is in H s, wit s > 3/. We perform te analysis in a unified manner. First, we prove a stability result for te continuous problem in proper norms. Tis estimate becomes useful in deriving te a posteriori estimates. As for te discretizations, we start by proving stability wit respect to a mes dependent norm. Tis discrete stability result implies stability in te continuous norms and yields quasi optimal a priori estimates witout additional regularity assumptions. For te stabilized metods, we use a tecnique first suggested by Gudi [5]. Te stabilized formulation of te classical Babuška s metod of Lagrange multipliers for approximating Diriclet boundary conditions is known to be closely related to Nitsce s metod [37]. Tis connection as been used for te contact problem in Hild Renard [9] and Couly Hild [16], and for te obstacle problem in [14]. We sow tat a similar relationsip olds ere as well and observe tat for te lowest order metod it leads to te penalty formulation. We end te paper by reporting on extensive numerical computations.. Problem definition. Consider finding te equilibrium position u = u(x, y) of an elastic, omogeneous membrane constrained to lie above an obstacle represented by g = g(x, y). Te membrane is loaded by a normal force f = f(x, y) and its boundary is eld fixed. Te problem can be formulated as (.1) u f 0 in Ω, u g 0 in Ω, (u g)( u + f) = 0 in Ω, u = 0 on Ω, were Ω R is a smoot bounded domain or a convex polygon, and were we assume tat g = 0 at Ω. Let V = H0 1 (Ω). Te problem (.1) can be recast as te following variational inequality: find u V suc tat (.) ( u, (v u) ) (f, v u) v V, were (.3) V = {v V : v g a.e. in Ω}. It is well known tat, given f L (Ω) and g H 1 (Ω) C(Ω), problem (.) admits a unique solution u V, cf. Lions Stampaccia [35], or Kinderlerer Stampaccia [34].

3 Given tat te second derivatives of u ave a jump across te free boundary separating te contact region from te region free of contact, one cannot expect te solution to be more regular tan C 1,1 (Ω), cf. Caffarelli [15]. However, te second derivatives are bounded if te data is smoot []. In particular, if f L (Ω) and g H (Ω), te solution u V H (Ω), cf. Brezis Stampaccia [8]. Introducing a non-negative Lagrange multiplier function λ : Ω R, we can rewrite te obstacle problem as (.4) u λ = f in Ω, u g 0 in Ω, λ 0 in Ω, (u g) λ = 0 in Ω, u = 0 on Ω. Te Lagrange multiplier is in te dual space to H 1 0 (Ω), i.e. wit te norm Q = H 1 (Ω), (.5) ξ 1 = sup v V v, ξ v 1, were, : V Q R denotes te duality pairing. Te corresponding variational formulation becomes: find (u, λ) V Λ suc tat (.6) ( u, v) v, λ = (f, v) v V, u g, µ λ 0 µ Λ, were Λ = {µ Q : v, µ 0 v V, v 0 a.e. in Ω}. Existence of a unique solution (u, λ) V Λ to te mixed problem (.6) and equivalence between formulations (.) and (.6) as been proven, e.g., in Haslinger et al. [8]. Let U = V Q and define te bilinear form B : U U R and te linear form L : U R troug B(w, ξ; v, µ) = ( w, v) v, ξ w, µ, L(v, µ) = (f, v) g, µ. Problem (.6) can now be written in a compact way as: find (u, λ) V Λ suc tat (.7) B(u, λ; v, µ λ) L(v, µ λ) (v, µ) V Λ. Our analysis is built upon te following stability condition. Note tat we often write a b (or a b) wen a Cb (or a Cb) for some positive constant C independent of te finite element mes. Teorem.1. For all (v, ξ) V Q tere exists w V suc tat (.8) B(v, ξ; w, ξ) ( v 1 + ξ 1 ) and (.9) w 1 v 1 + ξ 1. 3

4 Proof. Suppose te pair (v, ξ) V Q is given and let w = v q, were q V satisfies (.10) ( q, z) + (q, z) = z, ξ z V. Coosing te test function z = q, gives (.11) q, ξ = ( q, q) + (q, q) = q 1, and ence we obtain (.1) q 1 = ξ, q q 1 sup z V ξ, z z 1 = ξ 1. Te norm of ξ can be bounded from above using (.10) and te Caucy Scwarz inequality as follows: (.13) ξ 1 = sup z V z, ξ z 1 = sup z V ( q, z) + (q, z) z 1 q 1. Tis implies tat q 1 = ξ 1. Using now te results (.1) and (.13) and Poincaré s and Caucy Scwarz inequalities, we conclude tat B(v, ξ; w, ξ) = ( v, w) + v w, ξ = ( v, v) ( v, q) + q, ξ v 0 v 0 q 0 + q 1 ( v 1 + ξ 1 ). Finally, from te triangle inequality it follows tat w 1 = v q 1 v 1 + q 1 = v 1 + ξ 1. We will consider finite element spaces based on a conforming sape-regular triangulation C of Ω, wic we encefort assume to be polygonal. By E we denote te interior edges of Ω. Te finite element subspaces are Moreover, we define V V, Q Q. Λ = {µ Q : µ 0 in Ω} Λ. Remark.. Wen Q are piecewise polynomials of degree two or iger, te condition µ 0 becomes difficult to satisfy in practice. In tat case, one option would be to implement tis condition in discrete points, as is done in [3]. We do not pursue tis pat owever since it would lead to a nonconforming metod wit Λ Λ and require a separate analysis. We first consider metods corresponding to te continuous problem (.7). 3. Mixed metods. Te mixed finite element metod for problem (.7) reads as follows. Te mixed metod. Find (u, λ ) V Λ suc tat (3.1) B(u, λ ; v, µ λ ) L(v, µ λ ) (v, µ ) V Λ. 4

5 For tis class of metods, te finite element spaces ave to satisfy te Babuška Brezzi condition (3.) sup v V v, ξ v 1 ξ 1 ξ Q. Te Babuška Brezzi condition implies te following discrete stability estimate. Teorem 3.1. If condition (3.) is valid, ten for all (v, ξ ) V Q, tere exists w V, suc tat (3.3) B(v, ξ ; w, ξ ) ( v 1 + ξ 1 ) and (3.4) w 1 v 1 + ξ 1. Proof. Let w = v q, were q V is suc tat ( q, z ) + (q, z ) = z, ξ z V. By condition (3.) and te Caucy Scwarz inequality we ave z, ξ ( q, z ) + (q, z ) ξ 1 sup = sup q 1. z V z 1 z V z 1 Similarly as in te proof of Teorem.1, we get Finally, B(v, ξ ; w, ξ ) = ( v, w ) + ξ, q = v 0 ( v, q ) + q 1 v 0 + q 1 ( v 1 + ξ 1 ). w 1 = v q 1 v 1 + q 1 v 1 + ξ 1. We will use te tecnique of bubble functions to define a family of stable finite element pairs. Wit b K P 3 (K) H 1 0 (K) we denote te bubble function scaled to ave a maximum value of one and define (3.5) B l+1 (K) = {z H 1 0 (K) : z = b K w, w P l (K)}, were P l (K) denotes te space of omogeneous polynomials of degree l. Let k 1 be te degree of te finite element spaces defined by { {v V : v K P 1 (K) B 3 (K) K C } for k = 1, (3.6) V = {v V : v K P k (K) B k+1 (K) K C } for k, and let (3.7) Q = { {ξ Q : ξ K P 0 (K) K C } for k = 1, {ξ Q : ξ K P k (K) K C } for k. 5

6 Note tat te approximation orders of te finite element spaces are balanced, i.e. (3.8) inf v V u v 1 = O( k ) and inf ξ Q λ ξ 1 = O( k ), wen u H k+1 (Ω) and λ H k 1 (Ω). We will use te following discrete negative norm in proving te stability condition: (3.9) ξ 1, = K ξ 0,K ξ Q. Analogously to te Stokes problem [43], we will need te following auxiliary result. Note tat te result olds independently of te coice of te finite element spaces. Lemma 3.. Tere exist positive constants C 1 and C suc tat (3.10) sup v V v, ξ v 1 C 1 ξ 1 C ξ 1, ξ Q. Proof. Te continous stability (Teorem.1) implies tat tere exists w H 1 0 (Ω) and C > 0 suc tat (3.11) w, ξ C w 1 ξ 1 for all ξ Q. Let w V be te Clément interpolant [17] of w. Since ξ L (Ω) te duality pairing equals to te L -inner product. Ten (3.11) and te Caucy Scwarz inequality give (3.1) w, ξ = w w, ξ + w, ξ = (w w, ξ ) K + C w 1 ξ 1 w w 0,K ξ 0,K + C w 1 ξ 1 = K w w 0,K K ξ 0,K + C w 1 ξ 1 1 ( K w w 0,K ξ 1, + C w 1 ξ 1. From te properties of te Clément interpolant, we ave ( K w w 0,K C w 1,K and w 1 C w 1 wic togeter wit (3.1) sows tat w, ξ C w 1 ξ 1, + C w 1 ξ 1 C w 1 ξ 1, + C w 1 ξ 1 C (C ξ 1 C ξ 1, ) w 1. Dividing by w 1 provides te claim. Using tis result one proves te following. 6

7 Lemma 3.3. If we ave stability in te discrete norm, i.e. (3.13) sup w V w, ξ w 1 ξ 1, ξ Q ten te Babuška Brezzi condition (3.) olds. Proof. Suppose (3.13) olds. Ten by Lemma 3., for t > 0 we ave (3.14) w, ξ w, ξ w, ξ sup = t sup + (1 t) sup w V w 1 w V w 1 w V w 1 t(c 1 ξ 1 C ξ 1, ) + (1 t)c 3 ξ 1, = tc 1 ξ 1 + (C 3 C 3 t C t) ξ 1,. Tus, if we coose t = 1 C 3(C + C 3 ) 1, te second term on te rigt and side of (3.14) is positive and ence w, ξ C 1 C 3 sup w V w 1 (C + C 3 ) ξ 1 ξ Q. Te advantage in using te intermediate step in proving te stability in te mesdependent norm is tat te discrete negative norm can be computed elementwise in contrast to te continuous norm wic is global. Lemma 3.4. Te finite element spaces (3.6) and (3.7) satisfy te Babuška Brezzi condition (3.). Proof. We begin by sowing stability in te discrete norm 1, and ten apply Lemma 3.3 to get te stability in te continuous norm. Given ξ Q, we can define w V by (3.15) w K = b K Kξ K. Ten we estimate (3.16) w, ξ = (w, ξ ) K = K b K Kξ dx ξ 1,. Moreover, using te inverse inequality and te definition (3.15) we get (3.17) w 1 K w 0,K K ξ 0,K = ξ 1,. Combining estimates (3.16) and (3.17) proves stability in te discrete norm. Finally, we apply Lemma 3.3 to conclude te result. Lemma 3.5. Te following inverse estimate olds ξ 1, ξ 1 ξ Q. Proof. In te preceeding lemma, we sowed tat w, ξ sup ξ 1, ξ Q. w V w 1 Te assertion tus follows from te fact tat w, ξ w 1 ξ 1. 7

8 Remark 3.6. Note tat te above inverse inequality is valid in an arbitrary piecewise polynomial finite element space Λ, since one can always use te bubble function tecnique to construct a space V in wic te discrete stability inequality is valid. Te a priori error estimate now follows from te discrete stability estimate of Teorem 3.1. Teorem 3.7. Te following error estimate olds ( u u 1 + λ λ 1 inf u v 1 + inf λ µ 1 + u g, µ ). v V µ Λ Proof. Let (v, µ ) V Λ be arbitrary. In view of te stability estimate, tere exists w V suc tat (3.18) w 1 u v 1 + λ µ 1 and (3.19) ( u v 1 + λ µ 1 ) B(u v, λ µ ; w, µ λ ). Given te discrete problem statement and te bilinearity of B, by adding and subtracting B(u, λ; w, µ λ ), we obtain (3.0) B(u v, λ µ ; w, µ λ ) = B(u, λ ; w, µ λ ) B(v, µ ; w, µ λ ) B(u v, λ µ ; w, µ λ ) + L(w, µ λ ) B(u, λ; w, µ λ ) = B(u v, λ µ ; w, µ λ ) + u g, µ λ B(u v, λ µ ; w, µ λ ) + u g, µ λ, were in te last two lines we ave used te continuous problem, i.e. 0 u g, λ λ. Te continuity of te bilinear form B and te bound (3.18) imply (3.1) B(u v, λ µ ; w, µ λ ) ( u v 1 + λ µ 1 ) ( u v 1 + λ µ 1 ). Combining te estimates (3.19), (3.0) and (3.1) gives ( u v 1 + λ µ 1 ) ( u v 1 + λ µ 1 )( u v 1 + λ µ 1 ) + u g, µ λ. Applying Young s inequality to te first term on te rigt and side and completing te square gives u v 1 + λ µ 1 u v 1 + λ µ 1 + u g, µ λ. Since u g, λ = 0, te triangle inequality yields u u 1 + λ λ 1 u v 1 + u v 1 + λ µ 1 + λ µ 1 ( inf u v 1 + inf λ µ 1 + ) u g, µ. v V µ Λ 8

9 Remark 3.8. Tis error estimate is known in te literature, cf. [7, Teorem 5, Remark 3] and [8, Lemma 4.3, Remark 4.9]. However, we perform te analysis more in te spirit of Babuška [1, ] tan tat of Brezzi [9] wic seems to be te common approac. Next we derive te a posteriori estimate. We define te local error estimators η K and η E by η K = K u + λ + f 0,K, η E = E u n 0,E. Furter, we define η = ηk + ηe, E E S m = (g u ) (g u ) +, λ, were w + = max{w, 0} denotes te positive part of w. Teorem 3.9. Te following a posteriori estimate olds u u 1 + λ λ 1 η + S m. Proof. By te stability of te continuous problem (Teorem.1), tere exists w V suc tat (3.) w 1 u u 1 + λ λ 1 and (3.3) ( u u 1 + λ λ 1 ) B(u u, λ λ ; w, λ λ). Let w V be te Clément interpolant of w. Te problem statement gives 0 B(u, λ ; w, 0) + L( w, 0). It follows tat ( ) u u 1 + λ λ 1 B(u, λ; w, λ λ) B(u, λ ; w, λ λ) B(u, λ; w, λ λ) B(u, λ ; w, λ λ) B(u, λ ; w, 0) + L( w, 0) L(w, λ λ) B(u, λ ; w w, λ λ) + L( w, 0) L(w w, λ λ) B(u, λ ; w w, λ λ). Opening up te rigt and side and combining terms, results in ( u u 1 + λ λ 1 ) (f, w w) g, λ λ ( u, (w w)) + w w, λ + u, λ λ = ( u + λ + f, w w) K ( u n, w w) E + u g, λ λ. E E 9

10 Te first two terms are estimated as usual; recall tat te Clément interpolant satisfies ( K w w 0,K + E w w 0,E w 1. Te last term is bounded as follows: (3.4) (3.5) u g, λ λ = g u, λ E E 1 (g u ) +, λ = (g u ) +, λ λ + (g u ) +, λ (g u ) + 1 λ λ 1 + (g u ) +, λ. To derive lower bounds, let f Q be te L -projection of f and define osc K (f) = K f f 0,K osc(f) = osc K (f). A function w H 1 0 (U), wit U Ω, we extend by zero into Ω \ U, and and for functions in µ Λ we ten define (3.6) µ 1,U = sup v H 1 0 (U) v, µ v 1,U. (3.7) (3.8) (3.9) (3.30) and We let ω(e) be te union of te two elements saring E. Lemma For all v V and µ Q, it olds K v + µ + f 0,K u v 1,K + λ µ 1,K + osc K (f), ( K v + µ + f 0,K u v 1 + λ µ 1 + osc(f), 1/ E v n 0,E u v 1,ω(E) + λ µ 1,ω(E) + ( E E E v n 0,E K ω(e) u v 1 + λ µ 1 + osc(f). Proof. Using te bubble function b K P 3 (K) H 1 0 (K), we define γ K by γ K = Kb K ( v + µ + f ) in K, and γ K = 0 in Ω \ K. Testing wit γ K in (.6 yields (3.31) ( u, γ K ) K γ K, λ = (f, γ K ) K. Using tis and te norm equivalence in polynomial spaces, we obtain K v + µ + f 0,K K b K ( v + µ + f ) 0,K osc K (f), (3.3) = ( v + µ + f, γ K ) K = ( v + µ, γ K ) K + (f, γ K ) K + (f f, γ K ) K = ( v + µ, γ K ) K + ( u, γ K ) K γ K, λ + (f f, γ K ) K = ( (u v ), γ K ) K + γ K, µ λ + (f f, γ K ) K. 10

11 Te rigt and side above can be estimated as follows ( (u v ), γ K ) K + γ K, µ λ + (f f, γ K ) K (u v ) 0,K γ K 0,K + µ λ 1,K γ K 1,K + osc K (f) 1 K γ K 0,K. By inverse inequalities, we ave (3.33) γ K 1,K K γ K 0,K = K b K ( v + µ + f ) 0,K K v + µ + f 0,K. Combining (3.3) (3.33) gives te first estimate (3.7). To prove (3.8), we write γ = γ K and sum te inequality (3.3) over all elements. Tis yields (3.34) K v + µ + f 0,K { } ( (u v ), γ K ) K + γ K, µ λ + (f f, γ K ) K = ( (u v ), γ) + γ, µ λ + (f f, γ) ( u v 1 γ 1 + µ λ 1 γ 1 + osc(f) K γ 0,K Summing estimates (3.33) over K C, leads to (3.8). Next, let b E be te bubble on E, and denote w E = b E X E ( v n ), were X E is te standard extension operator onto H 1 0 (ω(e)) cf. Braess [7]. By scaling and Poincaré s inequality we ave (3.35) v n 0,E b E v n 0,E 1/ E w E 0,ω(E) 1/ E w E 0,ω(E) 1/ E w E 1,ω(E). Using tis, Green s formula, and te variational formulation (.6), yield (3.36) Hence, it olds (3.37) and (3.35) gives (3.38) v n 0,E b E v n 0,E = ( v n, w E ) E = ( v, w E ) ω(e) + ( v, w E ) ω(e). = ( v, w E ) ω(e) + ( (v u), w E ) ω(e) + ( u, w E ) ω(e) = ( v, w E ) ω(e) + ( (v u), w E ) ω(e) + (f, w E ) ω(e) + λ, w E = ( v + µ + f, w E ) ω(e) + ( (v u), w E ) ω(e) + λ µ, w E. v n 0,E v + µ + f 0,ω(E) w E 0,ω(E) + (u v ) 0,ω(E) w E 0,ω(E) + λ µ 1,ω(E) w E 1,ω(E), 1/ E v n 0,E E v + µ + f 0,ω(E) + (u v ) 0,ω(E) + λ µ 1,ω(E), 11

12 and (3.9) follows from te already proved estimate (3.7). In order to prove (3.30), we use (3.36) to obtain (3.39) E E E v n 0,E E E E ( v + µ + f, w E ) ω(e) + E E E ( (v u), w E ) ω(e) + E E E λ µ, w E. Since eac E E contains two elements of C, we get from (3.35) (3.40) E ( v + µ + f, w E ) ω(e) E E ( ( E v + µ + f 0,ω(E) E w E 0,ω(E) E E E E ( ( K v + µ + f 0,K E v n 0,E E E, (3.41) and (3.4) E ( (u v ), w E ) ω(e), E E ( ( (u v ) 0,ω(E) E E (u v ) 0 ( E E E v n 0,E E E w E ω(e), E w E, λ µ = E w E, λ µ E E E E λ µ 1 E w E 1 E E λ µ 1 ( E E E v n 0,E Te asserted estimate now follows by combining (3.39) (3.4) and (3.8). Coosing v = u and µ = λ above, we obtain te local lower bounds (3.43) (3.44) η K u u 1,K + λ λ 1,K + osc K (f), η E u u 1,ω(E) + ( λ λ 1,K + osc K (f) ), K ω(e) and te global bound (3.45) η u u 1 + λ λ 1 + osc(f). 1

13 Remark Tese estimates ave appeared in te literature, cf. [3, 6, 13]. For completeness, we ave given a proof based on our approac. Remark 3.1. In proving Lemma 3.10, we never used te fact tat (u, λ ) solves te mixed problem. Te estimates tus old also for te stabilized metods tat will be presented in te next section. In fact, tey turn out to be crucial for te a priori error analysis of tese metods. 4. Stabilized metods. From te Stokes problem, it is known tat te tecnique of using stabilizing bubble degrees of freedom can be avoided by te so-called residual-based stabilizing [33, 1]. Below we will sow tat tis approac applies also to te present problem. Te resulting formulation, stability and error estimates are valid for any finite element pair (V, Λ ). Let us start by introducing te bilinear and linear forms S and F by S (w, ξ; v, µ) = K( w ξ, v µ) K, F (v, µ) = K(f, v µ) K, and ten define te forms B and L troug B (w, ξ; v, µ) = B(w, ξ; v, µ) α S (w, ξ; v, µ), L (v, µ) = L(v, µ) α F (v, µ), were α > 0 is a stabilization parameter. Note tat wit te assumption f L (Ω) it olds tat u + λ L (Ω), even if u L (Ω) and λ L (Ω). Hence it olds (4.1) S (u, λ; v, µ ) = F (v, µ ) (v, µ ) V Λ. Tis motivates te following stabilized finite element metod. Te stabilized metod. Find (u, λ ) V Λ suc tat (4.) B (u, λ ; v, µ λ ) L (v, µ λ ) (v, µ ) V Λ. In our analysis, we need an inverse inequality wic we write as: tere exists a positive constant C I suc tat (4.3) C I K v 0,K v 0 v V. Te following stability condition olds. Teorem 4.1. Suppose tat 0 < α < C I. It ten olds: for all (v, ξ ) V Q, tere exists w V, suc tat (4.4) B (v, ξ ; w, ξ ) ( v 1 + ξ 1 ) and (4.5) w 1 v 1 + ξ 1. 13

14 Proof. Let (v, ξ ) V Q be arbitrary. Wit te assumption 0 < α < C I, te inverse estimate (4.3) gives B (v, ξ ; v, ξ ) = v 0 α K v 0,K + α ξ 1, (1 α ) v 0 + α ξ 1, C I C 3 ( v 0 + ξ 1,), wic guarantees stability wit respect to te mes-dependent norm for functions in Λ. To prove stability in te H 1 -norm, we let q be te function for wic te supremum in Lemma 3. is obtained, viz. (4.6) q, ξ q 1 C 1 ξ 1 C ξ 1,. By omogeneity we can assign te equality (4.7) q 1 = ξ 1. Using te above relations, estimate (4.3), and te Young s inequality, wit ε > 0, gives B (v, ξ ; q, 0) = ( v, q ) + q, ξ α K( v + ξ, q ) K v 0 q 0 + C 1 ξ 1 C ξ 1, ξ 1 ( ( ( α K v 0,K K q 0,K α ξ 1, K q 0,K (1 + α ) v 0 q 0 + C 1 ξ 1 C ξ 1, ξ 1 α ξ 1, q 0 C I CI (1 + α ) v 0 ξ 1 + C 1 ξ 1 C ξ 1, ξ 1 α ξ 1, ξ 1 C I CI ( C 1 ε (1 + α ) + C + α) ξ 1 1 ((1 + α ) v 0 + (C + α ) ) ξ 1, C I ε C I C I C 4 ξ 1 C 5 ( v 0 + ξ 1,), were ε as been cosen small enoug. Hence (4.8) B (v, ξ ; v + δq, ξ ) δc 4 ξ 1 + (C 3 δc 5 ) ( v 0 + ξ ) 1, and te assertion follows by coosing 0 < δ < C 3 /C 5. Next, we derive te a priori estimate. We follow our analysis for te Stokes problem, see [45], and use a tecnique introduced by Gudi [5]. Te key ingredient is a tool from te a posteriori error analysis, namely te estimate (3.8) of Lemma Teorem 4.. Te following a priori estimate olds u u 1 + λ λ 1 ( inf u v 1 + inf λ µ 1 + u g, µ ) +osc(f) v V µ Λ Proof. Teorem 4.1 implies tat tere exists w V suc tat w 1 u v 1 + λ µ 1 14

15 and ( u v 1 + λ µ 1 ) B (u v, λ µ ; w, µ λ ). We ten estimate, similarly to te bound (3.0) of Teorem 3.7, B (u v, λ µ ; w, µ λ ) = B (u, λ ; w, µ λ ) B (v, µ ; w, µ λ ) B(u v, λ µ ; w, µ λ ) + L(w, µ λ ) B(u, λ; w, µ λ ) α K( (u v ) (λ µ ), w (µ λ )) K B(u v, λ µ ; w, µ λ ) + u g, µ λ + α K v + µ + f 0,K w + µ λ 0,K. Te first two terms are as in Teorem 3.7. Te last term is estimated using te triangle and Caucy Scwarz inequalities, Lemma 3.10, and te inverse inequality of Lemma 3.5: K v + µ + f 0,K w + µ λ 0,K ( ( K v + µ + f 0,K K w 0,K ( ( + K v + µ + f 0,K K µ λ 0,K ( u v 1 + λ µ 1 + osc(f))( w 0 + µ λ 1, ) ( u v 1 + λ µ 1 + osc(f))( w 0 + µ λ 1 ). For te a posteriori estimate we define (4.9) S s = (g u ) (u g) +, λ. Teorem 4.3. Te following a posteriori estimate olds u u 1 + λ λ 1 η + S s. Proof. By te continuous stability tere exists w V wit te properties (4.10) w 1 u u 1 + λ λ 1, and (4.11) ( u u 1 + λ λ 1 ) B(u u, λ λ ; w, λ λ). Using te problem statement we ave 0 B (u, λ ; w, 0) + L ( w, 0), 15

16 were w is te Clément interpolant of w. It follows tat (4.1) ( u u 1 + λ λ 1 ) B(u, λ; w, λ λ) B(u, λ ; w, λ λ) L(w, λ λ) B(u, λ ; w, λ λ) B (u, λ ; w, 0) + L ( w, 0) L(w w, λ λ) B(u, λ, w w, λ λ) + α K( u λ f, w) K. Te first two terms are estimated similarly as in te proof of Teorem 3.9 wit te exception of te term u g, λ λ. For te stabilized metod, u g, λ 0, and terefore u g, λ λ (g u ) +, λ g u, λ = (g u ) +, λ λ + (u g) +, λ (g u ) + 1 λ λ 1 + (u g) +, λ. Te last term in (4.1) is bounded using te Caucy Scwarz inequality and inverse estimate as ( K( u λ f, w) K K u + λ + f 0 w 0 and by te properties of te Clément interpolant we ave tat w 0 w 1. Note tat we ave not explicitly defined te finite element spaces, and ence te metod is stable and te estimate olds for all coices of finite element pairs. Te optimal coice is dictated by te approximation properties and is (4.13) V = {v V : v K P k (K) K C }, and (4.14) Q = { {ξ Q : ξ K P 0 (K) K C } for k = 1, {ξ Q : ξ K P k (K) K C } for k. Remark 4.4. For te lowest order mixed metod, i.e. for te metod (3.1) wit k = 1 in (3.6) and (3.7), a local elimination of te bubble degrees of freedom gives te stabilized formulation wit k = 1, for wic we ave S (w, ξ; v, µ) = K(ξ, µ) K and F (v, µ) = K(f, µ) K. Tis is in complete analogy wit te relationsip between te MINI and te Brezzi Pitkäranta metods for te Stokes equations, cf. [1, 39]. Note also tat no upper bound needs to be imposed on α in tis case. 5. Iterative solution algoritms. Te contact area, i.e. te subset of Ω were te solution satisfies u = g, is unknown and must be solved as a part of te solution process. Let us first consider te mixed metod (3.1). Te weak form corresponding to problem (3.1) reads: find (u, λ ) V Λ suc tat (5.1) (5.) ( u, v ) (λ, v ) = (f, v ), v V, (u g, µ λ ) 0, µ Λ. 16

17 Testing te inequality (5.) wit µ = 0 and µ = λ leads to te system ( u, v ) (λ, v ) = (f, v ), v V, (u g, µ ) 0, µ Λ, (u g, λ ) = 0. We consider te case of low order elements wit 1 k 3, and let ξ j, j {1,..., M}, be te Lagrange (nodal) basis for Q. Wen writing µ = M j=1 µ jξ j, we ten ave te caracterization { M } (5.3) Λ = µ = µ j ξ j µ j 0. j=1 By letting ϕ j, j {1,..., N}, be te basis for V, and writing u = N j=1 u jϕ j, we arrive at te finite dimensional complementarity problem (5.4) (5.5) (5.6) (5.7) were Au B T λ = f, Bu g, λ T (Bu g) = 0, λ 0, A R N N (A) ij = ( ϕ i, ϕ j ), B R M N (B) ij = (ξ i, ϕ j ), f R N (f) i = (f, ϕ i ), g R M (g) i = (g, ξ i ), u R N (u) i = u i, λ R M (λ) i = λ i. Remark 5.1. For iger order metods wit k > 3 and a nodal basis te inequalities λ 0 and λ 0 are not equivalent and anoter solution strategy is required if one wants te solution space to span all positive piecewise polynomials. Following e.g. Ulbric [46] te tree constraints (5.5) (5.7) can be written as a single nonlinear equation to get (5.8) Au B T λ = f, λ max{0, λ + c(g Bu)} = 0, wit any c > 0. Application of te semismoot Newton metod to te system (5.8) leads to Algoritm 1 [30]. In te algoritm definition we use a notation similar to Golub Van Loan [4] were, given a matrix C and a row position vector i, we denote by C(i, :) te submatrix formed by te rows of C marked by te index vector i. Similarly, b(i) consists of te components of vector b wose indices appear in vector i. Note tat te linear system to be solved at eac iteration step (Step 8) as te saddle point structure. For tis class of problems tere exists numerous efficient solution metods, cf. Benzi et al. [5]. Let us next consider te stabilized metod (4.). Te respective discrete weak formulation is: find (u, λ ) V Λ suc tat ( u, v ) (λ, v ) α K( u +λ, v ) K = (f, v ) + α K(f, v ) K, (u g, µ λ ) + α k( u +λ +f, µ λ ) K 0, 17

18 Algoritm 1 Primal-dual active set metod for te mixed problem 1: k = 0; λ 0 = 0 : Solve Au 0 = f 3: wile k < 1 or λ k λ k 1 > TOL do 4: s k = λ k + c(g Bu k ) 5: Let i k consist of te indices of te nonpositive elements of s k 6: Let a k consist of te indices of te positive elements of s k 7: λ k+1 (i k ) = 0 8: Solve [ ] [ ] [ ] A B(a k, :) T u k+1 f B(a k, :) 0 λ k+1 (a k = ) g(a k ) 9: k = k : end wile old for every (v, µ ) V Λ. Troug similar steps as in te mixed case we arrive at te algebraic system (5.9) (5.10) (5.11) (5.1) were A α R N N B α R M N C α R M M A α u B T α λ = f α, B α u + C α λ g α, λ T (B α u + C α λ g α ) = 0, λ 0, (A α ) ij = ( ϕ i, ϕ j ) α K( ϕ i, ϕ j ) K, (B α ) ij = (ξ i, ϕ j ) + α K(ξ i, ϕ j ) K, (C α ) ij = α K(ξ i, ξ j ) K, f α R N (f α ) i = (f, ϕ i ) + α K(f, ϕ i ) K, g α R M (g α ) i = (g, ξ i ) α K(f, ξ i ) K. Te system corresponding to (5.8) reads (5.13) A α u B T α λ = f α, λ max{0, λ + c(g α B α u C α λ)} = 0, wic leads to Algoritm. Note tat te inversion of te matrix C α is performed on eac element separately, and tat equation to be solved in Step 6 is symmetric and positive-definite. It as a condition number of O( ) and ence standard iterative solvers can be used. 6. Nitsce and penalty metods. Consider te stabilized metod and recall tat te stability and error estimates old for any finite element subspace Λ for te reaction force. Let Ω c denote te contact region and assume, for te time being, tat 18

19 Algoritm Primal-dual active set metod for te stabilized problem 1: k = 0 : Solve A α u 0 = f α 3: λ 0 = C 1 α (g α B α u 0 ) 4: wile k < 1 or λ k λ k 1 > TOL do 5: Let a k consist of te indices of te positive elements of λ k 6: Solve ( Aα + B α (a k, :) T C α (a k, a k ) 1 B α (a k, :) ) u k+1 7: λ k+1 = max{0, C 1 α (g α B α u k+1 )} 8: k = k + 1 9: end wile = f α + B α (a k, :) T C α (a k, a k ) 1 g α (a k ) its boundary lies on te inter-element edges. We will derive te Nitsce s formulation by te following line of argument. Noting tat te functions of Λ are discontinuous, we may eliminate te variable λ locally on eac element. Testing wit (0, µ ) in te stabilized problem (4.) gives (u, µ λ )+α K( u +λ, µ λ ) K (g, µ λ ) α K(f, µ λ ) K for every µ Λ. Since te contact area Ω c is assumed to be known, tis reads (u, µ ) + α K( u + λ, µ ) K = (g, µ ) α K(f, µ ) K K Ω c K Ω c giving locally (6.1) λ K = (α K) 1 (Π g Π u ) K Π f K Π u K K Ω c, were Π is te L -projection onto Λ. Testing wit (v, 0) in (4.) and substituting (6.1) into te resulting equation gives te problem: find u V suc tat ( u, v ) + (Π u, Π v ) K + (Π u, Π v ) K + α 1 α K Ω c K Ω c K (Π u, Π v ) K + α K( u, v ) K K Ω c K Ω c K((I Π ) u, (I Π ) v ) K K Ω\Ω c = (f, v ) Ω\Ω c + ((I Π )f, v ) Ω c + α K((I Π )f, v ) K + K Ω c + α K Ω\Ω c (Π g, Π v ) K + α 1 K(f, v ) K, K Ω c 19 K Ω c K (Π g, Π v ) K

20 for every v V. Now we are free to coose Q = {ξ Q : ξ K P k (K) K C }. Ten te formulation simplifies to: find u V suc tat ( u, v ) + α K Ω\Ω c K Ω c (u, v ) K + K( u, v ) K = (f, v ) Ω\Ω c + K Ω c K Ω c (g, v ) K + α 1 ( u, v ) K + α 1 K Ω c K Ω c K (g, v ) K + α K (u, v ) K K Ω\Ω c K(f, v ) K, olds for every v V. Note tat te only ting tat now remains of te discrete Lagrange multiplier is te rule to determine te contact region, i.e. te elements K for wic formula (6.1) yields a positive value for λ. Tis motivates te formulation of Nitsce s metod in te general case were te contact region is arbitrary. Given v V and te local mes lengts K, we define te L (Ω) functions and v by (6.) K = K, and v K = v K, K C, respectively. Te discrete contact force is ten defined as (6.3) λ (x, y) = max{0, ( (α ) 1 (g u ) f u ) (x, y) } and te contact region is (6.4) Ω c = { (x, y) Ω λ (x, y) > 0 }. Te Nitsce s metod. Find u V and Ω c = Ωc (u ), suc tat ( u, v ) + (u, v ) Ω c + ( u, v ) Ω c + α 1 ( u, v ) Ω c α( u, v ) Ω\Ω c = (f, v ) Ω\Ω c + (g, v ) Ω c + α 1 ( g, v ) Ω c + α( f, v ) Ω\Ω c, olds for every v V. Te iteration in Algoritm corresponds now to solving te problem by updating te contact force troug (6.5) λ k+1 (x, y) = max{0, ( (α ) 1 (g u k ) f u k ) (x, y) } and computing te contact area from (6.6) Ω c,k+1 = { (x, y) Ω λ k+1 (x, y) > 0 }, wit te stopping criterion (6.7) λ k λ k 1 1, T OL. 0

21 For te lowest order metod, wit (6.8) V = {v V : v K P 1 (K) K C }, te Nitsce s metod reduces to (6.9) ( u, v ) + α 1 ( u, v ) Ω c = (f, v ) Ω\Ω c + α 1 ( g, v ) Ω c, wic (except for (f, v ) Ω\Ω c instead of (f, v ) Ω ) is te standard penalty formulation, cf. Scoltz [41]. Note tat te a posteriori estimate of Teorem 4.3 still olds wen te reaction force is computed from (6.3). Our conclusion is tat te stabilized metod can be implemented in a straigtforward way using te above Nitsce s formulation. In practice, one can replace f and g in (6.5) wit teir interpolants in V. 7. Numerical results. Let Ω = {(x, y) R : x + y < 4} and consider problem (.4) wit (7.1) f(x, y) = 1, { 1 r if r < 0.9, g(r) = c 1 r + c oterwise, were r = x + y is te distance from te origin and c 1, c are cosen suc tat te obstacle is C 1 in te wole domain. Te radial symmetry reduces (.4) to te ordinary differential equation (7.) u r + 1 u r r = 1, a < r <, u() = 0, u(a) = g(a), u (a) = g (a), were te unknowns are te function u = u(r) and te radius a of te contact area. Evidently λ = 0 for r > a, and wen r < a te solution (u, λ) satisfies (7.3) u = g, and λ = 1 g. Solving (7.) leads to u(r) = ( 1 + a log r ) ( r 4 1 ) + g (r)a log r, and a Te solution u as a step discontinuity in te second derivative in radial direction. Hence, it is globally only in H 5/ ε (Ω), ε > 0, but smoot in bot te contact subregion and its complement. First, te prescribed problem is solved by mixed and stabilized metods using two mes families: one tat follows te boundary of te true contact region (a conforming family of meses) and an arbitrary mes (nonconforming mes) see Fig. 1 for examples of te two different types of meses. In Fig., te analytical solution is compared against te discrete solutions obtained by te P P 0 and P 1 P 0 stabilized metods. Note tat te P 1 P 0 metod does not (even for te conforming mes) yield a reaction force converging in L. For te displacement we only give one picture for eac mes type since bot metods give similar results. Te mes is refined uniformly and te errors of te displacement u in te H 1 - norm and of te Lagrange multiplier λ in te discrete H 1 -norm are computed and 1

22 tabulated. Te resulting convergence curves are visualized in Fig. 3 as a function of te mes parameter = max K C K. Te parameter p stands for te rate of convergence O( p ). Te stabilization parameters were cosen troug trial-and-error as α = 0.1 and α = 0.01 for te P P 0 and P 1 P 0 stabilized metods, respectively. Te numerical example reveals tat te limited regularity of te solution due to te unknown contact boundary limits te convergence rate to O( 3/ ) and tat te H 1 -error of te lowest order metods are not affected by it. Wen comparing te convergence rates of te Lagrange multiplier it can be seen tat te lowest order mixed metod does not initially perform as well as te lowest order stabilized metod. Troug local elimination of te bubble functions (cf. Remark 4.4 above) tis can be traced to a smaller effective stabilization parameter causing a larger constant in te a priori estimate. If te stabilization parameter of P 1 P 0 metod is furter decreased, te performance of te two metods will be identical. It is furter investigated weter te limited convergence rate due to te unknown contact boundary can be improved by an adaptive refinement strategy. Based on te a posteriori estimate of te stabilized metod we define an elementwise error estimator as follows: E K (u, λ ) = K u + λ + f 0,K + 1 K u n 0, K + (g u ) + 1,K + (u g) + λ dx. Refining te triangles wit 90% of te total error we create an improved sequence of meses. See Fig. 4 for examples of te resulting meses. We repeatedly adaptively refine te mes and compute te solution and error of P P 0 stabilized metod. Te resulting convergence rates wit respect to te number of degrees of freedom are given in Fig. 5. Note tat te for a uniform mes te relationsip between te number of degrees of freedom and te mes parameter is N. Hence, we see tat by te adaptivity we regain te optimal rate of convergence wit respect to te degrees of freedom for te P P 0 metods. 8. Summary. We ave introduced families of bubble-enriced mixed and residualbased stabilized finite element metods for discretizing te Lagrange multiplier formulation of te obstacle problem. We ave sown tat all metods yield stable approximations and proven te respective a priori and a posteriori error estimates. Te lowest order metods ave been tested numerically against an analytical solution and sown to lead to convergent solution strategies wit optimal convergent rates. Acknowledgements. Funding from Tekes te Finnis Funding Agency for Innovation (Decision number 3305/31/015) and te Finnis Cultural Foundation is gratefully acknowledged. K REFERENCES [1] I. Babuška, Error-bounds for finite element metod, Numer. Mat., 16 (1970/1971), pp [], Te finite element metod wit Lagrangian multipliers, Numer. Mat., 0 (1973), pp [3] L. Banz and A. Scröder, Biortogonal basis functions in p-adaptive FEM for elliptic obstacle problems, Computers and Matematics wit Applications, 70 (015), pp

23 Fig. 1. Te two different mes families: first conforming to te true contact boundary (upper panel), te oter a general nonconforming family (lower panel). [4] L. Banz and E. P. Stepan, A posteriori error estimates of p-adaptive IPDG-FEM for elliptic obstacle problems, Appl. Numer. Mat., 76 (014), pp [5] M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (005), pp [6] D. Braess, A posteriori error estimators for obstacle problems anoter look, Numer. Mat., 101 (005), pp [7], Finite elements, Cambridge University Press, Cambridge, tird ed., 007. Teory, fast solvers, and applications in elasticity teory, Translated from te German by Larry L. Scumaker. [8] H. R. Brezis and G. Stampaccia, Sur la régularité de la solution d inéquations elliptiques, Bull. Soc. Mat. France, 96 (1968), pp [9] F. Brezzi, On te existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recerce Opérationnelle Sér. Rouge, 8 (1974), pp [10] F. Brezzi, W. W. Hager, and P.-A. Raviart, Error estimates for te finite element solution of variational inequalities, Numer. Mat., 8 (1977), pp [11], Error estimates for te finite element solution of variational inequalities. II. Mixed metods, Numer. Mat., 31 (1978/79), pp [1] F. Brezzi and J. Pitkäranta, On te stabilization of finite element approximations of te Stokes equations, in Efficient solutions of elliptic systems (Kiel, 1984), vol. 10 of Notes Numer. Fluid Mec., Friedr. Vieweg, Braunscweig, 1984, pp [13] M. Bürg and A. Scröder, A posteriori error control of p-finite elements for variational inequalities of te first and second kind, Comput. Mat. Appl., 70 (015), pp

24 Fig.. A comparison of analytic (upmost panel) and discrete (lower panels) solutions. Te discrete solution was computed using nonconforming (left panel) and conforming (rigt panel) meses and stabilized P P0 (tird row) and P1 P0 (fourt row) metods. 4

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26 EK(u,λ) u u 1,K +K λ λ 0,K EK(u,λ) u u 1,K +K λ λ 0,K EK(u,λ) u u 1,K +K λ λ 0,K Fig. 4. Tree meses arising from te adaptive refinement strategy. Te local error estimators (left panels) are compared to te true local error (rigt panels). Apl. Mat., 6 (1981), pp [8] J. Haslinger, I. Hlaváček, and J. Nečas, Numerical metods for unilateral problems in solid mecanics, in Handbook of numerical analysis, Vol. IV, Handb. Numer. Anal., IV, Nort-Holland, Amsterdam, 1996, pp [9] P. Hild and Y. Renard, A stabilized Lagrange multiplier metod for te finite element approximation of contact problems in elastostatics, Numer. Mat., 115 (010), pp

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