and it has also been combined with adaptive grid generation ë20ë and domain decomposition ë21ë. The functions a i : æærær 2!R, in è1.1è are assumed to

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1 Mixed Finite Element Method for onlinear Elliptic Problems: the h-p -Version Miyoung Lee and Fabio A. Milner Department of Mathematics, Purdue University West Lafayette, I , U.S.A. Abstract. Mixed ænite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed. Existence and uniqueness of the approximate solution are demonstrated using a æxed point argument. Convergence and stability of the method are proved both with respect to mesh reænement and increase of the degree of the approximating polynomials. The analysis is carried out in detail using Raviart-Thomas-edelec spaces as an example. umerical results for minimal surface problems are obtained using Brezzi-Douglas-Marini spaces. Graphs of the approximate solutions are presented for two different problems. Key words. nonlinear elliptic problem, mixed method, minimal surfaces, B.D.M space. AMS subject classiæcation. 6512, 6515, 6522, 6530, 35J Introduction We shall consider the numerical solution of the following nonlinear Dirichlet problem, which appears in many engineering applications: è1.1è, P 2 i èx;u;ruè+a 0 èx;u;ruè = 0; x 2 æ; uèxè =,gèxè; x where æ is a rectangular domain in R 2 with The combination of the h and p versions of the ænite element method is in widespread use, for example, in structural mechanics and æuid dynamics, 1

2 and it has also been combined with adaptive grid generation ë20ë and domain decomposition ë21ë. The functions a i : æærær 2!R, in è1.1è are assumed to be as regular as needed for our arguments. We shall also assume that the quasilinear operator associated with è1.1è is elliptic in the following sense. Let èx;u;zè and æèx;u;zè denote, respectively, the minimum and maximum eigenvalues of the matrix A = ; which we shall assume to be j ii;j=1;2 Then, for all 2 R 2, 6= 0, and for all èx;u;zè 2 æ ærær 2, we have 0 éèx;u;zèkk 2 T A æèx;u;zèkk 2 : The variable x will normally be omitted in this notation. For 1 qé1 and k any nonnegative integer, let W k;q èæè = f 2 L q èæè : D æ 2 L q èæè if jæjkg denote the standard Sobolev space endowed with its usual norm kæk k;q;æ, with the obvious modiæcation in the case q = 1. Let H k èæè = W k;2 èæè denote L 2 -based Sobolev spaces with the norm kæk k = kæk k;2. In particular, the notation kæk 0 will mean kæk L 2 èæè or kæk L 2 èæè2. For 0 s é 1, let W s;q èæè, W s;q è@æè, H s èæè and H s è@æè denote the fractional order Sobolev spaces endowed with the norms kæk s;q;æ, kæk s;q;@æ, kæk s;æ and kæk s;@æ. The subscript æ in the norm will be omitted. Let be normed by and let V = Hèdiv;æè=fv 2 H 2 èæè 2 : div v 2 L 2 èæèg kvk V = kvk 0 + k div vk 0 ; W = L 2 èæè: The mixed ænite element method ë5ë approximates at the same time the solution of è1.1è, u, and the æux è1.2è =,aèu; ruè =,èa 1 èu; ruè;a 2 èu; ruèè; which is frequently the quantity that needs to be found in the problem at hand representing, for example, a velocity æeld or a stress tensor. By the implicit function theorem è1.2è can be inverted to obtain ru as a function of u and, say ru =,bèu; è: 2

3 We now set fèu; è =,a 0 èu; ruè =,a 0 èu;,bèu; èè in è1.1è. Then, the mixed weak form of è1.1è we shall consider consists of ænding è;uè 2 V æ W such that è1.3è èbèu; è;vè, èdiv v; uè =é g;væ né 8v 2 V; èdiv ;wè =èfèu; è;wè 8w 2 W; where n denotes the unit outer normal vector Let us consider a quasi-uniform family ft g of decompositions of æ by rectangles E èwith boundary elements allowed to have one curved edgeè. For each decomposition, let p 2 0 be the degree of the approximating piecewise polynomials used for this decomposition. ext, let V æ W V æ W be the Raviart-Thomas-edelec space of index p 0 associated with this decomposition ë11, 19, 23ë. The mixed ænite element method is a discrete form of è1.3è and consists of ænding è ;u è 2 V æ W such that è1.4è èbèu ; è;vè, èdiv v; u è = ég;væné 8v 2 V ; èdiv ;wè = èfèu ; è;wè 8w 2 W : We shall use an L 2 -projection onto W, P : L 2! W, given by è1.5è èp w, w; è =0; 2 W ;w 2 W: We shall also use the R-T projection of V onto V, : V! V ë23ë, which is locally deæned èon every element Eè èsee ë23ëè. Lemma 1.1 Let : V! V be the Raviart-Thomas projection. Let s 2 and réf1=2; 3=2, 3=sg. Then, for v 2 èh r èæèè 2, è1.6è k v, vk 0;s Ch minfp +1;rg,1+2=s p 5=2,r,4=s kvk r ; where C is a constant independent of h, p, and s but depends on r. Proof The result follows from a simple argument that combines known estimates on each element which are summed over all elements and combined with Corollary 2.1 in ë18ë and Lemma 3.2 in ë24ë. 3

4 Lemma 1.2 Let P be L 2 -projection onto W deæned by è1.5è and let = min fp +1;rg, s 2, m 3=2, 3=s. Then, for w 2 H r èæè, è1.7è kw, P wk 0;s Ch 2=s,1+ p,r+3=2,3=s kwk; where C is a constant independent of h, p, and w. Proof The proof is similar to that of Lemma 1.1, using è1.7è in ë18ë and Lemma 4 in ë3ë. We will also use the following inverse-type inequalities. Lemma 1.3 Let 2 L s èæè T W èor 2 L s èæè 2 T V è, 1 r s 1. Then, è1.8è Proof in ë22ë. kk 0;s Qh 2=s,2=r p 4=r,4=s kk 0;r : The proof follows easily, using inequalities è1.9è in ë18ë and è4.6è The theoretical results in this paper are essentially a combination of those in ë15ë and ë22ë. However, it is possible to choose h asymptotically bounded by some power of p in such a way that the regularity required of the solution of è1.1è is less than needed for the p-version of the method. We shall indicate this reduction along the way. The second half of the paper will be devoted to addressing some of the programming problems associated with the implementation of the method and presenting results from simulations for minimal surfaces. In section 2 we shall linearize the system è1.1è and analize the linearization. In section 3 we will show that è1.4è is uniquely solvable by using a æxed point argument, and that its solution è ;u è converges to è;uè in V T L 4 èæè 2 æ L 4 èæè. Then in section 4 we derive error estimates in L 2 for the approximation. The last four sections will contain, respectively, the description of a ewton iteration algorithm which may be used to handle the nonlinearities of the problem, programming techniques for BDM mixed ænite elements, numerical results obtained for minimal surface problems, and some concluding remarks. 4

5 2. Solvability of the Linearized Problem Following ë15ë and ë22ë, for 2 W, 2 V weintroduce ærst and second order Taylor expansions, è2.1è fè; è, fèu; è =,f u èu; èèu, è, f èu; èè, è+ ~ Qf èu, ;, è =, ~ f u è; èèu, è, ~ f è; èè, è: Similarly, for b =èb 1 ;b 2 è, bè; è, bèu; è =,b u èu; èèu, è, b èu; èè, è+ Qb ~ èu, ;, è =, ~ b u è; èèu, è, ~ b è; èèu, èè, è: We obtain our ærst error equations by subtracting è1.4è from è1.3è: è2.2è èbèu; è, bèu ; è;vè, èdiv v; u, u è=0 8v 2 V ; èdiv è, è;wè=èfèu; è, fèu ; è;wè 8w 2 W : Recall, ë11ë, that div æ = P æ div : H 1 èæè 2! W : Combining è2.1è-è2.2è with = u and =, we obtain the following form of the error equations, which we will need for our æxed point argument: èbèu; èë, ë;vè, èdiv v; P u, u è+è, 1 ëp u, u ë;vè =èbèu; èë, ë+, 1 ëp u, uë+ ~ Qb èu, u ;, è;vè 8v 2 V ; èdiv è, è;wè, è, 2 ë, ë;wè, èæëp u, u ë;wè =è,, 2 ë, ë, æëp u, uë, ~ Qf èu, u ;, è;wè 8w 2 W : Here we have set, just as in ë15ë and in ë22ë, Bèu; è =b èu; è =A,1 èu; è,, 1 = b u èu; è,, 2 = f èu; è, and æ = f u èu; è. ext we deæne, just as in ë15ë and ë22ë, M : H 2 èæè! L 2 èæè by è2.3è Mw =,div èaèu; èrw + Aèu; è, 1 wè+aèu; è, 2 ærw,èæ,, T 2 Aèu; è, 1èw and its formal adjoint M æ by M æ =,div èaèu; èr + Aèu; è, 2 è+aèu; è, 1 ær,èæ,, T 2 Aèu; è, 1è: 5

6 From ë11, 15ë we know that the restrictions of the operator M and M æ to H 2 èæè T H 1 0 èæè have bounded inverses, if we assume that èu; è can be extended to a pair è~u; ~è deæned on a domain æ 0 with a C 2 -boundary, such that æ æ 0 and measèæ 0, æè is arbitrary small ë1, 14ë. Then, for any è 2 L 2 èæè there is a unique 2 H 2 èæè T H 1 0èæè such that M = è èrespectively, M æ = èè and kk 2 Ckèk 0 if we assume that, for example, the zero order term of M æ is nonnegative, that is è2.4è æ, T 2 Aèu; è, 1, div èaèu; è, 2 è; where u 2 C 0;1 èæè and 2 C 0;1 èæè 2 ë12, 22ë. In order to be able to employ our duality arguments, we shall assume the structure condition è2.4è. ote that this condition is reduced to æ 0if, 2 =0. Let æ : V æ W! V æ W be given by æè; è =ès; qè where ès; qè is the solution of the following linear system: èbèu; èë, së;vè, èdiv v; P u, qè+è, 1 ëp u, që;vè =èbèu; èë, ë+, 1 ëp u, uë+ ~ Qb èu, ;, è;vè 8v 2 V ; èdiv è, sè;wè, è, 2 ë, së;wè, èæëp u, që;wè =è,, 2 ë, ë, æëp u, uë, ~ Qf èu, ;, è;wè 8w 2 W : ote that the left hand side of this system corresponds to the mixed method for the operator M given by è2.3è. We will show next that this system is uniquely solvable, so that the map æ is well-deæned. Thus, the problem we want to study is that of ænding èy; qè 2 W æ V such that, for a pair of given l 2 L 2 èæè 2 and m 2 L 2 èæè, è2.5è èbq;vè, èdiv v; yè+è, 1 y; vè = èl; vè 8v 2 V ; èdiv q; wè, è, 2 q; wè, èæy;wè = èm; wè 8w 2 W : Lemma 2.1 Let q 2 V, l 2 L 2 èæè 2, and m 2 L 2 èæè. If y 2 W satisæes the relation è2.5è, then for suæciently large p or small h, kyk 0 Q hh p,1=2 kqk 0 + h 2 p,2 kdiv qk 0 + klk 0 + kmk 0 i : Proof The proof follows exactly as in ë15, 22ë using Lemmas Lemma 2.2 If q 2 V satisæes è2.5è, then kqk 0 + kdiv qk 0 C ëkyk 0 + klk 0 + kmk 0 ë : 6

7 Proof To bound kqk 0, choose v = q, w = y in è2.5è and add the resulting equations. The choice w = div q in è2.5è gives the bound for kdiv qk 0. Lemma 2.3 There exists one and only one solution of the system è2.5è. Proof Existence follows from uniqueness since the system is linear. Assume l = 0;m = 0. Then Lemma 2.1 implies kyk 0 Qh p,1=2 kqk V ; where kqk V = kqk 0 + kdiv qk 0. By Lemma 2.2, we have kqk V ckyk 0 èkyk 0 Qh p,1=2 kqk V Qh p,1=2 kyk 0 ; which implies y = 0 for large p, or small h. Then, q = 0 as needed. ow it is clear from Lemma 2.3 that the functional æ is well-deæned. 3. Existence and Uniqueness The solvability of è1.4è is now equivalenttoshowing that æ has a æxed point. We state this result in the following theorem. Theorem 3.1 For p suæciently large or h suæciently small, æ has a æxed point. In order to prove this, we shall need the following duality lemma. Lemma 3.1 Let! 2 V, l 2 L 2 èæè 2, and m 2 L 2 èæè. If 2 W satisæes the relation è3.1è èbèu; è!; vè, èdiv v; è+è, 1 ;vè =èl; vè 8v 2 V ; èdiv!; wè, è, 2!; wè, èæ;wè =èm; wè 8w 2 W ; then, for any, 2 4, ", there exists a positive constant C = Cè; u; ;, 1 ;, 2 ;æ;æ;"è, independent of p and h, such that kk 0: Cëh 2= p1=2,2= k!k 0 + h 2=+1 p,1,2= kdiv!k 0 + klk 0 + kmk 0 : ow let V = V with the stronger norm kvk V = kvk 0;4 + kdiv vk 0, and let W = W with the stronger norm kwk W = kwk 0;4. It follows from the Brouwer æxed point theorem that Theorem 3.1 is true if we can show the following. 7

8 Theorem 3.2 For æ é 0 suæciently small èdependent on p and h è, æ maps the ball of radius æ of V æw ; centered at è ;P uè, into itself. Proof Let k,qk V æ and kp u,yk W æ. We apply Lemma 3.1 with = P u, y,! =, q, and l = Bèu; èë, ë+, 1 ëp u, uë+ ~ Qb èu, ;, è m =,, 2 ë, ë, æëp u, uë, ~ Qf èu, ;, è: Using Lemmas and the relation 2ab a 2 + b 2, it follows that for suæciently large p or suæciently small h, è3.2è kp u, yk 0 ëh p,1=2 k, qk 0 +h 2 p,2 kdiv è, qèk 0 + klk 0 + kmk 0 ë Cëæ 2 + p 1=2,r ë where C = Cèkk r ; kuk m ;, 1 ; ~ Qb ; ~ Qf è. ext, by applying Lemma 2.2 to è3.1è, we have k, qk 0 + kdiv è, qèk CëkP u, yk 0 + klk 0 + kmk 0 ë Cëh r p 1 2,r + æ2 ë: è3.3è Also, using è1.8è we see that k, qk V Ch,1=2 p ëh r p1=2,r while è3.2è and è3.3è imply that kp u, yk W Ch,1=2 Hence, + æ 2 ë, p ëæ 2 + h r p1=2,r ë. kp u, yk W + k, qk V 2Ch,1=2 p ëæ 2 + h r p1=2,r ë; where C = Cèkk r ; kuk m ;, 1 ; Qb ~ ; Qf ~ è. Finally, we want to choose p ;h, and æ so that I =ë4ch r,1=2 p 3=2,r, ë is not empty. This requires that r é 5=2 èor r = 5=2 and h 1 4C h1=2 p,1 suæciently smallè, a severe regularity constraint. Then, for any æ 2 I, we have kp u, yk W æ and k, qk V æ as needed. However, given "é0 small, if we let k =3=è2"è, 1+" and impose the constraint h ép,k on the mesh sizes of the decompositions of æ, then for p large enough that p èk+1èr,k, C 2 and for r 1+", the interval I is not empty. Thus we see that existence of solutions is really guaranteed as 8

9 long as ré1. ext we shall prove a uniqueness result which holds provided that the coeæcients a i, i =0; 1; 2 of è1.1è are three times continuously diæerentiable. Theorem 3.3 If p is suæciently large or h suæciently small, there isa unique solution of è1.4è near the solution fu; g of è1.3è. Proof Let è i ;u i è 2V æw, i =1; 2 be solutions of è1.4è and let U = u 1, u 2 ; æ = 1, 2 ; i =, i ; i = u, u i ; i =1; 2: Rewrite è2.2è as èbèu; è, bèu 2 ; 2 è;vè, èdiv v; Uè =èbèu; è, bèu 1 ; 1 è;vè; 8v 2V ; Then, using è2.1è we obtain èdivæ;wè=èfèu 1 ; 1 è, fèu 2 ; 2 è;wè; 8w 2W : èbèu; èæ;vè,èdiv v; Uè+è, 1 U; vè =è~ Qb è 2 ; 2 è, ~ Qb è 1 ; 1 è;vè; 8v 2V ; èdiv æ;wè,è, 2 æ;wè,èæu;wè =è~ Qf è 2 ; 2 è, ~ Qf è 1 ; 1 è;wè; 8w 2W ; and then it follows from Lemma 2.2 that kæk 0 + kdiv æk 0 CëkUk 0 + k ~ Qb è 1 ; 1 è, ~ Qb è 2 ; 2 èk 0 +k ~ Qf è 1 ; 1 è, ~ Qf è 2 ; 2 èk 0 ë: Also, by Lemma 2.1, we see that kuk 0 Qfh p,1=2 kæk 0 + h 2 p,2 kdiv æk 0 + k ~ Qb è 1 ; 1 è, ~ Qb è 2 ; 2 èk 0 +k ~ Qf è 1 ; 1 è, ~ Qf è 1 ; 1 èk 0 g: ext we want to show that k Qb ~ è 1 ; 1 è, Qb ~ è 2 ; 2 èk 0 Ch 2r,1 p5,2rèkæk 0 + kuk 0 è; k Qf ~ è 1 ; 1 è, Qf ~ è 2 ; 2 èk 0 Ch 2r,1 p5,2rèkæk 0 + kuk 0 è; so that, if ré5=2, or r =5=2 and h is suæciently small, or r 1+" with the additional condition h ép 1,3=2"," given in the proof of Theorem 3.2, we will have è3.4è k ~ Qb è 1 ; 1 è, ~ Qb è 2 ; 2 èk 0 Cp,"2 èkæk 0 + kuk 0 è; k ~ Qf è 1 ; 1 è, ~ Qf è 2 ; 2 èk 0 Cp,"2 èkæk 0 + kuk 0 è: 9

10 To show è3.4è we use the mean value theorem on the quadratic forms ~ Qb and ~ Qf, just as in ë15, 22ë. It follows that, for example, k Qf ~ è 2 ; 2 è, Qf ~ è 1 ; 1 èk 0 Cëk 1 k 0;1 + k 2 k 0;1 + k 1 k 2 0;1 +k 1 k 0;1 + k 1 k 0;1 k 1 k 0;1 + k 2 k 0;1 + k 2 k 2 0;1ëëkæk 0 + kuk 0 ë: è3.5è ote that, from the inverse estimates è1.8è, it follows that kp u, u k 0;1 p kp u, u k 0;4 p h,1=2 k, k 0;1 p k, k 0;4 p h,1=2 So, we have æ Khr,1 p1=2,r ; æ Khr,1 p1=2,r : k i k 0;1 = ku i, u i k 0;1 ku i, P u i k 0;1 + kp u i, u i k 0;1 K h p,m+3=2 kuk m + p,"2 ; è3.6è k i k 0;1 = k i, i k 0;1 k i, i k 0;1 + k i, i k 0;1 K h p 5=2,r kk r + p,"2 ; and using è3.6è in è3.5è we obtain è3.4è: k ~ Qb è 1 ; 1 è, ~ Qb è 2 ; 2 èk 0 Kh " p3=2," ëkæk 0 + kuk 0 ë; k ~ Qf è 1 ; 1 è, ~ Qf è 2 ; 2 èk 0 Kh " p 3=2," ëkæk 0 + kuk 0 ë: By Lemmas 2.1 and 2.2, this concludes the proof of the theorem, provided ré5=2, or r =5=2 and h is suæciently small, or ré1 and h = Oèp,k è as in Theorem L 2 -Error Estimates In this section we shall bound the error of the solution of the h-píversion for problem è1.1è. In order to simplify the notation, from now on we shall omit the subindex on the parameters h and p. Theorem 4.1 Assume that the coeæcients of è1.1è are smooth enough that u 2 H r+1 èæè and 2 H r èæè 2, where r é 5=2, or r = 5=2 and h is sufæciently small, or r é 1 and h = Oèp,k è as in Theorem 3.2. Then, for suæciently large p or suæciently small h, the following estimates hold. 10

11 ku, u k 0 Qfh r p 1=2,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 +1èg; k, k 0 Qfh r p 1=2,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 +1èg; kdiv è, èk 0 Qfh r p 1,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 +1èg; Proof Let =,, = u,u, =,, and = P u,u. ext, rewrite è2.2è in the form èbèu; è; vè, èdiv v; è+è, 1 ;vè =èbèu; èë, ë+, 1 ëp u, uë+ ~ Qb è;è;vè 8v 2 V ; èdiv ; wè, è, 2 ; wè, èæ;wè =è,, 2 ë, ë, æëp u, uë, ~ Qf è;è;wè 8w 2 W : Then, just as in Theorem 5.1 in ë15ë, we can see that kk 0 + kdiv k 0 Cëkk 0 + kk 2 0;4 + kk 2 0;4 + k, k 2 0;4 +k, k 0 + kp u, uk 2 0;4 + kp u, uk 0 ë Cëkk 0 + h,1=2 pkk 0;4 kk 0;2 + kk 0;4 h,1=2 pkk 0 +k, k 2 0;4 + k, k 0 + kp u, uk 2 0;4 + kp u, uk 0 ë Cëkk 0 + h r,1 p 5=2,r kk 0;2 + h r,1 p 5=2,r kk 0 +k, k 2 0;4 + k, k 0 + kp u, uk 2 0;4 + kp u, uk 0 ë Cëkk 0 + h r,1 p 5=2,r kk 0 + h r,1 p 5=2,r kk 0 +h r p 1 2,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 + 1èë: For p large enough or h small enough, we obtain the bound è4.1è kk 0 + kdiv k 0 Cëkk 0 + h r p 1=2,r kk r èkk r +1è +h r+1 p,r,1 kuk r+1 èkuk r+1 + 1èë: Likewise, by Lemma 2.1, kk 0 Qëhp,1=2 kk 0 + h 2 p,2 kdiv k 0 + klk 0 + kmk 0 ë Qëhp,1=2 kk 0 + h 2 p,2 kdiv k 0 + h r,1 p 5=2,r kk 0 + h r,1 p 5=2,r kk 0 +h r p 1=2,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 + 1èë; 11

12 and if p is suæciently large, it follows that è4.2è kk 0 Qëhp,1=2 kk 0 + h 2 p,2 kdiv k 0 + h r,1 p 5=2,r èkk 0 + kk 0 è +h r p 1=2,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 +1èg: Substituting è4.2è into è4.1è, we ænd that è4.3è kk V Qëh r p 1=2,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 +1èg; and substituting è4.3è into è4.2è we arrive at the estimate è4.4è kk 0 Qfh r p 1=2,r kk r èkk r +1è+h r+1 p,r,1 kuk r+1 èkuk r+1 +1èg: By combining è4.3è and è4.4è with Lemmas 1.1 and 1.2 we have ku, u k 0 Qfh r p1=2,r kk r èkk r +1è+h m p,m kuk m èkuk m +1èg; k, k 0 Qfh r p 1=2,r kk r èkk r +1è+h m p,m kuk m èkuk m +1èg; kdiv è, èk 0 Qfh r p 1,r kk r èkk r +1è+h m p,m kuk m èkuk m +1èg; as needed. 5. ewton's Method Following ë22ë, we compute a ewton approximation of f ;u g using a sequence f m ;u m g 1 m=0 in V æ W satisfying the following relations: For v 2 V and w 2 W, èbèu m ; m èè m+1, m è;vè, èdiv v; u m+1 è +è, 1 èu m ; m èèu m+1, u m è;vè=è,bèu m ; m è;vè+ ég;væné; è5.1è èdiv m+1 ;wè, è, 2 èu m ; m èè m+1, m è;wè,èæèu m ; m èèu m+1, u m è;wè=èfèu m ; m è;wè; where Bèu m ; m è=b èu m ; m è,, 1 èu m ; m è=b u èu m ; m è,, 2 èu m ; m è= f èu m ; m è, and æèu m ; m è=f u èu m ; m è. We shall show that the algorithm è5.1è is well deæned and that it converges quadratically. To this end, we shall need a technical lemma. Let æ = sup kvk0;1 + kwk 0;1 kvk 0 + kwk 0 ote æ =Oèh,1 p 2 èby the inverse inequality è1.8è. : fv; wg 2V æ W,f0; 0g : 12

13 Lemma 5.1 Given 0 é 0, there exist positive constants p 0, h 0, æ 0, and C such that the following holds. If p 0 p, h h 0, and 2 L 1 èæè and 2 L 1 èæè 2 satisfy the following relations, kk 0;1 + kk 0;1 0 and èk, k 0 + k, uk V èæ æ 0 ; and if l æ 2 L 2 èæè 2 and m æ 2 L 2 èæè, then there exists a unique f æ ;u æ g 2 V æ W such that èbè; è æ ;vè, èdiv v; u æ è+è, 1 è; èu æ ;vè=èl æ ;vè 8v 2 V ; èdiv æ ;wè, è, 2 è; è æ æ ;wè, èæè; èu æ ;wè=èm æ ;wè 82W : è5.2è Furthermore, f æ ;u æ g satisæes the bound è5.3è ku æ k 0 + k æ k V Cëkl æ k 0 + km æ k 0 ë: Proof It suæces to show that è5.3è holds, since this will imply that the solution è5.2è is unique and hence it exists. First, rewrite è5.2è in the following form: For v 2 V and w 2 W, è5.4è èbèu; è æ ;vè, èdiv v; u æ è+è, 1 è; èu æ ;vè=èl æ ;vè +èèbèu; è, èbè; èè æ ;vè + èè, 1 èu; è,, 1 è; èèu æ ;vè; èdiv æ ;wè, è, 2 èu; è æ æ ;wè, èæèu; èu æ ;wè=èm æ ;wè +èè, 2 è; è,, 2 èu; èè æ æ ;wè+èèæè; è, æèu; èèu æ ;wè: Recall, Lemma 2.2, that è5.5è k æ k V Cëku æ k 0 + kl æ k 0 + km æ k 0 ë: Also, it follows from Lemma 2.1 that ku æ k 0 Cëhp,1=2 k æ k V + kbèu; è, Bè; èk 0 k æ k 0;1 +k, 1 èu; è,, 1 è; èk 0 ku æ k 0;1 + k, 2 èu; è,, 2 è; èk 0 k æ k 0;1 +kæèu; è, æè; èk 0 ku æ k 0;1 + kl æ k 0 + km æ k 0 ë C 1 ëhp,1=2 k æ k V +èk, k 0 + ku, k 0 è æ èku æ k 0 + k æ k 0 è+kl æ k 0 + km æ k 0 ë; which implies that ku æ k 0 C 2 ëèh p,1=2 + æ 0 èk æ k V + æ 0 ku æ k 0 + kl æ k 0 + km æ k 0 ë: 13

14 Let now æ 0 é 1 2C 2 è5.6è so that ku æ k 0 C 3 ëèh p,1=2 + æ 0 èk æ k V + kl æ k 0 + km æ k 0 ë; Then, substituting è5.6è into è5.5è, we see that, k æ k V C 4 ëèhp,1=2 + æ 0 èk æ k V + kl æ k 0 + km æ k 0 ë: ow, for hp,1=2 é 1 4C 4, and æ 0 é minf 1 4C 4 ; 1 2C 2 g; we have which implies, from è5.6è, k æ k V C 5 ëkl æ k 0 + km æ k 0 ë; ku æ k 0 C 6 ëkl æ k 0 + km æ k 0 ë; and the result follows. We can state now our result concerning the existence and convergence of the ewton iterates, which follows immediately from Lemma 5.1. Theorem 5.1 There exist positive constants p 0, h 0, æ 0, and ~ C, such that, if p 0 ép, héh 0 and æ ëku 0, u k V + k 0, k 0 ë æ 0 ; then f m ;u m g 1 m=0 is well deæned, and m = ku m, u k 0 + k m, k V a decreasing sequence satisfying is m+1 ~ Cæ 2 m : 6. Programming techniques with BDM Spaces We shall apply now the numerical methods described in Section 5 to approximate the solution of the minimal surface problem è6.1è,div è ru è1+jruj 2 è 1=2 è = 0 8x 2 æ; u =,g 8x The approximation we ænd is based on the BDM èbrezzi-douglas-mariniè spaces ë6ë which are based on polynomials of some æxed total degree, rather than on tensor products. Consequently, the local dimension of these spaces 14

15 is much smaller than those of the corresponding Raviart-Thomas-edelec spaces. We give now a brief description of the BDM spaces. Let p 1 and let T h = fe i g be such that the length of each side of E i is h. We base the ænite element space for the approximation of the scalar function u on polynomials of total degree not exceeding p, 1. Let W p,1 èe i è=p p,1 = 8 é é: X i+jp,1 9 é= c i;j x i y j ; c i;j 2R é; : The number of degrees of freedom for W p,1 èe i èispèp +1è=2. The ænite element space for the approximation of the vector function is based on polynomials of total degree p augmented by a space of polynomials of degree p + 1 of dimension two. Let V p èe i è=p p èe i è M Spanècurl x p+1 y;curl xy p+1 è; where P p èe i è=p p æp p. The number of degrees of freedom for V p èe i èis èp + 1èèp + 2è + 2. Hence, the total local number of degrees of freedom for the BDM space M p is 1:5p 2 +3:5p +4which is about half the size of the local number of degrees of freedom for the Raviart-Thomas-edelec space of the same index, 3p 2 +2p. ext, we deæne V = V p h = fv 2 Hèdiv ; æè : vj E i 2 V p èe i è;e i 2T h g W = W p,1 h = fw : wj Ei 2 W p,1 èe i è;e i 2T h g M p h = V p h æw p,1 h ; and we seek è ;u è 2M p h, the solution of è1.4è. The analysis of this resulting mixed method is facilitated by the existence of the L 2 -orthogonal projection è1.5è, locally deæned on each element E i of the decomposition T h by the relations è6.2è èw, P p,1 h w; zè Ei =0; z 2P p,1 èe i è; E i 2T h : Q p Let h be the projection analogous to the Raviart-Thomas projection, locally deæned on each element E i of the decomposition T h by the following 15

16 degrees of freedom: With fe i jg 4 j=1 being the sides of the element E i and n i j the exterior unit normal vector to E i along e i j, for any q 2 V, Z w q, Y è p è p X è6.3è q æ n i e i h j dx =0; w 2 a s x s ; a s 2R ; 1 j 4; j s=0 q, ZEi Y p è6.4è q æ v dx dy =0; v 2 P p,2 èe i è: h Here we shall consider æ = ë0; 1ë 2. For the minimal surface problem we consider, the mixed ænite element formulation è1.4è is simply èbèu ; è;vè, èdiv v; u è = ég;væ n é 8v 2 V ; èdiv v; wè = 0 8w 2 W : We use the iterative algorithm which was described in Section 5 in order to solve this nonlinear algebraic system. Since in this case, 1 =0,, 2 = 0, and æ = 0 in è5.1è, then we have the following simpler iteration algorithm: è6.5è èbèu n ; n èè n+1, n è;vè, èdiv v; u n+1 è =,èbèu n ; n è;vè+ ég;væ n é; v 2 V ; èdiv n+1 ;wè=0; w 2 W ; where èsee ë17ëè and Then, it follows that è6.6è Bèu n ; n è= bèu; è =,ru = Bèu; è = p 1,jj è! 1 1, è n 2 è 2 1 nn 2 è1,j n j 2 è 3=2 1 nn 2 1, è1 n è 2 : ow, let f i g m 1 i=1 and f jg m 2 j=1 be, respectively, bases of V and W. Assume that i and j are supported in E 2T h. We then have the following iteration algorithm: For any initial guess f 0 ;u 0 g2v æ W, è : è6.7è èbèu m ; m è m+1 ; i è, èdiv i ;u m+1 è=èbèu m ; m è m ; i è,èbèu m ; m è; i è+ ég; i æ n E é; i 2 V ; èdiv m+1 ; i è=0; i 2 W 16

17 We obtain è m ;u m è recursively and, in each iteration, use it to compute the coeæcients for the next iteration. In order to compute è m ;u m èwe need to solve the following linear system of equations, using any direct or iterative method ë13ë: where Sx + Yy = f 1 ; Y T x = 0; ësë i;j =èbèu n ; n è j ; i è; Y i;j = èdiv i ; j è; u = Xm 2 j=1 y j j ; = x =èx 1 ;x 2 ; :::; x m1 è T ; and y =èy 1 ;y 2 ; :::; y m2 è T : ow we write this system in the form Xm 1 i=1 x i i ; è6.8è where M = " S Y Y T 0 è ; b = Mz = b " f1 0 è ; and z = " x y è : We shall show next how to select a local basis of V to avoid diæculties with the necessary continuity in the normal direction across inter-element boundaries when we extend the local basis to a global basis. At the same time, we shall obtain a sparse matrix associated with the system of equations è6.8è at each iteration step. Recall that p is the degree of the approximating piecewise polynomials in V. We nowchoose a basis on the reference element R =ë,1; 1ë 2 based on the following obvious decomposition. V èrè = Span èl i èxèl j èyè; 0è : 0 i + j p M Span è0;l i èxèl j èyèè : 0 i + j p M Z x Z y Span,curl y L p èè d; curl x,1,1 L p èè d ; where L s is the Legendre polynomial of degree s. Then, we see that a basis for V èrè can be chosen as B x 0 ë B x 1 ëæææëb x p ë B y 0 ëæææëb y p ; 17

18 where, for 0 k p, 2, Bk x = L i èxèl k èyè; 0 ; 0 i + k p, 2 ë è,1è p,k+1 L p,k èxèl k èyè+è,1è p,k L p,k,1 èxèl k èyè; 0 B x p,1 = B x p = ë L p,k èxèl k èyè+l p,k,1 èxèl k èyè; 0 ; L 1 èxèl p,1 èyè, L p,1 èyè; 0 èx, 1èL p èyè; ë 1 2p +1 ël p,1èyè, L p+1 èyèë ë èx +1èL p èyè; L 1 èxèl p,1 èyè+l p,1 èyè; 0 1 2p +1 ël p,1èyè, L p+1 èyèë We obtain B y k in an analogous way. Also, it is obvious that a local basis for W can be chosen as fl i èxèl j èyè; 0 i + j p, 1g: We introduce now the Lagrange interpolation polynomials based on Gauss-Lobatto points ë7ë. For 1 i k + 1, let where fx j g k+1 j=1 `ki èxè = k+1 Y j=1;j6=i èx, x j è èx i, x j è ; is the set of Gauss-Lobatto points of degree k on ë,1; 1ë, ordered as,1 =x 1 éx 2 é æææ éx k+1 =1: We know that, for 0 k p,1, SpanfBkg x = Span ; 1 j p, k +1 ; and `p,k j èxèl k èyè; 0 SpanfBpg x 1 = Span `1 1èxèL p èyè;, 2è2p +1è ël p,1èyè, L p+1 èyèë 1 `12èxèL p èyè; 2è2p +1è ël p,1èyè, L p+1 èyèë : Also, note that there exists a nonsingular èp, k +1èæ èp, k +1è matrix M k è2æ2 if k=pè such that M k l k =g k : ; ; 18

19 where, for 0 k p, 1, while l k =èl 1 ;l 2 ; æææ;l p,k+1 è T where B x k consists of fl 1 ;l 2 ; æææ;l p,k+1 g and, for 0 k p, 1, g k = l p =èl 1 ;l 2 è T where B x p consists of fl 1 ;l 2 g; ë`p,k 1 èxèl k èyè; 0ë; ë`p,k 2 èxèl k èyè; 0ë; æææ; ë`p,k k+1 èxèl kèyè; 0ë T while g p = `11 èxèl 1 pèyè;, 2è2p +1è ël p,1èyè, L p+1 èyèë ; 1 `12èxèL p èyè; 2è2p +1è ël p,1èyè, L p+1 èyèë T : We proceed analogously for B y k, 0 k p, and we can ænd matrices ^Mk such that ^M k^lk = ^h k ; where, for 0 k p, 1, while ^lk =èl 0 1;l 0 2; æææ;l 0 p,k+1è T where B y k consists of fl0 1;l 0 2; æææ;l 0 p,k+1g; and, for 0 k p, 1, ^h k = ^lp =èl 0 1;l 0 2è T where B x p consists of fl 0 1;l 0 2g; ë0;`p,k 1 èyèl k èxèë; ë0;`p,k 2 èyèl k èxèë; æææ; ë0;`p,k k+1 èyèl kèxèë T ; while ^h p =, 1 2è2p +1è ël p,1èxè, L p+1 èxèë;`1 1èyèL p èxè 1 2è2p +1è ël p,1èxè, L p+1 èxèë;`12èyèl p èxè 19 ; T :

20 Then we replace the system è6.8è by the following equivalent one: M 0... M p ^M0... ^Mp 30 7B 5@ l 0 æ l p ^l 0 æ ^l p 1 = C A 0 g 0 æ g p ^h 0 æ ^h p 1 : C A To compute ësë i;j =èbèu m ; m è j ; i è for the system è6.7è we use the Gauss- Lobatto quadrature of degree 5p: S i;j = 5p+1 X i=1 5p+1 X j=1 w i w j èbèu m èx i ;y j è; m èx i ;y j èè j èx i ;y j è; i èx i ;y j èè where w k, 1 k 5p +1, is the k-th Gauss-Lobatto weight of degree 5p. We also use the Gauss-Lobatto quadrature of degree 5p to calculate èbèu m ; m è;vè and ég;væn é. Finally,we use the Gauss-Lobatto quadrature of degree p for èdiv v; u m+1 è since èdiv vèu m+1 is a polynomial of degree at most 2p, 1 and the Gauss-Lobatto quadrature of degree p is exact up to this degree. 7. umerical Results ; In this section we shall present some results from numerical simulations of minimal surfaces modeled by è6.1è. The approximations were obtained both by reæning the mesh and by increasing the degree of the approximating polynomials. The boundary data was chosen as è7.1è gèx; yè = 8 é é: logëcosèy, :5èë, logëcosè,:5èë; 0 y 1; x =0; logëcosèy, :5èë, logëcosè:5èë; 0 y 1; x =1; logëcosè:5èë, logëcosèx, :5èë; 0 x 1; y =1; logëcosè,:5èë, logëcosèx, :5èë; 0 x 1; y =0; so that the exact solution for for this problem is known and it is given by uèx; yè = log cosèy, 0:5è cosèx, 0:5è ; 20

21 which belongs to C 1 èæè. The knowledge of the exact solution allows us to compute the actual errors of each approximation we ænd, and thus we are able to compare the eæective rates of convergence in h èfor a æxed pè and in p èfor a æxed hè, with the theoretically predicted ones, Theorem 4.1. We present in Figures 7.1 and 7.2 approximations obtained by mesh reænement using p = 2. Figure 7.1 depicts the approximations obtained for mesh sizes h = 1=2 ètopè and mesh size h = 1=4 èbottomè, while Figure 7.2 depicts the approximation obtained for mesh size h =1=8 ètopè and the exact solution èbottomè. 21

22 Figure 7.1: h=1è2 and h=1è4 22

23 Figure 7.2: h=1è8 and exact solution 23

24 We also presentintables 7.1 and 7.2 the errors in the approximation of u and both in L 2 and in L 1, in correspondence with the numerical solutions depicted in Figures 7.1 and 7.2 using the boundary condition given by è7.1è. These give an idea of the eæective performance of the method with respect to mesh reænement è"h-version"è. The error in L 2 was computed using numerical integration. The convergence rate indicated on the tables was calculated under the standard assumption that the errors = u, u and =, are of the form Kh!. Then! is found from two approximations computed for diæerent values of h. In this case wehave chosen h =2, and we show the convergence rates computed, respectively, as! = lnè0 = 2 è ;! = lnè0 = 2 è : 2ln2 2ln2 mesh size h lnècosèy, :5è= cosèx, :5èè ku, uk 1 ku, uk E-01 1è E E-01 1è E E-02 1è E E-03 convergence rate h 2 h 2 Table 7.1: Error in the scalar function, u, u èp =2è mesh size h lnècosèy, :5è= cosèx, :5èè k, k 1 k, k è E-01 1è E-01 1è E E-02 convergence rate h 1:2 h 1:6 Table 7.2: Error in the vector function,, èp =2è Tables 7.3 and 7.4 give, respectively, the errors in the numerical solutions obtained for a æxed mesh of size h = 1=2, and for diæerent degrees of the approximating polynomials. These give an idea of the eæective performance of the method with respect to the increase of the degree of the approximating polynomials è"p-version"è. 24

25 degree of polynomial lnècosèy, :5è= cosèx, :5èè ku, uk 1 ku, uk E E E E E E E E E E E E-13 convergence rate p,14 p,14 Table 7.3: Error in the scalar function, u, u èh =1=2è degree of polynomial lnècosèy, :5è= cosèx, :5èè k, k 1 k, k E E E E E E E E E E E-12 convergence rate p,14 p,14 Table 7.4: Error in the vector function,, èh =1=2è The convergence rate indicated on these tables was calculated under the standard assumption that the errors = u, u and =, are of the form Kp,è. Then è is found from two approximations computed for diæerent values of p. In this case we have chosen p 2 = 3 and p 6 = 15, and we show the convergence rates computed, respectively, as è = lnè2 = 6 è ; è = ln 5 lnè 2 = 6 è : ln 5 Finally, we present in Figure 7.3 the minimal surfaces corresponding to the boundary condition gèx; yè = 8 é é:,èsin 2yè=20; 0 y 1; x =0; èsin 2yè=20; 0 y 1; x =1;,èsin 2xè=20; 0 x 1; y =1; èsin 2xè=20; 0 x 1; y =0: 25

26 Figure 7.3: very æne mesh, h=1è20, and high polynomial degree, p=15 26

27 The exact solution for this problem is not known. The ærst picture in Figure 7.3 depicts the approximation computed using a very æne mesh, h = 1=20, and p = 2. The second picture depicts the approximation computed on a coarse mesh, h =1=2, with polynomials of high degree, p = Conclusions and Future Directions We have shown that the h and the p versions of the mixed ænite element method can be combined for the numerical solution of quite general nonlinear second order elliptic problems in divergence form. The resulting methods have much better convergence than the p version without mesh reænement, both in the regularity required of the exact solution and although we have not indicated them in the CPU times needed for the computations. The theoretical results have some serious restrictions. First, they require either a very regular solution, u 2 H 7=2+", or extremely æne meshes, h é p,k. Secondly, there is a requirement that the domain be rectangular in order to be able to use approximation estimates for the Raviart-Thomas projection in the p version of the mixed method. Such estimates only exist on domains which are unions of rectangles and there does not seem to exist an easy way to extend them to domains with curved boundaries. As it frequently happens, numerical simulations seem to indicate that these restrictions may be artiæcial, that is, due to the nature of the proof of these results. In fact, using fairly coarse meshes and polynomials of low degree, one can still see the numerical approximations converge. The computational results are quite satisfactory. As indicated by the ægures, the numerical approximations are quite good even with relatively coarse meshes and polynomials of low degree. Several extensions of this work should be pursued. It would be highly desirable to have error estimates available on domains with curved boundary, as well as on non-convex domains. It would also be very useful to reduce the regularity required of the solution of the diæerential problem without having to require extremely æne meshes. References ë1ë R. A. Adams, Sobolev Spaces. 1975, Academic Press, ew York- San Francisco-London. 27

28 ë2ë I. Babuska and B. Guo, The h-p version of the ænite element method for problems with nonhomogeneous essential boundary condition, Comput. Meth. Appl. Mech. Engin., 74 è1989è, pp ë3ë I. Babuska and M. Suri, The h, p version of the ænite element method with quasiuniform meshes. RAIRO Model. Math. Anal. umer., 21 è1987è pp ë4ë J. Bergh and J. Líofstríom, Interpolation Spaces: An Introduction. 1976, Springer-Verlag, Berlin and ew York. ë5ë F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO, Anal. umer., 2 è1974è, pp ë6ë F. Brezzi,J. Douglas, Jr. and L. D. Marini, Two families of mixed ænite elements for second order elliptic problems, umer. Math, 47 è1985è, pp ë7ë C. Canuto, M. Y. Hussaini, A. Quateroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, ew York, ë8ë C. Canuto and A. Quateroni, Approximation results for orthogonal polynomials in Sobolev spaces Math. Comp., 38 è1982è, pp ë9ë P.G. Ciarlet, The ænite element method for elliptic problems. 1978, orth-holland, Amsterdam. ë10ë J. Douglas, Jr., H 1 -Galerkin methods for a nonlinear Dirichlet problem, in Mathematical Aspects of Finite Element MethodsèHeidelbergè, vol. 606 of Lecture otes in Mathematics, Springer-Verlag, Heidelberg, ë11ë J. Douglas, Jr. and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comp., 44 è1985è, pp ë12ë D. Gilbarg and. S. Trudinger, Elliptic Partial Diæerential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, ë13ë G. Golub and C. V. Loan, Matrix Computation 2nd ed., The Johns Hopkins University Press, Baltimore and London,

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