Global and Localised A Posteriori Error Analysis in the maximum norm for finite element approximations of a convection-diffusion problem

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1 FINITE ELEMENT CENTER PREPRINT Global and Localised A Posteriori Error Analysis in the maximum norm for finite element approximations of a convection-diffusion problem Mats Boman Chalmers Finite Element Center CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg Sweden 001

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3 CHALMERS FINITE ELEMENT CENTER Preprint Global and Localised A Posteriori Error Analysis in the maximum norm for finite element approximations of a convection-diffusion problem Mats Boman Chalmers Finite Element Center Chalmers University of Technology SE Göteborg Sweden Göteborg, January 001

4 Global and Localised A Posteriori Error Analysis in the maximum norm for finite element approximations of a convection-diffusion problem Mats Boman NO ISSN Chalmers Finite Element Center Chalmers University of Technology SE Göteborg Sweden Telephone: +46(0) Fax:+46(0) Printed in Sweden Chalmers University of Technology Göteborg, Sweden 001

5 GLOBAL AND LOCALISED A POSTERIORI ERROR ANALYSIS IN THE MAXIMUM NORM FOR FINITE ELEMENT APPROXIMATIONS OF A CONVECTION-DIFFUSION PROBLEM MATS BOMAN Abstract. We analyse nite element approximations of a stationary convection-diusion problem. We prove global and localised a posteriori error estimates in the maximum norm. For the discretisation we use the Streamline Diusion method. 1. Introduction In this note we study nite element solution of the problem (1.1) u + ru? u = f in u = 0 where,, and f are given functions and is a positive number. We assume that?r c > 0, where c 1, and kk L 1() C. Further, R d, d = 1 3, is a bounded, convex and polyhedral domain. We prove a posteriori error estimates in the maximum norm for the error u? U, where U is an approximation obtained by a nite element method. When 1 we use a standard Galerkin method for the discretisation and our a posteriori error estimate takes the form: ku? Uk L 1 C(1 + j log h min j) kh r(u)k L 1 where h = h(x) is the mesh function, and r(u) is a computable realisation of the residual R(U) = f? U? ru + U H?1 : In the elliptic case, 1, our result is essentially included in [3], [15] and []. These works also consider the non convex case. However our analysis also gives results for small. In this case we use a Streamline Diusion method, see [4], and our result is of the form: ku? Uk L 1 C(1 + j log h min j) 3 h h 1 r(u) 1= 3= L 1: This result is probably not optimal in the dependence. We believe that the factor h?3= should be replaced by h?1, which is the result that we obtain in one dimension, d = 1. In one dimension, d = 1, we prove an estimate with this dependence Mathematics Subject Classication. 65N30, 35J5. Key words and phrases. convection-diusion, a posteriori error estimate, residual, pointwise, maximum norm, adaptive, nite element, duality argument, streamline diusion. 1

6 MATS BOMAN It is known that if is small, then the solution u is nonsmooth in certain regions, typically near characteristic layers and boundary layers. When used in an adaptive algorithm, a global error bound (as the above) would always require heavy renement near such singular layers, even if we are only interested in the behavior in regions where the solution is smooth. In order to avoid this we prove, in a special case, that the above result may be localised to special regions 0, oriented along the streamlines and avoiding singular layers. More precisely we consider the following equation with R : u + u x? u = f in (1.) u = 0 Our localised result, essentially, takes the form ' h k'(u? U)k L 1 C(1 + j log h min j) 3= h min 1 r(u) 1= 3= L 1 where the cut-o function ' is such that ' 1 in 0 and ' decays exponentially with s= p where s is the distance to 0. For this result to be useful it is important that the residual r(u) can be made small in 0 without resolving the the singular layers. The standard Galerkin nite element method does not have this property. However, a priori error analysis indicates that this is possible for the Streamline Diusion method. In [13] it is shown that in regions 0, of the same type as above, and where the solution u of (1.) is assumed to be smooth, we have ku? Uk L 1( 0 ) Ch 5=4 max. This result is improved in [14] to O(h 11=8 max). Further in [18] it is proved, again with the assumption that u is smooth in 0, that for certain triangulations ku? Uk L 1( 0 ) Ch max. Numerical experiments indicate that this is valid also for more general meshes. However there are also numerical examples, using certain meshes, in which one only gets the convergence rate O(h 3= max). This has been further discussed in [17]. An important feature of these works is that they do not require that the singular layers are resolved. In the case when is small our work is related to [4], where an a posteriori error estimate in the L -norm is proved. The localised result is inspired by [13]. We also recall that there is a localised a priori error estimate in L for the Streamline Diusion method, see [1]. As in [4], [7], our a posteriori error estimates are proved by using a duality argument involving a continuous linear adjoint problem. The analysis relies on the regularity of the solution G of the adjoint problem. Since we consider estimates in the maximum norm we use an L 1 -L 1 duality argument, where G acts as a regularised Green function, a technique introduced in [15]. This paper is organised as follows. In Section we introduce the space discretisation and formulate our nite element method. In Section 3 we state our global error estimates. The localised result is stated and proved in Section 10. In sections 4{9 we prove the results of Section 3. In Section 4 we prove a lemma which splits the estimate of the error into a small a priori part and a part which can be estimated by a posteriori quantities. This lemma makes it possible to replace the exact Green function by a regularised Green function. In Section

7 5 we introduce the adjoint problem and perform the duality argument, expressing the a posteriori part of the error in terms of the residual R(U) and the solution G of the adjoint problem. In Section 6 we estimate the residual in terms of computable quantities and derivatives of G. In Section 7 we analyse the regularity of G. In Section 8 we prove a sharper regularity estimate for G in the case d = 1. In Section 9 we conclude the proofs of the theorems stated in Section 3. Throughout this paper we use the standard Lebesgue spaces L p (!) for! with the convention that L p = L p (), and the corresponding Sobolev spaces W k p (!), W k p = W k p (), H k = W k and H 1 0 = fu H 1 : = 0g. Moreover (u v)! = R! uv dx and (u v) = (u v). We use the notation f! = sup x! f(x) and f! = inf x! f(x). Further, we write D k v(x) = qp jj=k jd v(x)j, so that the W k p seminorm may be conveniently written kd k vk Lp.. The discretisation In this section we formulate a discretisation of (1.1) using the Streamline Diusion method, see [4]. For the discretisation let F = ft g be a family of triangulations, where a triangulation T = fkg is a partition of into open simplices K which are face to face so that = [ KT K. Let h K = diam(k) and let K denote the radius of the largest closed ball contained in K. We assume that F is nondegenerate, i.e., we assume that there is a constant c 0 such that for all triangulations T F we have (.1) max KT h K K c 0 : To each triangulation T F we associate a positive, piecewise constant function h(x), dened on by hj K = h K 8K T : We also need a measure of the \regularity" of a triangulation. We therefore introduce the quantity = (T ) as follows. Let T F be a triangulation and K be a simplex in T. We dene the set (.) and (.3) S K = fk 0 T : K 0 \ K 6= g = max KT max j1? h K 0 K S 0=h K Kj: For each triangulation T F we have an associated function space V = V (T ), consisting of all continuous functions on which are linear on each K T and vanish We discretise (1.1) by the Streamline Diusion method, see [4]: Find U V such that (.4) (U + r) + ( ru + r) + (ru r) = (f + r) 8 V: The streamline diusion coecient = (x) is dened by (.5) = c 1 max(0 h? ) 3

8 4 MATS BOMAN where c 1 0 is a constant. Note that c 1 = 0 gives the standard Galerkin method. 3. A posteriori error estimates In this section we state the global a posteriori error estimates. Our error estimates take the form of a small a priori term plus an a posteriori term. The a priori term is of the form Ch min, where can be chosen arbitrarily. However, enters also in the factor (3.1) L = 1 + j log h min j which multiplies the a posteriori term. Let u be the solution of (1.1) and let U be the solution of (.4) for some c 1 0. The term [@ U](x) denotes the jump of the exterior normal U at In the case when 1 we have the following result. Here we assume that is suciently small, see (.3), which is necissitated by the use of weighted estimates for the L -projection in the proof of the theorem (see [1]). The use of the L -projection, which is possible only for the standard Galerkin method, makes it possible to subtract an arbitrary function V in the residual. One can prove the theorem without using the L -projection, in which case the condition on can be removed, but then it is not possible to subtract the function in the residual. Theorem 3.1. Let 1. Let c 1 = 0 and let. Assume that (x) 0 and (x)? r (x) c > 0 for all x, where c 1. There exist constants, h and C such that, if and h min h, then (3.) ku? Uk L 1() Ch min + CL 3= kh (f? U? ru? )k L 1() where is an arbitrary function in V. + CL 3= max KT h kh?1 K [@ U]k L 1(@K) In the general case, allowing to be small, we have the following estimate. Theorem 3.. Let. Assume that (x) c, and (x)? r (x) c > 0 for all x, where c 1. Then there exist constants h and C such that if, h min h, then ku? Uk L 1() C h min + CL h 3= h min 1 (f? U? ru) 1= 3= L 1() (3.3) + CL 3= max KT min h h 1= kh?1 3= K [@ U]k L 1(@K) In one dimension we are able to prove the following estimate, which is sharper with respect to the -dependence. Theorem 3.3. Let d = 1 and let. Assume that (x) 0, (x)? x (x) c > 0 and j(x)j c for all x, where c 1, and that does not change sign. Then there exist :

9 5 constants h and C such that if, h min h, then ku? Uk L 1() C h min + CL min 1 h h (f? U? ru) (3.4) + CL max min h h kh?1 K KT [@ U]k L 1(@K) : L 1() 4. A split of the error into an a priori and an a posteriori part The main result of this section is a lemma which splits the estimate of the error e = u?u into two parts, see [15] and [7]. The rst part is then estimated through an a priori estimate. The remaining part is of the form j(!e g)j, where g is an approximate delta function and! is a weight-function. The presence of the weight-function! is motivated by the localisation argument in Section 10, where we need a more general version of the lemma. We assume that the weight-function! has the following properties: (4.1)! W 1 () 0 <! C 1 jd j!j C?j=! a:e: in j! SK! K C 3 8K T where S K is as in (.) and w SK = sup SK!, w K = inf K!. Note that the function!(x) 1 satises (4.1). We now dene the function g. Let x 0 and let g = g x0 be such that (4.) g dx = 1 supp g B(x 0 ) 0 g C?d : R d Here B(x 0 ) denotes the closed ball with center at x 0 and with small radius to be chosen. By direct calculation we have (4.3) kgk Lp C?d=p0 p 0 = p=(p? 1): We now state the main result of this section. Lemma 4.1. Let U V and u H0() 1 \ C () for some 0 < 1. Let e = u? U and let x 0 be such that j!e(x 0 )j = k!ek L 1(). Let g = g x0 be given by (4.) and let! satisfy (4.1). Let >. There exist a constant h > 0 such that, if h min h and h = min, then (4.4) k!(u? U)k L 1() Ch minkuk C () + j(!(u? U) g)j: Proof. Let B denote the union of all elements K T that intersect B(x 0 ). Extend e to be zero outside. By the mean value theorem there is an x 1 B(x 0 ) \ such that (!e)(x 1 ) = (!e g). We note that!e =!u?!u +!e, where : C()! V is the Lagrange interpolation operator. Thus (4.5) j(!e)(x 0 )? (!e)(x 1 )j j(!u)(x 0 )? (!u)(x 1 )j + kd(!u)k L 1(B) + kd(!e)k L 1(B):

10 6 MATS BOMAN We have (4.6) j!(x 0 )u(x 0 )?!(x 1 )u(x 1 )j j!(x 0 )u(x 0 )?!(x 1 )u(x 0 )j + j!(x 1 )u(x 0 )?!(x 1 )u(x 1 )j kd!k L 1()kuk L 1() + kuk C ()k!k L 1() C 1 kuk C ()(C?1= 1? + 1) C 1 kuk C () where we used (4.1) and the estimate kuk L 1() Ckuk C (). The last inequality in (4.6) is valid if 1= 1=(1?) (4.7) : C Further, (4.8) kd(!u)k L 1(B) k(d!)uk L 1(B) + k!duk L 1(B) C 1 C?1= kuk L 1(B) + C 1 kd(u? u(x 0 ))k L 1(B) C 1 C?1= kuk L 1(B) + CC 1 h?1 min ku? u(x 0)k L 1(B) CC 1 C?1= kuk C () + 1+ CC 1 kuk C () = CC 1 kuk C () 1? C?1=?1 + h min CC 1 kuk C () where we used (4.1), the stability of in the L 1 -norm and an inverse estimate. The last inequality in (4.8) is valid if (4.9) Likewise, (4.10) 1= 1=(1?) and hmin : C kd(!e)k L 1(B) C?1= k!ek L 1(B) + k!dek L 1(B) max KB! K(C?1= kek L 1(K) + kdek L 1(K)) max KB! K(C?1= kek L 1(K) + Ch?1 min kek L1(K)) max KB! K C?1= k!ek L 1(K) + Ch?1! k!ek min L1(K) K C 3 (C?1= + Ch?1 min )k!ek L1() 1 k!ek L1()

11 where we used (4.1), the stability of in the L 1 -norm and an inverse estimate. The last inequality in (4.10) is valid if (4.11) 1 4 1= (C 3 C )?1 and 1 4 h min(c 3 C)?1 : Let now h = for h min min h. We see that, since = > 1, for suciently small h the conditions (4.7), (4.9) and (4.11) are satised and we get, using (4.5), (4.6), (4.8) and (4.10), (4.1) which concludes the proof. k!ek L 1() j(!e)(x 1 )j + j(!e)(x 0 )? (!e)(x 1 )j j(!e g)j + CC 1 kuk C () + 1 k!ek L1() We have the following lemma which gives a rough estimate of the Holder-norm of a solution u of (1.1). Lemma 4.. Let be a convex polyhedral domain in R d, d = 1 3. Let u be a solution of (1.1) with? 1 r 0. For any 0 < 1= there is a constant C such that (4.13) kuk C C? kfk L : Proof. By a standard energy argument we get (4.14) Further (4.15) kuk L C? kfk L : kd uk L Ckuk L C? kfk L where we used elliptic regularity. By Sobolev's inequality we get (4.13) for <? d=, d = A duality argument In this section we express the quantity (e g) in terms of the residual R(U) and the solution G of an adjoint problem, using a duality argument. In order to do so we use the following adjoint problem: (5.1) Multiplying by e = u? U gives (5.) G? r (G)? G = g in (e g) = (e G? r (G)? G) G = 0 = (u + ru? u G)? (U + ru G)? (ru rg) = (f? U? ru G)? (ru rg) = hr(u) Gi where we used (1.1) and where the residual R(U) H?1 is dened by (5.3) hr(u) vi = (f? U? ru v)? (ru rv) 8v H 1 0: 7

12 8 MATS BOMAN We thus have (5.4) which we will use with g given by (4.). (e g) = hr(u) Gi 6. Estimates of the residual In this section we state and prove some estimates of the residual R(U). One of the estimates involve a weight-function! with properties as in (4.1). We note that the estimate is valid in the unweighted case,! 1. The weighted estimate will be used in the localisation argument in Section 10. The estimates are also weighted with powers of, anticipating the regularity estimates of G in Section 7-8. We recall that there is an interpolant, see [16] and the discussion in [11], such that for v W 1 \ H0, 1 (6.1) kvk L1 (K) Ckvk L1 (S K ) 8K T kv? vk L1 (K) Ch i KkD i vk L1 (S K ) i = 1 8K T kd(v? v)k L1 (K) Ch i?1 K kdi vk L1 (S K ) i = 1 8K T : Lemma 6.1. Let U be the solution of (.4). Let! be as in (4.1) and let v W 1 \ H 1 0. There exist a constant C such that jhr(u)!vij C! min 1 (f? U? ru) (6.) h h 1= 3= L 1() kvk L1 () + 1= kdvk L1 () + 3= kd vk L1 () h + C max! min h h?1 [@ KT 1= U] 1= kdvk L1 () + 3= kd vk L1 () 3= L 1(@K) Proof. We rst note that by expanding derivatives and using (4.1) and (6.1) we get the following inequalities (6.3) K k(!v)k L 1 (K) Ckvk L1 (S K ) K kd(!v)k L 1 (S K ) C(?1= kvk L1 (S K ) + kdvk L1 (S K )) K kd (!v)k L1 (S K ) C(?1 kvk L1 (S K ) +?1= kdvk L1 (S K ) + kd vk L1 (S K )): We also recall the inverse estimate (6.4) We now have (6.5) krk L1 (K) Ch?1 K kk L 1 (K) 8 V: hr(u)!vi = (f? U? ru!v)? (ru r(!v)) = (f? U? ru!v? (!v)? r(!v))? (ru r((!v)? (!v))) = I? II :

13 9 where we used (.4). Here (6.6) jij! min 1 max h h 1= (f? U? ru) 3= 1 1= h 3= h L 1() (!v? (!v)? r(!v)) L1 () where (6.7) max 1 1= h 3= h X KT = III: K (!v? (!v)? r(!v)) max 1 1= h K 3= h K L1 ()!v? (!v)? r(!v) L1(K) Further, since is constant on K, (6.8) K k!v?(!v)? r(!v)k L 1 (K) K (k!vk L 1 (K) + k(!v)k L1 (K) + kk L 1kD(!v)k L1 (K)) K (k!vk L 1 (K) + Ck!vk L1 (S K ) + Ch?1 K k!vk L 1 (S K )) C K k!vk L 1 (S K ) Ckvk L1 (S K ) where we used (4.1), (6.1) and (6.4) combined with the fact that Ch, see (.5). We also have the estimate (6.9) K k!v? (!v)? r(!v)k L 1 (K) K (k!v? (!v)k L 1 (K) + kk L 1kD(!v? (!v))k L1 (K) + kk L 1kD(!v)k L1 (K)) Ch K K kd(!v)k L 1 (S K ) Ch K (?1= kvk L1 (S K ) + kdvk L1 (S K )) where we used (4.1), (6.1), (6.3) combined with Ch. In a similar way (6.10) K k!v? (!v)? r(!v)k L 1 (K) C K h K(kD (!v)k L1 (S K ) +?1 kd(!v)k L1 (S K )) Ch K(?3= kvk L1 (S K ) +?1 kdvk L1 (S K ) + kd vk L1 (S K )) where we used (4.1), (6.1), (6.3) and the inequality C h which follows from (.5). Let A = fk T : min(1 1= 3= ) = 1g, B = fk T : max(1 1= 3= ) = 1= )g and let h h h h h

14 10 MATS BOMAN C = fk T : max(1 1= 3= ) = 3= )g. By combining (6.8), (6.9), and (6.10) we get h h h III = X KA + X KB K k!v? (!v)? r(!v)k L 1 (K) 1= K h k!v? (!v)? r(!v)k L 1 (K) (6.11) + X KC 3= K h k!v? (!v)? r(!v)k L 1 (K) Therefore (6.1) C X KA kvk L1 (S K ) + C X KB(kvk L1 (S K ) + 1= kdvk L1 (S K )) + C X KB(kvk L1 (S K ) + 1= kdvk L1 (S K ) + 3= kd vk L1 (S K )) C(kvk L1 () + 1= kdvk L1 () + 3= kd vk L1 ()): jij C! min 1 We now estimate II. We rst note that h h 1= 3= (f? U? ru) L 1() (kvk L1 () + 1= kdvk L1 () + 3= kd vk L1 ()): (ru r(!v? (!v))) = X KT (ru r(!v? (!v))) K (6.13) where (6.14) Further, (6.15) j([@ U]!v? j 1= max h 3= h = X KT (@ U!v? =? 1! min X KT max h(!v? (!v)) ([@ U]!v? h h 1= 1= h?1 [@ 3= U] L 1(@K) h 3= h(!v? (!v)) h L1 (@K) C K max 1= h 3= h k!v? (!v)k L1 (K) + h K kd(!v? (!v))k L1 (K) L1 (@K):

15 11 where we used a trace inequality. We here have that (6.16) K (k!v? (!v)k L 1 (K) + h K kd(!v? (!v))k L1 (K)) Ch K K kd(!v)k L 1 (S K ) Ch K?1= kvk L1 (S K ) + kdvk L1 (S K ) where we used (6.1) and (6.3). But we also have the estimate (6.17) K (k!v? (!v)k L 1 (K) + h K kd(!v? (!v))k L1 (K)) Ch K K kd (!v)k L1 (S K ) Ch K?1 kvk L1 (S K ) +?1= kdvk L1 (S K ) + kd vk L1 (S K ) where we again used (6.1) and (6.3). By combining (6.14), (6.16), and (6.17) we get the following estimate (6.18) j(ru r(!v? (!v)))j C max KT X KB + X KC 1= C max! min KT which proves the lemma.! min h 1= h 3= h?1 [@ U] L 1(@K) h!?1 K (k!v? (!v)k L 1 (K) + h K kd(!v? (!v))k L1 (K)) 3= h!?1 K (k!v? (!v)k L 1 (K) + h K kd(!v? (!v))k L1 (K)) h h h?1 [@ 1= 3= U] L 1(@K) kvk L1 () + 1= kdvk L1 () + 3= kd vk L1 () In the proof of Theorem 3.1, where 1, we will use the following version of Lemma 6.1. The proof of this lemma is similar to the proof of Lemma 6.1, but uses the L -projection instead of the interpolant, which is possible since we use the standard Galerkin method in this case. In order to use the L -projection one needs a regularity assumption on the triangulation. This is expressed through the parameter in (.3), see [5], [1]. Lemma 6.. Let U be the solution of (.4) with c 1 = 0. Let v W 1 \ H 1 0 and let the residual R(U) be dened by (5.3). For suciently small there exist a constant C such that (6.19) jhr(u) vij C h + max KT where is an arbitrary function in V. (f? U? ru? ) L 1() h kh?1 K [@ U]k L 1(@K) kd vk L1 ()

16 1 MATS BOMAN The following lemma will be used in the proof of Theorem 3.3 where d = 1. The proof of this lemma is an obvious modication of the proof of Lemma 6.1. Lemma 6.3. Let U be the solution of (.4). Let v W 1 \ H 1 0 be dened by (5.3). Then there exist a constant C such that (6.0) jhr(u) vij C min 1 h h + max KT min h h and let the residual R(U) L 1() (f? U? ru) kh?1 K [@ U]k L 1(@K) kvk L1 () + kdvk L1 () + kd vk L1 () 7. The regularity of the solution G of the adjoint problem In this section we study the regularity of the solution G of the adjoint problem (5.1). We have the following lemma. Lemma 7.1. Let G be the solution of (5.1) with g given by (4.). Assume that (x) c > 0 and (x)? r (x) c > 0 for all x, where c 1. For any > 0 there is a constant C such that, if, then (7.1) where L = 1 + j log j. kgk L1 1=c kdgk L1 CL 1=?1= kd Gk L1 CL 3=?3= Remark. Another way to get estimates for the regularity of G is to use an interpolation inequality. We have the following result: Let v W p () \ H 1 0(), 1 p 1. Then for any > 0, we have (7.) kdvk Lp() kvk W p () + C?1 kvk Lp() see Theorem 7.7 in [9]. An argument based on this inequality would give the estimates (7.3) kdgk L1 CL?1 kd Gk L1 CL? : Proof. We rst estimate kgk L1. Multiplying (5.1) by G, > 0, which is a regularisa- p+g tion of sign(g), we get, after an integration by parts, G rg (7.4) G p + G + + G ( + G ) 3= We want to let! 0 + here. In order to do so we note that rg (7.5) G : rg rg ( + G ) dx = g 3= L1 kk ( + G ) 3= L 1kDGk L G L : ( + G ) 3= G p : + G

17 By a standard energy argument, using that? 1 r 0 which follows from 0 and? r 0, we obtain (7.6) Further (7.7) Here G (+G ) 3 theorem, R (7.8) 1 and G (+G ) 3 kdgk L C?1 kgk L C?1?d= < 1: G ( + G ) 3= G (+G ) 3 L = G ( + G ) 3 dx:! 0 as! 0 +, so that by the dominated convergence! 0 as! 0 +. By letting! 0 + in (7.4) we conclude that kgk L1 kgk L1 1: Since c, where c 1, this gives the estimate for kgk L1. We now estimate kdgk L1. Since? r 0, the maximum principle gives G 0 in. Therefore log(1 + G) 0 is well dened in. We note that log(1 + G) = 0 We also note that the maximum principle combined with the assumption? r c, where c 1, and the denition of g gives the estimate (7.9) kgk L 1 c?1 kgk L 1 C?d : We now multiply (5.1) with log(1 + G), integrate by parts using r log(1 + G) = rg, to 1+G get (7.10) (G log(1 + G)) + G where, by integration by parts, we have rg G = (1 + G) 1 + G (7.11) = =? rg + rg 1 + G rg dx + rg = (g log(1 + G)) 1 + G rg 1 + G dx? r log(1 + G) dx (r )G dx + r log(1 + G) dx (r ) log(1 + G) dx where we used that G and log(1 + G) = 0 vanish We thus have (7.1) G log(1 + G) dx + jrgj 1 + G dx = (g log(1 + G)) + (r G? log(1 + G)) (g log(1 + G)) + ( G) kgk L1 log(1 + kgk L 1) + kgk L1 CL where we used G log(1 + G) 0, the condition? r 0, kgk L1 = 1, (7.9), (7.8) and where L = 1 + j log j. By (7.1) combined with the fact that G log(1 + G) 0 we 13

18 14 MATS BOMAN get (7.13) jrgj dx CL: 1 + G Further, by Cauchy's inequality, k p p rgk L1 rg p = L1 p 1 + G (7.14) 1 + G 1 + G CL 1= (1 + kgk L1 ) 1= CL 1= where we used the estimate kgk L1 C. Hence jrgj 1= k1 + Gk L1 L1 (7.15) kdgk L1 CL 1=?1= : The estimate of kd Gk L1 will be obtained from an estimate of kgk Lp for p near 1. In order to estimate kgk Lp, using (5.1), we need to estimate k(? r )Gk Lp and k rgk Lp for 1 p. These L p estimates are obtained by interpolation between L 1 and L. We rst estimate k(?r)gk L1. As in (7.4) we multiply by a regularisation of sign(g) and get G G rg rg (? r )G p? rg p + dx + G + G ( + G ) 3= (7.16) and we conclude = g G p + G (7.17) k(? r )Gk L1 kgk L1 + kk L 1kDGk L1 CL 1=?1= where we used (7.15). We also need an estimate for k(? r )Gk L, which we get from (7.9) as follows: (7.18) k(? r )Gk L k? r k L kgk L 1 C?d : We now estimate k(? r )Gk Lp and k rgk Lp, for 1 < p <. By Holders inequality, combined with (7.17) and (7.18) we get (7.19) k(? r )Gk Lp k(? r )Gk 1?=p0 L 1 k(? r )Gk =p0 L CL 1=?1=?d=p0 where p 0 = p=(p? 1). Further (7.0) k rgk Lp kk L 1kDGk Lp kk L 1kDGk 1?=p0 L 1 kdgk =p0 L CL 1=?1=?=p0?d=p0

19 15 where we used (7.6) and (7.15). By (5.1) we get (7.1) kgk Lp kgk Lp + k(? r )Gk Lp + k rgk Lp C?d=p0 + CL 1=?1=?d=p0 + CL 1=?1=?=p0?d=p0 CL 1=?1=?=p0?d=p0 : We now use the condition, which gives (7.) and therefore (7.3)?=p0? 1 p0 kgk Lp CL 1=?3=?c=p0 where c = = + d. We recall the following elliptic regularity estimate: If is a smooth domain or a convex domain, then (7.4) kd vk Lp Cp 0 kvk Lp 8v W p \ H < p : The p-dependence in (7.4) is classical in in the case of a smooth domain. In the case of a convex domain we argue as in [7]. Let v = T f be the solution of the Dirichlet problem?v = f in, v = 0 and let D ij be a partial derivative of second order. It is well known, [10], that the operator D ijt is bounded on L, i.e, it is strong type ( ) this is the case p = of (7.4). Moreover D ijt is weak type (1 1) this is an unpublished result of Dahlberg, Verchota, and Wol, a proof can be found in [8] and a generalisation in [6]. An application of Marcinkiewicz interpolation theorem now yields (7.4). We now use (7.4) to get an estimate for kd Gk L1. By Holder's inequality and (7.4) we get (7.5) The choice p 0 = j log j gives (7.6) which proves the lemma. kd Gk L1 CkD Gk Lp Cp 0 kgk Lp CL 1=?3= p 0?c=p0 : kd Gk L1 CL 1=?3= j log j CL 3=?3= We now prove a lemma similar to Lemma 7.1 with worse -dependence, but with more general coecients,. We use this lemma in the proof of Theorem 3.1, where we assume that 1. Lemma 7.. Let G be the solution of (5.1) with g given by (4.). Assume that (x) 0 and (x)? r (x) c > 0 for all x, where c 1. For any > 0 there is a constant C such that, if, then (7.7) where L = 1 + j log j. kd Gk L1 CL?

20 16 MATS BOMAN Proof. We rst note that the proof of (7.14) is only based on the condition 0 and? r c, where c 1. We thus have (7.8) k 1= DGk L1 CL 1= (1 + kgk L1 ) 1= but we no longer get an estimate estimate for kgk L1 by (7.8). Instead we use the inequality kgk L1 CkDGk L1 in (7.8). Hence (7.9) k 1= DGk L1 CL 1= + CL 1= kdgk 1= L 1 CL 1= + CL?1= + 1 1= kdgk L1 where we used the inequality ab 1 a + 1 b. We conclude that (7.30) kdgk L1 CL?1 and consequently, by using the inequality kgk L1 CkDGk L1 again, we get (7.31) kgk L1 CL?1 : By the same argument as in Lemma 7.1, but using (7.31) and (7.30) instead of (7.8) and (7.15), we now get (7.7). 8. Regularity of the adjoint problem in 1-dimension In one dimension, d = 1, our regularity estimate, for the solution G of the adjoint problem (5.1), is sharper with respect to the -dependence. Lemma 8.1. Assume that that d = 1 and (x) 0, (x)? x (x) c > 0, j(x)j c, where c 1, for all x, and that does not change sign. Let G be the solution of (5.1). Then (8.1) where L = 1 + j log j. kgk L1 CL kg x k L1 CL kg xx k L1 CL?1 Proof. We rst prove an estimate for k(? x )Gk L1. Note that as in (7.8) we have kgk L1 1. As in the proof of Lemma 7.1 we multiply (5.1) with log(1 + G) and similarly we get the estimate, see (7.1), (8.) G log(1 + G) dx? x G dx + G x 1 + G dx = (g? x log(1 + G)) (kgk L1 + k x k L1 ) log(1 + kgk L 1) CL

21 where we used kgk L1 = 1, (7.9), and where L = 1 + j log j. We note that by (8.) we get G log(1 + G)? x G dx log(1+g)1 (8.3) + G log(1 + G)? x G dx CL: log(1+g)1 Since log(1 + G) 1 implies G, we conclude (8.4) G log(1 + G)? x G dx CL + k x k L1 CL log(1+g)1 where we also used that G log(1 + G) 0 in. Therefore (8.5) (? x ) G dx = CL log(1+g)1 log(1+g)1 (? x ) G dx + G log(1 + G)? x G log(1+g)1 (? x ) G dx dx + kgk L1 + k x k L1 where we used (8.3), the assumption 0 and the estimate kgk L1 1. Since? x 0 and G 0, this proves (8.6) k(? x )Gk L1 CL: G We now estimate kg x k L1. Multiply (5.1) by?sign() p x, where > 0. Recall that +G x sign() is constant and an integration by parts, (8.7)?sign() b a pg x +G x is a regularisation of sign(g x ). Let = (a b), we get, after (? x )G G x p dx + + G x + sign()? sign() b a G x jjg x p dx + G x G x (b) G x (b) p + Gx (b)? G G x (a) x(a) G xx G x ( + G x) dx =? G g sign() p x : 3= + G x b G Since G 0 and jp x j 1 this implies +G x a p + Gx (a) 17 (8.8) b a G x jjg x p dx k(? x )Gk L1 + (jg x (b)j + jg x (a)j) + G x + b a G x G xx ( + G x) 3= dx + kgkl1 :

22 18 MATS BOMAN We now show that (8.9) b a G x By Holder's inequality we have (8.10) b a G x G xx ( + G x) 3= dx! 0 as! 0 + : G xx A standard energy argument gives (8.11) Further, (8.1) ( + G x) dx kgxx k 3= L G x L : ( + G x) 3= kg xx k L C? kgk L < 1: G x ( + G x) 3= b = L a G x ( + G x) 3 dx where G x 1, and G (+G x x )3 (+G R! 0 as! 0 +, so that by the dominated convergence x )3 b theorem, we conclude that G x! 0 as! 0 +, and (8.9) follows. a (+G x In order to estimate (jg )3 x (b)j + jg x (a)j) we argue as follows. Multiply (5.1) by 1 to get (8.13) which, after an integration by parts, gives (8.14) (G 1)? ((G) x 1)? (G xx 1) = (g 1) kgk L1? G x (b) + G x (a) = 1: By the maximum principle we have that G 0 in [a b], hence G x (a) 0 and G x (b) 0. Consequently, by (8.14), we get (8.15) (jg x (b)j + jg x (a)j) 1: Let now! 0 + in (8.8). Using (8.6) and kgk L1 () = 1 we obtain (8.16) By assumption jj c, where c 1, so that (8.17) The inequality kgk L1 CkG x k L1 (8.18) gives Finally, by the equation (5.1), we get (8.19) which implies kg xx k L1 CL?1. kg x k L1 CL: kg x k L1 CL: kgk L1 CL: kg xx k L1 k(? x )Gk L1 + kg x k L1 + kgk L1 CL

23 9. Conclusion of proofs of Theorems 3.1, 3.3, and 3. In this section we conclude the proofs of Theorem 3.1, Theorem 3., and Theorem 3.3. We recall that the parameter is given by the denition (4.). In order to use Lemma 4.1 we have the conditions (9.1) h = min and h min h, where h is determined by conditions in the proof of Lemma 4.1, is the Holder exponent of the solution u of (1.1) and > can be chosen arbitrarily. Further, in order to use Lemma 7.1 and Lemma 7. we have the condition (9.) where > 0 can be chosen arbitrarily. The choice of the parameters and?1 aects the constant C in the theorems: the larger the values of and?1, the larger the value of C. Let now h min( h ). Assume that h min 1. Since = 1, we have h = min h min h and the condition (9.) is satised. As a remark we note that the proof of Lemma 4.1 indicates that = 1=, see (4.11), is a natural choice. We also note that by Lemma 4. we have 1=. This implies h min. The choice is natural, since the a posteriori part is of order in h. In the proof of Theorem 3.1, Theorem 3. and Theorem 3.3, enters in Lemma 7.1, Lemma 7. and Lemma 8.1 through the logarithmic factor L = 1 + j log j. We now note that the weight function! 1 satises (4.1). Therefore by Lemma 4.1, with the above condition on, we have (9.3) kek L 1 Ch minkuk C + j(e g)j C h min + jhr(u) Gij where we also used Lemma 4. and (5.4). By combining (9.3), Lemma 6. and Lemma 7. we get Theorem 3.1. By combining (9.3), Lemma 6.1 and Lemma 7.1 we get Theorem 3.. By combining (9.3), Lemma 6.3 and Lemma 8.1 we get Theorem A localisation result In this section we state and prove a localised a posteriori error estimate in a special case. The estimate is localised in the following sense. The error and the residual are multiplied by a cut-o function ', where ' is such that ' 1 inside 0 and decays exponentially with the distance from 0 outside 0. The set 0 can not be chosen arbitrarily. It has to be oriented along the streamlines with a cut-o in the downstream direction. The point of this result is that it indicates that it is possible to cut-o certain regions where the residual can be very large, as in a boundary layer or a characteristic layer. We now dene the problem that we will consider. Let R. Let u be the solution of (10.1) u + u x? u = f in u = 0

24 0 MATS BOMAN In [13] it is proved, for the equation (10.1), that, for the Streamline-Diusion method, in regions 0, of the type mentioned above, with the additional assumption that the the solution u of (10.1) is smooth in 0 one have the a priori estimate kek L 1( 0 ) Ch 5=4 max, see the discussion in Section 1. It is important to note that this result show that the error in 0 can be made small without resolving the singular layers. We now dene the cut-o function '. Let (10.) (t) = 1 t e?(s) ds where (s) = s for t 1 and (s) =?(s? ) for t < 1. By direct calculation we nd that 00 is smooth except at t = 1, where 00 has a jump, and that (10.3) (t) > 0 8t e?1 (t) e?1 t 1 (t) = e?t t 1 0 (t) < 0 8t j 0 (t)j (t) 8t j 00 (t)j? 0 (t) t 6= 1: Let 0 = f(x y) : x A B 1 y B g and dene the cut-o function (10.4) '(x y) = ( x? A p ) ( y? B 1 3 p ) (B? y 3 p ): We note that ' C 1 () with a jump in the second derivatives 0 and ( ' SK if SK 0 (10.5) ' K e Ch p otherwise: We note that (10.5) guarantees that, if Ch p in n 0 then ' satises the condition (4.1), which makes it possible to use Lemma 4.1 and Lemma 6.1 with! = '. In order to localise we use the following adjoint problem: (10.6) 'G? ('G) x? ('G) = 'g in G = 0 where g is given by (4.). The duality argument reads as follows: (10.7) ('e g) = (e 'g) = (e 'G)? (e ('G) x )? (e 'G) = (e 'G) + (e x 'G) + (re r('g)) = (f 'G)? (U 'G)? (U x 'G)? (ru r('g)) = hr(u) 'Gi where now R(U) H?1 () is takes the form (10.8) hr(u) vi = (f? U? U x v)? (ru rv) 8v H 1 0:

25 In order to put (10.6) in standard form, see (5.1), we expand the derivatives, collect terms and then divide by ' to get the equation (10.9) where (10.10) G? r (G)? G = g = ~ + r ~ = 1? ' x '? ' xx '? ' yy ' =? 1 + ' x ' ' y : ' In order to prove the localised result we need to analyse the regularity of the solution G of (10.9). The regularity estimates we need follows from Lemma 7.1 if we can show that the coecients and satises the conditions of Lemma 7.1. We have the following result: Lemma Let ',, and be given by (10.4) and (10.10). Assume that 1=9. Then 1=3 (10.11)? r 5=9: Proof. We rst estimate (10.1) By denition and (10.3) we have that? 'x ' (10.13)? r = ~ = 1? ' x '? ' xx '? ' yy ' : 'xx 0 and j j p j 'x j and thus ' '? ' x '? ' xx ' (1? p ) ' x 0: ' It is convenient to use the notation p(y) = ( y?b 1 3 p ) and q(y) = ( B?y ). We have (10.14) ' yy = p yyq + p y q y + pq yy ' pq jpyy j + jp yj jq y j p p q + jq yyj q 4 9 since by denition jq y j q 3 p, jp yj 3 p, jp yyj p 9 and, jq yyj q. By combining (10.1), 9 (10.13), and (10.14) we conclude that? r 5=9. We now estimate = ~ + r. After dierentiation and rearrangement of terms we get 3 p 1 (10.15) where as above (10.16) = ~ + r = 1? ' x ' + ' xx '? ' x ' + ' yy '? ' y '? ' x ' + ' xx '? ' x ' (1? 3p ) ' x ' 0

26 MATS BOMAN since 1=9. Further, (10.17) ' yy '? ' y ' = p yyq + p y q y + pq yy pq? (p yq) + p y qpq y + (pq y ) (pq) = p yyq + pq yy? (p yq) + (pq y )? p yq y pq (pq) pq pyy p + q yy q? py p + qy q where we used that?p y q y 0. Now, by the same argument as in (10.14) we get (10.18) pyy p + q yy q jpyy j p 6=9:? py qy + p q + jq yyj + py q p By combining (10.16), (10.17), and (10.18) we conclude that qy + q (10.19) 1 + ' yy '? ' y 1? 6=9 = 1=3: ' Note that the proof of Lemma 7.1 is partly based on the maximum principle. Since the classical maximum principle requires continuous coecients and since the coecient in (10.9) has a jump discontinuity, we refer to the generalised weak maximum principle for weak solutions, see [9] Theorem 8.1. By Lemma 10.1 and Lemma 7.1 we get Lemma 10.. Let G be the solution of (10.9) with g given by (4.). Assume 1=9. Then there is a constant C such that (10.0) kgk L1 () CL kdgk L1 () CL 1=?1= kd Gk L1 () CL 3=?3= where L = 1 + j log j. We now have the following localised a posteriori error estimate: Theorem Assume that h?1= C in n 0, where C 1. Let ' be given by (10.4). Let. Let u be the solution of (10.1) and let U be the corresponding solution

27 3 of (.4). There exist constants h and C such that if h min h then (10.1) where (10.) k'(u? U)k L 1() C h min + CL 3= ' min 1 + CL 3= max KT min L = 1 + j log h min j: h h 1= h (f? U 3= x ) L 1() k'h?1 3= K [@ U]k L 1(@K) h 1= Proof. We rst note that ' satises the condition (4.1) which makes it possible to use Lemma 4.1 and Lemma 6.1. With the same argument, concerning the conditions on, as in Section 9 and the argument which gives (9.3) we get by combining Lemma 4.1, with! replaced by ', (10.7), Lemma 6.1, with! replaced by ', and, nally Lemma 10. the proof of Theorem References 1. M. Boman, A posteriori error analysis in the maximum norm for a penalty nite element method for the time dependent obstacle problem, To appear.. E. Dari, R. G. Duran, and C. Padra, Maximum norm error estimators for three-dimensional elliptic problems, Siam J. Numer. Anal. 37 (000), 683{ K. Eriksson, An adaptive nite element method with ecient maximum norm error control for elliptic problems, Math. Models Methods Appl. Sci. 4 (1994), 313{ K. Eriksson and C. Johnson, Adaptive streamline diusion nite element methods for stationary convection-diusion problems, Math. Comp. 60 (1993), 167{ , Adaptive nite element methods for parabolic problems II: Optimal error estimates in L1L and L1L1, SIAM J. Numer. Anal. 3 (1995), 706{ M. Fassihi, L p integrability of the second order derivatives of Green potentials in convex domains, Preprint 1998-, Department of Mathematics, Chalmers University of Technology and Goteborg University. 7. D. A. French, S. Larsson, and R. H. Nochetto, Pointwise a posteriori error analysis for an adaptive penalty nite element method for the obstacle problem, Preprint 000:58, Department of Mathematics, Chalmers University of Technology and Goteborg University. 8. S. J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc. 119 (1993), 5{ D. Gilbarg and N. S. Trudinger, Elliptic Partial Dierential Equations of Second Order, Springer- Verlag, Berlin, P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, J. Ivarsson, A posteriori error analysis of the discontinuous Galerkin method for the time dependent Ginzburg-Landau equations, To appear. 1. C. Johnson, U. Navert, and J. Pitkaranta, Finite element methods for linear hyperbolic problems, Comput. Methods. Appl. Mech. Engrg. 45 (1984), 85{ C. Johnson, A. H. Schatz, and L. B. Wahlbin, Crosswind smear and pointwise errors in streamline diusion nite element methods, Math. Comp. 49 (1987), 5{ K. Niijima, Pointwise error estimates for a streamline diusion nite element method, Numer. Math. 56 (1990), 707{719.

28 4 MATS BOMAN 15. R. H. Nochetto, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp. 64 (1995), 1{. 16. L. R. Scott and S. hang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483{ G. hou, How accurate is the streamline diusion method?, Math. Comp. 66 (1997), 31{ G. hou and R. Rannacher, Pointwise superconvergence of the streamline diusion nite element method, Numerical Methods for Partial Dierential Equations 1 (1996), 13{145.

29 Chalmers Finite Element Center Preprints 1999{01 On Dynamic Computational Subgrid Modeling Johan Homan and Claes Johnson 000{01 Adaptive Finite Element Methods for the Unsteady Maxwell's Equations Johan Homan 000{0 A Multi-Adaptive ODE-Solver Anders Logg 000{03 Multi-Adaptive Error Control for ODEs Anders Logg 000{04 Dynamic Computational Subgrid Modeling (Licentiate Thesis) Johan Homan 000{05 Least-Squares Finite Element Methods for Electromagnetic Applications (Licentiate Thesis) Rickard Bergstrom 000{06 Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method Peter Hansbo and Mats G. Larson 000{07 A Discountinuous Galerkin Method for the Plate Equation Peter Hansbo and Mats G. Larson 000{08 Conservation Properties for the Continuous and Discontinuous Galerkin Methods Mats G. Larson and A. Jonas Niklasson 000{09 Discontinuous Galerkin and the Crouzeix-Raviart element: Application to elasticity Peter Hansbo and Mats G. Larson 000{10 Pointwise A Posteriori Error Analysis for an Adaptive Penalty Finite Element Method for the Obstacle Problem D A French, S Larson and R H Nochetto 000{11 Global and Localised A Posteriori Error Analysis in the maximum norm for nite element approximations of a convection-diusion problem Mats Boman These preprints can be obtained from Department of Mathematics, Chalmers University of Technology and Goteborg University, SE{41 96 Goteborg, Sweden address: boman@math.chalmers.se