FINITE ELEMENT APPROXIMATION OF CONVECTION DIFFUSION PROBLEMS USING GRADED MESHES

Size: px

Start display at page:

Download "FINITE ELEMENT APPROXIMATION OF CONVECTION DIFFUSION PROBLEMS USING GRADED MESHES"

Byron Holt
5 years ago
Views:

1 FINITE ELEMENT APPROXIMATION OF CONVECTION DIFFUSION PROBLEMS USING GRADED MESHES RICARDO G. DURÁN AND ARIEL L. LOMBARDI Abstract. We consider the nmerical approximation of a model convection-diffsion eqation by standard bilinear finite elements. Using appropriately graded meshes we prove optimal order error estimates in the -weighted H -norm valid niformly p to a logarithmic factor in the singlar pertrbation parameter. Finally we present some nmerical examples showing the good behavior of or method.. Introdction As is well known the nmerical approximation of convection-diffsion eqations reqires some special treatment in order to obtain good reslts when the problem is convection dominated de to the presence of bondary or interior layers. A lot of work has been done in this direction see for example the books [4 5] and their references). There are in principle two ways to proceed: to se some kind of pwind or to se adapted meshes appropriately refined. This last possibility seems very reasonable when the layers are de to the bondary conditions and so their location is known a priori. In this paper we analyze the approximation of the soltion of a model convection-diffsion eqation. We prove that sing appropriate graded meshes the soltion is well approximated by the standard piecewise bilinear finite element method in the -weighted H -norm. defined as Precisely we consider the problem v = v L ) + v L )..) + b + c = f = in on where = ) and > is a small parameter. We prove that on appropriate graded meshes h C log/) N where h is the standard piecewise bilinear approximation of on a graded mesh T h where h > is a parameter arising in the definition of the mesh) N denotes the nmber of nodes and C is a constant independent of and N. Observe that this error estimate is almost optimal i. e. the order with respect to the nmber of nodes is the same as that obtained for a smooth fnction on niform meshes and p to a logarithmic factor the estimate is valid niformly in the pertrbation parameter. Conseqently the graded meshes seems an interesting alternative to the well known Shishkin meshes which provide also optimal order [6]. Indeed from some nmerical experiments the graded meshes procedre seems to be more robst in the sense that the nmerical reslts are not strongly affected by variations of parameters defining the meshes. The rest of the paper is organized as follows. In Section we introdce the graded meshes and prove the error estimates and in Section 3 we present some nmerical reslts. We end the paper with some conclsions. 99 Mathematics Sbject Classification. 65N3. Key words and phrases. Convection-diffsion graded meshes finite elements. Spported by ANPCyT nder grant PICT 3-59 by Universidad de Benos Aires nder grant X5 and by Fndación Antorchas. The first athor is a member of CONICET Argentina.

2 RICARDO G. DURÁN AND ARIEL L. LOMBARDI. Error estimates In this section we consider the nmerical approximation of problem.). We assme that the fnctions b = b b ) c and f are smooth on and that.) b i < γ with γ > for i =. Then the soltion will have a bondary layer of width O log ) at the otflow bondary {x x ) : x = or x = [5]. We also assme the following compatibility conditions f ) = f ) = f ) = f ) = i+j f ) = for i + j. x i xj Under these conditions it can be proved that C 4 ) C ) and precise estimates of the derivatives that will be sefl for or prposes are known. Using the graded meshes defined below we will obtain an almost optimal estimate in the -weighted H -norm for the error between the soltion of Problem.) and its standard piecewise bilinear finite element approximation. Given a parameter h > and a constant σ > we introdce the partition {ξ i M i= of the interval [ ] given by ξ = ξ = σh.) ξ i+ = ξ i + σhξ i for i M ξ M = where M is sch that ξ M < and ξ M + σhξ M. We assme that the last interval ξ M ) is not too small in comparison with the previos one ξ M ξ M ) if this is not the case we jst eliminate the node ξ M ). In practice it is natral to take h i := ξ i ξ i to be monotonically increasing. Therefore it is convenient to modify the partition by taking h i = h for i sch that ξ i < and starting with the graded mesh after that. In this way we obtain the following alternative partition ξ = ξ i = iσh for i <.3) σh + ξ i+ = ξ i + σhξ i for σh + i M ξ M = with M as in the other case. For any of these choices of ξ i we introdce the partitions T h of defined as T h = {R ij M ij= where R ij = ξ i ξ i ) ξ j ξ j ). Associated with T h we introdce the standard piecewise bilinear finite element space V h and its corresponding Lagrange interpolation operator Π. First we have to prove some weighted a priori estimates for the soltion. The following pointwise estimates are immediate conseqences of Theorem. of [6]: k x k x x ) C + ) γx k e.4) k k x k x x ) C + ) γx k e.5) k x x ) x x C + γx e + γx e + ) γx e e γx.6). for all x x ) and k. For the proof of or error estimates we will need to decompose as = 3 where

3 3.7) = { R ij : ξ i < c log = { R ij : ξ i c log ξ j < c log 3 = { R ij : ξ i c log ξ j c log with a constant c > /γ. As a conseqence of the estimates.4).5) and.6) we obtain the following lemma. Lemma.. There exists a constant C sch that we have the following a priori estimates.8) 3 x L ).9) x.) x L ) x 3 C and L ) x.) x x x L ) C and 3 C L ) x C and x C L ) x x 3 C L ) x C and x x x C L ).) x x C L ).3) x i x i x j C if c > /γ i j =. L 3) Proof. Let s prove for example the first ineqality in.). From.6) we have x x x dx dx C x + γx e + γx e + γx 4 e e γx C + + C + ) se γs ds + se γs ds e γs ds+ ) e γs ds obtaining the desired ineqality. For ineqality.3) with i = j = we have also from.6) x 3 x x dx dx x + γx c log c log e + γx e + + ) γx 4 e e γx dx dx C 3 + ) e γs ds + e γs ds c log c log if c /γ. C ) dx dx

4 4 RICARDO G. DURÁN AND ARIEL L. LOMBARDI For ineqality.3) with i = j = we have from the estimate.4) that x 3 x dx dx C x + ) γx c log 4 e dx C 3 + ) s e γs ds c log [ = C 3 + 4γ 8γ 3 c log + 4γ log ) ] + γc C whenever c > /γ. The other ineqalities follow by similar argments and so we omit the details. We can now prove the error estimates for the Lagrange interpolation on or graded meshes. Theorem.. Let be the soltion of Problem.). If T h are the meshes given by the partitions.) or.3) then Π L ) Ch and Π) L ) Ch with a constant C independent of h and. In particlar.4) Π Ch. Proof. For simplicity we give the proof for the case of the partitions given in.). However it is not difficlt to see that the same argments apply in the other case. Recalling that R ij = ξ i ξ i ) ξ j ξ j ) for i j N and h i = ξ i ξ i we have from standard error estimates see for example []) {.5) Π L R ij) C h i + h j. L R ij) Now we decompose the error as x N Π L ) = Π L R + N j) Π L R + i) j= i= Then sing.5) and the definition of the mesh we have { Π L R ) Ch4 4 and { Π L R j) Ch4 4 x { Π L R ij) Ch4 x x { Π L R i) Ch4 x x x L R j) L R ij) L R i) L R ) + + x x x L R ij) + 4 x x + 4 x x L R j) L R ij) N ij= L R i) L R ) Π L R ij). for j for i j for i and therefore ptting all together and sing the a priori estimates.8) and.) we obtain Π L ) Ch. Now to bond the other part of the norm we se the known estimate see for example []) {.6) Π) x C h i + h j. L R ij) x x x L R ij) L R ij)

5 Then proceeding as in the case of the L norm we can easily obtain { Π) x Ch L R ) x + L R ) x x { Π) x Ch L R j) x + x L R j) x x L R j) { Π) x Ch x + x x x and Π) x L R ij) L R i) x { Ch x x L R ij) L R i) + x x L R ij) L R i) L R ) for j for i j for i. Therefore mltiplying by smming p and sing the a priori estimates.8).9).) and.) we obtain Π) x Ch. L ) Clearly a similar estimate holds for Π) x and so the theorem is proved. Now we consider the nmerical approximation of problem.). The weak form of this problem consists in finding H ) sch that a v) = fv dx v H ) where av w) = v w + b v w + c vw) dx. Assme that there exists a constant µ independent of sch that.7) c div b µ >. Observe that as pointed ot in [5 page 67] this is not an important restriction becase nder the assmption.) a change of variable e η x v for sitable chosen η leads to a problem satisfying.7). It is known that the bilinear form a is coercive in the -weighted H -norm niformly in [5] i. e. there exists β > independent of sch that 5.8) β v av v) v H ). However the continity of a is not niform in and this is why the standard theory based on Cea s lemma can not be applied to obtain error estimates valid niformly in. The finite element approximation h V h is given by a h v) = fv dx v V h. In the following theorem we prove an almost optimal error estimate in the -weighted H -norm. The constants will depend on the coercivity constant β on the constants in the estimates given in Lemma. on the L - norms of b and c on the constant σ introdced in the definition of the meshes and on the constant γ appearing in the estimates.4).5) and.6) also on the c sed in the partition of introdced in.7) bt this constant depends only on γ). We will not state all these dependencies explicitly. Theorem.. Let be the soltion of Problem.) and h V h its finite element approximation. If T h are the meshes given by the partitions.) or.3) then.9) h Ch log with a constant C independent of h and.

6 6 RICARDO G. DURÁN AND ARIEL L. LOMBARDI Proof. Let e = h Π V h. In view of Theorem. it is enogh to bond the norm of e. Again we consider only the case of T h given by.). In the other case the reslt can be obtained by simple modifications. From.8) and sing the error eqation a h e) = we have β e ae e) = a Π e) C { Π e + b Π)e C Π + β e + C b Π)e where we have sed the generalized arithmetic-geometric mean ineqality. Then it is enogh to prove that.) b Π)e Ch log + δ e for a small δ to be chosen. To prove this estimate we se the decomposition of introdced in.7). Since e vanishes at the bondary we know from the Poincaré ineqality that e L ) C log e. x Therefore since b is bonded we have L ) b Π)e C Π) L ) log e L ) C log Π) L + ) δ e L ) and so sing Theorem. we obtain.) b Π)e Ch log + δ e L. ) Clearly the same argment can be applied to obtain an analogos estimate in. Finally for R ij 3 we have shown in the proof of Theorem. that and Π) x Π) x and so sing.3) we obtain L R ij) L R ij) { Ch x x L R ij) { Ch x x x + x x x L R ij) + x 3 b Π)e Ch + δ e L 3). x L R ij) L R ij) Therefore ptting together the estimates in and 3 we obtain.) and the proof concldes by choosing δ small enogh depending on the coercivity constant β. To show that the error estimate is almost optimal we have to restate the ineqality.9) in terms of the nmber of nodes in the mesh. This is the objective of the following corollary. As we mentioned before the se of the partition given in.) may prodce too small intervals in the bondary layer region increasing the nmber of nodes in an nnecessary way. Therefore in practice it is more natral to se the meshes based on the partitions given in.3) and so we will consider only this case. Corollary.3. Let be the soltion of Problem.) and h V h its finite element approximation. If T h are the meshes given by the partitions.3) and N is the nmber of nodes in T h then with a constant C independent of h and. h C log /) N

7 7 Proof. We have to show that h C log/) N. Let M be the nmber of points ξ i in the partition given in.3) sch that ξ i and M be the nmber of points sch that ξ i >. Clearly M is bonded by C/h. To bond M let s call ξ k the smallest of the points sch that ξ i >. Assming σh we have M = ξi+ ξ i+ ξ i ) M i=k = σh M i=k ξ i dξ = ξi+ ξi+ σhξ i ) dξ M i=k ξ i ξ dξ σh ξ i M i=k ξ dξ = σh log. Therefore the proof concldes by recalling that M = M + M and N = M. ξi+ σhξ i+ ) dξ Observe that the estimate given in the corollary is almost optimal. Indeed the order is the same as that obtained in the approximation of a smooth fnction by piecewise bilinear elements on a niform mesh and the factor log/) is not significant in practice. Remark.4. Some slight variations of the meshes T h cold be more convenient. For example the following grading giving a lower nmber of nodes can be sed ξ i+ = ξ i + σhξi α with α = log. Of corse also in this case we can take a niform partition at the beginning and start with the grading after. With this choice of α the same error estimates can be proved by a simple modification of or argments assming that α > / which is valid for small ) sing now the estimates C α C α L ) and xα x L ) xα x x i x i x j C if c > 3/γ. L 3) which can be proved sing the same argments of Lemma. and observing that α = e. 3. Nmerical experiments In this section we present some nmerical examples which show the good behavior of the standard piecewise bilinear finite element method on graded meshes for convection-diffsion eqations. Althogh for simplicity we have restricted the analysis in the previos section to Dirichlet bondary conditions in or examples we have considered more general bondary conditions. Also we have inclded two examples which do not satisfy condition.7) Examples ) and )) to test the method in cases not covered by the theory. We have solved the problem + b + c = f = D in Γ D n = g in Γ N with different choices of coefficients b and c and data D and g namely ) b = ) c = Γ D = [ ] { Γ N = { [ ] D = g = and f = ) b = ) c = Γ D = [ ] { Γ N = { [ ] D = on { [ ] and D = on { [ ] g = and f = 3) b = ) c = Γ D = D = and f = in ξ i

8 8 RICARDO G. DURÁN AND ARIEL L. LOMBARDI 4) b = ) ) c = ) Γ D = D = and [ ) e x e y fx y) = x + y e e )] e x+y. For the examples ) and ) which present a bondary layer only on [ ] we have sed meshes graded only along the x axis. We have made several experiments sing the gradings giving in.) and.3) and the variant explained in Remark.4. Also we have considered different vales of the constant σ appearing in the definition of the meshes and we have observed that the reslts do not change significantly for vales of σ of the order of therefore we have taken σ = for or examples. No significant differences were observed with the different choices of meshes and in all cases no oscillations in the approximate soltions have been observed see Figre where the nmerical soltions obtained in the for cases for = 6 are shown). Therefore we will show only reslts obtained with the meshes graded with α = defined in Remark.4 and starting the grading log after x..8.6 y.4. ) ).5 y..4 x y.6.8 y x 3) 4) x Figre For the example 4) we know the exact soltion which is given by [ x y) = x ) e x y )] e y e x+y e e therefore we know the exact error and so the order of convergence in terms of the nmber of nodes can be compted.

9 In Table we show the -weighted H -norm of the error for different vales of N for Problem 4) with = 4 and = 6. The orders compted from this tables is for the first case and.574 for the second one as predicted by the theoretical reslts. 9 N Error = 4 Table N Error = 6 With the following reslts we want to point ot an advantage of the graded meshes over the Shishkin meshes: the graded meshes designed for a given work well also for larger vales of. Indeed this follows from the error analysis. This is not the case for the Shishkin meshes as shown by the following example. This might be of interest if one want to solve a problem with a variable. Table shows the vales of the -weighted H -norm of the error for different vales of solving the problem with the mesh corresponding to = 6 sing graded meshes and Shishkin meshes. Error Error Graded meshes N = 44 Shishkin meshes N = 69 Table We consider one more example similar to tests ) and ). Precisely we take = 6 b = ) c = and fx y) = wx) + vy) where wt) = t e e t e ; vt) = e e t e t + e +t e t The soltion of this problem is x y) = wx)vy) and it presents exponential bondary layers along x = of width O log ) and along y = of width O log ). Althogh the bondary layers presented near y = are weaker than the one near x = we have sed the grading indicated in Remark.4 for the three bondary layers. Of corse a different refinement with a lowest nmber of nodes cold be sed near the weaker layers bt we wanted to show that or procedre works well also for cases in which the eqation becomes reaction dominant. In Table 3 we show the reslts obtained in this case. We observe that the nmerical order is Finally jst to see the different strctres we show in Figre a Shishkin mesh and one of or graded meshes having the same nmber of nodes. For the sake of clarity we have pictred only the part of the meshes corresponding to /) /) and = 3. t ) +.

10 RICARDO G. DURÁN AND ARIEL L. LOMBARDI N Error Table Figre 4. Conclsions We have proved that the optimal order in the -weighted H -norm p to logarithmic factors is obtained sing appropriate graded meshes and standard finite element methods of lowest order for convection diffsion problems. In both theoretical and nmerical experiments we have worked with rectanglar elements bt it is not difficlt to see that analogos reslts can be obtained for linear trianglar elements. The nmerical experiments showed that no oscillations appear in the nmerical soltion and the predicted order of convergence is observed. We believe that graded meshes are an interesting alternative to the Shishkin meshes that have been widely analyzed for this kind of problems. In particlar nmerical experiments show that the graded mesh method is more robst in the sense that the nmerical reslts does not depend strongly on parameters defining the mesh instead the reslts obtained with Shishkin meshes depend in a significant way on the parameter defining the point where the mesh change its size indeed if this parameter is slightly moved from its optimal choice the nmerical soltion may present oscillations. Also we have observed that the graded meshes designed for some vale of the singlar pertrbation parameter work well also for larger vales of this parameter while this is not the case for the Shishkin meshes. This might be of interest in problems with a variable. We have performed the analysis and the nmerical experiments for a model problem in a sqare domain. However we believe that similar reslts cold be obtained for more general domains. Also similar problems in three dimensional domains can be analyzed in a similar way. However in that case a mean average interpolation shold be sed to prove the error estimates becase as it is known the estimates for the Lagrange interpolation in H are not independent of the relations between different edges of an element in 3D. On the other hand the graded meshes defined in Remark.4 and sed in or nmerical experiments satisfy the local reglarity conditions reqired in [3] for the error estimates

11 proved in that paper for the mean average interpolant introdced there and therefore that operator cold be sed for the analysis. These generalizations will be the object of frther research. Acknowledgment: We want to thank Thomas Apel for helpfl comments on a previos version of or reslts which lead to a considerable simplification and improvement of or paper. In particlar he pointed ot to s the argment sed in the proof of Corollary.3. References. T. Apel Anisotropic finite elements: Local estimates and applications Series Advances in Nmerical Mathematics Tebner Stttgart S. C. Brenner and L. R. Scott The Mathematical Theory of Finite Element Methods Texts in Applied Mathematics 5 Springer Berlin R. G. Drán A. L. Lombardi Error estimates on anisotropic Q elements for fnctions in weighted Sobolev spaces Math. Comp. to appear. 4. K. W. Morton Nmerical Soltions fo Convection-Diffsion Problems Chapman & Hall London H. G. Roos M. Stynes L. Tobiska Nmerical Methods for Singlarly Pertrbed Differential Eqations Springer- Verlag Berlin M. Stynes E. O Riordan A niformly convergent Galerkin Method on a Shishkin mesh for a convection-diffsion problem J. Math. Anal. Appl ) Departamento de Matemática Facltad de Ciencias Exactas y Natrales Universidad de Benos Aires 48 Benos Aires Argentina. address: rdran@dm.ba.ar Departamento de Matemática Facltad de Ciencias Exactas y Natrales Universidad de Benos Aires 48 Benos Aires Argentina. address: aldoc7@dm.ba.ar

h-vectors of PS ear-decomposable graphs

h-vectors of PS ear-decomposable graphs Nima Imani 2, Lee Johnson 1, Mckenzie Keeling-Garcia 1, Steven Klee 1 and Casey Pinckney 1 1 Seattle University Department of Mathematics, 901 12th Avene, Seattle,