A MESH ADAPTATION METHOD FOR 1D-BOUNDARY LAYER PROBLEMS

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 3, Number 4, Pages c 2012 Institute for Scientific Computing and Information A MESH ADAPTATION METHOD FOR 1D-BOUNDARY LAYER PROBLEMS ANDRÉ FORTIN, JOSÉ M. URQUIZA, AND RICHARD BOIS Abstract. We present a one-dimensional version of a general mes adaptation tecnique developed in [1, 2] wic is valid for two and tree-dimensional problems. Te simplicity of te one-dimensional case allows to detail all te necessary steps wit very simple computations. We sow ow te error can be estimated on a piecewise finite element of degree k and ow tis information can be used to modify te grid using local mes operations: element division, node elimination and node displacement. Finally, we apply te wole strategy to many callenging singularly perturbed boundary value problems were te one-dimensional setting allows to pus te adaptation metod to its limits. ey words. ierarcical error estimation, mes adaptation, singular perturbation, boundary layer problems. 1. Introduction Adaptive numerical metods ave now a long istory in efficiently computing approximations to ordinary or partial differential equations. Teir goal is to reac a given level of precision at a minimal cost wic often means a minimal number of degrees of freedom (DOFs). Tis also results in minimizing te size of te resulting algebraic systems of equations to be solved. In te case of partial differential equations (PDEs) and te finite element metod one of te most important approximation metods for PDEs te two key ingredients are an a posteriori error estimator and a mes adaptation procedure. Te adaptation metod uses te information from te error estimator to modify te mes in order to reac te desired error level. For eac of tese two steps, tere exist a large variety of possibilities. Our goal ere is not to review all or even several of tese metods and we refer te interested reader to [3, 4, 5, 6, 7] for starting points. In tis paper, we present a one-dimensional version of a general tecnique developed in [1, 2] for two and tree-dimensional problems. Te presented metod does not depend on te PDE itself (unlike some oter error estimations, based on residuals, as in [8]) and can be applied to finite element solutions of any degree. Te adaptation metod is based on an error estimator tat can be easily calculated on eac element. As we sall see, our error estimator as a nice interpretation linking te finite element error to te classical Lagrange interpolation error. Once te error as been estimated, local operations are used to modify te mes: element division, node elimination and node displacement. Tis indirectly means tat te node positions are also unknown and must terefore be determined troug an iterative metod. Te first main goal of te present paper is to take advantage of te one-dimensional setting to give all te details of te adaptive strategy: te explicit construction of Received by te editors May, 2012 and, in revised form, September Matematics Subject Classification. 65G20 65N12 65N50. Tis researc was supported by NSERC. 408

2 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 409 te error estimator, te computation of te local errors on a patc of adjacent elements and te decision process for te local mes modifying operations. Tis is done for two-point boundary value problems. Te second main goal is to illustrate te efficiency of te metod troug a variety of problems, some of tem aving proved to be particularly callenging steady convection-diffusion-reaction equations. Wen te diffusion coefficient is small compared to te convection velocity, solutions present steep variations localized in so-called boundary layers. Finite element approximations ten present unpysical oscillations in tese regions unless te mes size is very small and tus te resulting algebraic system to be solved is considerably large. Part of tis problem can be circumvented by modifying te variational formulation and adding stabilization terms but oscillations may remain or oversooting (or undersooting) penomena may appear. In [9], stabilization metods are combined wit graded meses adapted to te solution to improve te numerical solution but using suc and-tailored meses necessitates a precise knowledge of te solution. Tis is igly unrealistic for more general applications suc as fluid flow problems in several space dimensions. In our numerical examples, we sow tat te automatic adaptation strategy presented ere, wic does not use an a priori knowledge of te solution (nor of te PDEs being solved) compares favorably wit tese and-made meses. Finally, we also test our strategy for approximating solutions presenting discontinuities, an even more callenging problem. Te major advantage of te one-dimensional case is tat it can be pused to te limit witout unduly increasing te computational burden. As we sall see in te numerical results, we end up wit elements of lengt as small as in a unit domain. Tis would ardly be possible in 2D and even less in 3D. Tis article is organized as follows. In Section 2 we introduce te one-dimensional convection-diffusion equation and its standard Galerkin and stabilized Petrov-Galerkin formulations. Te error estimator and its interpretation as an interpolation error are presented in details in Section 3. In Section 4, eac step and te wole metodology of te adaptation strategy is presented in te one-dimensional setting. In Section 5 we present our numerical tests, wic compare our mes adaptation strategy to uniform meses or to and-made graded meses for convection-diffusion problems presenting boundary layers or even discontinuous solutions. 2. Position of te problem As a test problem, we consider te classical convection-diffusion-reaction equation of te general form (1) d ( d(x) du ) (x) + b(x) du (x) + c(x)u(x) = f(x) on te domain Ω = [α, β] wit Diriclet boundary conditions u(α) = u α and u(β) = u β. More general boundary conditions can also be considered. Convection-diffusion problems ave been studied for a long time since it is well known tat standard Galerkin finite element formulations (or centered finite differences) are unable to produce appropriate solutions wen advection is dominant, unless very fine meses are used. Some form of stabilization is needed, suc as te Streamline-Upwind- Petrov-Galerkin (SUPG) formulation wic goes back to te work of Huges and Brooks [10, 11]. We will solve equation (1) by te finite element metod. Tis requires a mes T wic is a partition of te interval [α, β] wit N elements denoted = [x i, x i+1 ] wit lengt = x i+1 x i. Let V te space of Lagrange continuous functions

3 410 A. FORTIN, J. URQUIZA AND R. BOIS wose restriction to any element of T belongs to te space P (k) () of polynomials of degree k over te element. We also introduce te space V 0 of te functions of V vanising at x = α and x = β. Multiplying equation (1) by v (k) β (x) + τ b(x) dv(k) ( ( d d(x) du(k) α T τ ( α d ( wit v (k) (x) V 0, we get ) + b(x) du(k) d(x) du(k) ) + c(x)u(k) (x) f(x) + b(x) du(k) Te first diffusion term is integrated by parts ( ) β d d(x) du(k) + c(x)u(k) β v (k) (x) = α ) f(x) d(x) du(k) v (k) (x) + ) dv (k) b(x) dv(k) = 0. and te boundary terms vanis due to te Diriclet boundary conditions (v (k) (α) = v (k) (β) = 0). Te diffusion stabilization term (2) ( ) d τ d(x) du(k) b(x) dv(k) remains as it is, toug it vanises in te linear case (k = 1) wen d(x) is constant. Tis is probably wy it is sometimes neglected in te quadratic case but tis may ave undesirable consequences. Te SUPG-stabilized variational formulation takes te form: find u (k) (3) β α d(x) du(k) T τ ( V suc tat for all v (k) ( dv (k) + d ( b(x) du(k) d(x) du(k) ) V 0 + c(x)u(k) (x) f(x) + b(x) du(k) ) + c(x)u(k) v (k) (x) + f(x) ) b(x) dv(k) = 0. Determining an optimal coice for te parameter τ as been te object of a vast literature. We will use te formula proposed by Roos [12] ( 1 1 ) if P τ = = b > 1, 2 b P 2d 0 if P 1, were P is te local Peclet number and b and d are te values of b(x) and d(x) at te center of te element. Oter similar forms could be used. To obtain te classical Galerkin formulation, we simply put τ = 0. Under fairly general conditions, one can expect te error to satisfy u u (k) 1,Ω = β α ( du du(k) (x) (x) ) 2 1/2 C 1 k u k+1,ω for te H 1 -seminorm wile for te L 2 -norm, one sould observe ( 1/2 β u u (k) 0,Ω = (u(x) u (k) ) (x))2 C 0 k+1 u k+1,ω α

4 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 411 for te Galerkin formulation and u u (k) 0,Ω C 0 k+1/2 u k+1,ω for te SUPG formulation (see Roos-Stynes-Tobiska [13]). Note tat, in te latter case, O( k+1 ) convergence is often observed in practice but it is sown in Zou [14] tat tere exists two-dimensional examples were te convergence rate is only k + 1/2. In te above error estimates, C 0 and C 1 are constants, independent of and u. Moreover, is te mes size but tis definition makes sense only on quasi uniform meses. For te adapted meses tat will be considered in tis paper, it is more appropriate to express tese convergence rates as functions of te number of degrees of freedom (DOFs). For instance, for a linear solution, te number of DOFs is N +1 wile it is 2N + 1 for a quadratic solution and so fort. We tus ave N 1 and consequently (4) u u (k) 0,Ω C 0 N (k+1) u k+1,ω and u u (k) 1,Ω C 1 N k u k+1,ω. 3. Error estimator Let u be te solution of Equation (1), usually unknown. We terefore suppose tat u (k) V is a finite element approximation of degree k to te exact solution u. Our objective now is to estimate te error u u (k) were is some appropriate norm. Our error estimator is based on te ypotesis tat a more accurate approximation ũ (k+1) of degree k+1 of te solution can be built from u (k) in a post-processing step. Te precise construction of ũ (k+1) will soon be described but for now we suppose its existence. Te error is ten simply approximated using (5) u u (k) ũ(k+1) u (k). Our construction of ũ (k+1) requires not only te finite element solution u (k) but it also necessitates a continuous approximation of degree k of its derivative tat will be denoted g (k) and wic will also belong to V Approximating te derivatives at nodes. Since u (k) is of Lagrange type, its derivative is discontinuous at element interfaces. Terefore du(k) / V and some form of projection is needed. We tus want to evaluate te derivatives at eac node of te mes, including tose at te boundary between two adjacent elements were du (k) is not defined. Tese nodal values will be denoted g (k) i. In te linear case, we consider for eac node, te polynomial of degree 2 (p (2) (x)) interpolating u (1) (x) at te tree nodes of te two adjacent elements: u (1) u (1) p (2) (x) i 1 i x i 1 x i x i+1 and te value of te recovered derivative at tis node is simply g (1) i = dp(2) (x i). Once tis is done for eac node, a piecewise linear approximation (denoted g (1) (x)) of te derivative u (x) is obtained. In te quadratic case, we determine for eac

5 412 A. FORTIN, J. URQUIZA AND R. BOIS node x i (including mid-element nodes) of te mes, a polynomial of degree 3 using te 4 closest nodes and we set g (2) i = dp(3) (x i). In te general case, a similar procedure can be applied to obtain te nodal values g (k) i of te recovered derivative (x). Te above procedure for te recovery of derivatives at nodes works well but for te two and tree-dimensional cases, we use a more sopisticated metod proposed by Zang and Naga [15] wic requires te solution of a least square problem around eac node. Te metod is sown to be superconvergent on more or less regular meses. g (k) 3.2. Construction of te estimator. Te construction of û (k+1) is performed on an element by element basis. Indeed, on eac element of te mes, we look for an approximation of te form: (x) = u (k) (x) + c (k+1) φ (k+1) (x), û (k+1) were c (k+1) is an unknown coefficient and φ (k+1) (x) is a basis function of degree k + 1. Te new solution is tus written as te sum of te finite element solution of degree k and a correction term of degree k + 1. Fort tis reason, we say tat te estimator is ierarcical as in Bank and Smit [16]. Te function φ (k+1) (x) vanises at all te nodes of te element. For instance, in te linear and quadratic cases, let φ (2) (ξ) = 1 ξ 2 and φ (3) (ξ) = ξ(1 ξ 2 ) be te quadratic and cubic ierarcical basis functions defined on te reference element [ 1, 1]. Transformed on te element = [x i, x i+1 ], we get (6) φ (2) (x) = 4 2 (x x i )(x i+1 x), φ (3) (x) = 8 3 (x x i )(x x m )(x i+1 x), were x m = (x i + x i+1 )/2 is te mid-element node for a quadratic element. Tese functions are illustrated in Figure 1. Te idea for te construction of û (k+1) is ten very simple. Since û (k+1) is an approximation of u and g (k) is an approximation of its derivative, ten te derivative of order k + 1 of û (k+1) sould coincide wit te derivative of order k of g (k) d k+1 û(k+1) k+1 = dk k g(k). Since u (k) is a polynomial of order k, it disappears from te left-and side and we get d k+1 û(k+1) k+1 = c (k+1) Te unknown coefficient can now be evaluated as c (k+1) = d k+1 φ(k+1) k+1. d k g (k) k d k+1 φ (k+1) k+1 In te linear and te quadratic cases, simple computations give (7) c (2) were g (k) i = 8 (g(1) i+1 g(1) i ) and c (3) = 12 (g(2) i+1 2g(2) m + g (2) i ), denotes te nodal values of te recovered derivative at node i and g (k) m is te estimated derivative at te mid-element node x m. We now ave our error estimator..

6 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 413 x i x m x i+1 φ (2) (x) x i x m x i+1 φ (3) (x) Figure 1. Quadratic (left) and cubic (rigt) basis functions φ (k) k (x) on te element [x i, x i+1 ] In te adaptation process, te L 2 -norm and te H 1 -seminorm of te estimated error will be needed. A simple calculation gives xi+1 ( ) 2 (8) ũ (2) u(1) 2 0, = c (2) φ(2) (x) 3 = 120 (g(1) i+1 g(1) i ) 2 and xi+1 ( (9) ũ (2) u(1) 2 1, = x i for te linear case wile one gets (10) ũ (3) and (11) ũ (3) u(2) 2 0, = x i xi+1 x i xi+1 ( u(2) 2 1, = x i c (2) d φ(2) (x) ) 2 = 12 (g(1) i+1 g(1) i ) 2 ( ) 2 c (3) φ(3) (x) 3 = 1890 (g(2) i+1 2g(2) m + g (2) i ) 2 c (3) in te quadratic case. Note tat u (k) d φ(3) (x) ) 2 = 45 (g(2) i+1 2g(2) m + g (2) i ) 2 is not needed in te estimation of te error Interpretation of te error estimator. Te error is tus estimated as u u (k) ũ(k+1) u (k) = c(k+1) φ (k+1) In te linear case, recalling tat g (1) is a piecewise linear approximation of u (x), we ave from (7) and (6) u(x) u (1) (x) c(2) φ(2) (x) = 1 (g (1) i+1 2 g(1) i )(x x i )(x i+1 x), = 1 (g (1) i+1 g(1) i ) (x x i )(x x i+1 ), 2 (x). 1 2 u (x m )(x x i )(x x i+1 ) + O( 2 ). Similarly, g (2) (x) is now a piecewise quadratic approximation of u (x) and u(x) u (2) (x) c(3) φ(3) (x) 1 3! u (x m )(x x i )(x x m )(x x i+1 ) + O( 3 ) in te quadratic case. Te error estimator on te element is noting but an approximation of te classical Lagrange interpolation error.

7 414 A. FORTIN, J. URQUIZA AND R. BOIS 4. Mes adaptation Te global strategy driving our mes adaptation is to try to reac a target level e Ω of error in L 2 -norm. Oter norms can also be considered but our experience sows tat te L 2 -norm provides te best results. More specifically, we would like to ave β α (u u (k) )2 = e 2 Ω. We would also like to obtain some form of equidistribution of tis error in te sense tat it sould be constant u u (k) = c everywere in te domain. Using te L 2 -norm, tis means tat e 2 Ω = β α u u (k) 2 = c 2 (β α) and tus c 2 = e2 Ω (β α). Tis also implies tat on eac element = [x i, x i+1 ], te local error e sould satisfy e 2 = xi+1 x i u u (k) 2 = e2 Ω (b a). If tis target value can be reaced on eac element, ten te global error e Ω will be attained on te wole interval [α, β]. Te mes will terefore be modified in order to acieve tis goal. To obtain a new mes, only local operations on te mes (node displacement, node elimination and element refinement) are used. Element refinement and node elimination are used to reac te target level of error e Ω. Node displacement is used to minimize te L 2 -norm of te derivative of te error (te energy norm). Minimizing te norm of te derivative leads to an equidistribution of te error in te sense tat if te norm of te derivative of te error is small, tis means tat te error is essentially constant. Moreover, tis also leads to optimal position of te nodes as described in D Azevedo and Simpson [17]. We end up wit a mes minimizing te error per unit lengt. In practice, te true error is obviously replaced by te estimated error described in Section 3.2. We now describe te local operations on te mes. We will present te situation in te linear case (k = 1) and we will terefore use formula (8) and (9) for te L 2 -norm and H 1 -seminorm of te estimated errors. In te quadratic case, tese formulas are replaced by (10) and (11) respectively Node elimination. Node elimination is applied only to nodes at te boundary between two elements. It is not applied to nodes inside an element suc as te mid-element node in a quadratic discretization. Te same is true for node displacement. For eac node of te mes, we consider te two adjacent elements: i 1 i x i 1 x i x i+1 If te sum of te estimated errors is smaller tan te sum of te local target errors 3 i g(1) i g (1) i i 120 g(1) i+1 g(1) i 2 < e2 Ω β α ( i 1 + i ), te node is removed and a new element is created (te elements sould be renumbered)

8 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 415 x i 1 x i+1 Oterwise, noting is done and we move on to te next node. In practice, te nodes of te mes are swept many times until no node is eliminated or until te maximum number of iterations is attained Element division. On an element i : i x i x i+1 if te estimated error is larger tan te local target error 3 i 120 g(1) i+1 g(1) i 2 > e2 Ω β α i, ten a new node is created and tus te element is divided into two new elements 1 and 2 (te elements sould be renumbered): 1 2 x N+1 x i x i+1 Oterwise, noting is done and we move on to te next element. At te new node, te nodal value g (k) N+1 is obtained by interpolation using g(k) (x). In practice, te elements of te mes are swept many times until no node is created or until te maximum number of iterations is attained Node displacement. For eac interior node x i, we consider once again te patc consisting of two adjacent elements: i 1 i x i 1 x i x i+1 Te node is moved in order to minimize te H 1 -seminorm of te error on te patc ( i 1 min g (1) x i g (1) i 1 i ) i 12 g(1) i+1 g(1) i 2, using an optimal descent or a Fibonacci metod for instance. i 1 i x i 1 x i x i+1 At te new position, te nodal value of g (k) i is simply interpolated using g (k) (x). Here again, te nodes of te mes are swept many times until no node is moved or until te maximum number of iterations is reaced. Te above local operations on te mes are systematically used in te euristic Algoritm 4.1. Step 3, in particular, can take many forms depending on te order cosen to perform te different local operations on te mes. In practice, we start wit element division followed by node displacement and ten, node elimination, followed by anoter node displacement. For eac operation, te nodes (or te elements) are swept many times until tere is no more cange in te mes. Te starting point is, unless oterwise specified, a more or less refined uniform mes. Obviously, starting from two different initial meses will result in very similar meses but not exactly te same ones. Despite tis, te algoritm seems very robust as will be demonstrated in te following section.

9 416 A. FORTIN, J. URQUIZA AND R. BOIS Algoritm 4.1 GLOBAL MESH ADAPTATION INPUT: Initial mes M 0, maximum number of iterations max_it, e Ω. OUTPUT: Final mes and computed solution. i = 0; wile Adaptation process modified te mes and i max_it do STEP 1 : Solve 1 to obtain a solution u (k) on mes M i ; STEP 2 : Use te derivative recovery metod to compute g (k) on mes M i ; STEP 3 : Adapt te mes M i using local operations to obtain M i+1 ; Minimize te H 1 -seminorm of te error using node displacement (error equidistribution and element optimality); Control te L 2 -norm of te error (or te number of elements) using element refinement and node elimination; STEP 4 : i i + 1; end wile 5. Numerical experiments We conducted numerical experiments in order to assess te performance of our adaptive metod. Te global strategy is presented in Algoritm 4.1. First, we sow tat optimal convergence rates can be obtained as a function of te number of nodes (see Equation 4). More difficult problems suc as singularly perturbed problems as described in Franz and Roos [9] are ten tested Convergence rates. We start wit a rater simple problem in order to verify tat our adaptive metod results in a significant gain in accuracy wit respect to uniform meses and tat te convergence rates are tose expected from te teory. We consider te problem (12) u (x) = 8ar4 x 2 (1 + r 2 x 2 ) 3 + 2ar 2 (1 + r 2 x 2 ) 2, in te interval [ 5, 5], wit a = 1 and r = 10. Te analytical solution (see Figure 2) is a u(x) = (1 + r 2 x 2 ), were te parameter a gives te maximum value of u wile r controls te steepness of u in te neigborood of te origin. A large value of r gives a steeper variation of u in a narrower interval around x = 0. In te numerical results, we set a = 1 and r = 10. Appropriate Diriclet boundary conditions are applied at bot ends of te interval. A standard Galerkin formulation was used in tat case (no SUPG). We first use uniform meses and te results are summarized in Table 1 for different norms as a function of te number of DOFs. Te L 2 -norm and H 1 - seminorm are computed as usual wile for te L -norm, eac element is subdivided into 10 subintervals and te maximum error is evaluated on tis partition i.e. not only at te computational nodes. Te number of DOFs is also indicated. Recall tat for quadratic elements, tis number includes mid-element nodes and is basically twice te number of elements. For te linear approximation u (1), quadratic and linear convergence rates are observed in L 2 -norm and H 1 -seminorm respectively wile cubic and quadratic rates are observed for te quadratic solution u (2). We ave also indicated te L -norm to present a more complete picture.

10 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 417 Table 1. Equation 12: Error in different norms for uniform meses Linear solution (k = 1) Exact Exact Number of L 2 norm H 1 -seminorm L -norm DOFs Quadratic solution (k = 2) For our adaptive metod, according to Algoritm 4.1, we give as input a target error e Ω and a maximum number of iterations max_it = 10. Te initial mes M 0 is a 100 element uniform mes. Table 2 presents te evolution of te errors for te adapted meses. Te target error e Ω and te final estimated error in L 2 -norm are also indicated. Clearly, as te number of DOFs increases, te estimated and exact errors become very close sowing te efficiency of te estimator. Figure 3 compares tese L 2 -errors for bot uniform and adapted meses on a log-log scale. Clearly, te adapted meses presents te same (optimal) rates of convergence as te uniform meses but wit an error level almost two orders of magnitude smaller in te linear case and tree in te quadratic case. Important gains are also observed for te H 1 -seminorm. Tese differences can be made even more spectacular by increasing te value of te parameter r in te problem definition. As an example, te numerical solutions (linear and quadratic) are illustrated in Figure 4 for a target error e Ω = As expected, for te same target error, te quadratic solution requires a muc smaller number of DOFs (81 instead of 171), tat is 40 quadratic elements versus 170 linear elements Finding a needle in a aystack. We consider te problem (13) u (x) = sin(x) 8ar4 (x π 2 )2 (1 + r 2 (x π 2 )2 ) 3 + 2ar 2 (1 + r 2 (x π 2 )2 ) 2, wose solution is (14) u(x) = sin(x) + a (1 + r 2 (x π 2 )2 ). Here again, a standard Galerkin metod is used. We will work in te interval [0, π] wit Diriclet boundary conditions provided by (14). We will also set a = and r = suc tat te solution is a sine function plus a very small amplitude

11 418 A. FORTIN, J. URQUIZA AND R. BOIS Figure 2. Equation 12: Analytical solution u(x) = x 2 Figure 3. Equation 12: adapted meses Convergence rates for uniform and Figure 4. Equation 12: Numerical solutions wit linear and quadratic adapted meses (e Ω = 10 4 )

12 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 419 Table 2. Equation 12: Error in different norms for adapted meses Linear adaptation (k = 1) Target Estimated Exact Exact Number of L 2 -norm L 2 -norm L 2 norm H 1 -seminorm L -norm DOFs Quadratic adaptation (k = 2) perturbation centered at x = π/2 (barely visible) of te exact same form as in Section 5.1. Taking r = forces te perturbation to take te form of a needle. As can be expected, uniform meses ave some difficulties capturing tis solution. Te particular form of te rigt-and side in Equation 13 introduces a very strong perturbation tat cannot be properly captured unless te mes is very fine in te neigborood of x = π/2. Figure 5 presents different intermediate solutions in comparison wit te analytical solution togeter wit te respective number of DOFs. Note tat te perturbation in te solution (see te second term in Equation 14) is visible only on te finest mes. It terefore takes uniform quadratic elements ( DOFs) to correctly represent te solution. Some of te intermediate solutions are completely wrong. Tis problem can be seen as aving two scales since te L 2 -norm of te solution is 1.25 wile te L 2 -norm of te perturbation is around It is terefore necessary to use a target value e Ω for te L 2 -norm of te error muc smaller tan Te iterative process was started wit a 100 element uniform mes and tus te size of te elements ( = π/100) is muc larger tan te widt of te perturbation. Using e Ω = 10 7 produces a solution were te perturbation is clearly visible wit only 223 DOFs (not illustrated). Figure 6 presents te results for e Ω = 10 9 resulting in a 501 element mes (1003 DOFs). Te L 2 -norm of te error on tis adapted mes is similar to te most refined uniform mes. For a similar accuracy, te adapted mes requires 64 times less DOFs. It is wort mentioning tat our adaptive metod is also an iterative metod wose unknowns are te nodes of te mes. Convergence is terefore not guaranteed. Interestingly, in tis example te intermediate adapted meses also passes troug completely erroneous intermediate solutions similar to tose in Figure 5 but ends up wit a very good solution. Tis sows some robustness in te iterative process Failure of te SUPG metod. We consider te following problem (see [9]) ( (15) ɛu (x) + x 1 ) u (x) = x 1 2 2,

13 420 A. FORTIN, J. URQUIZA AND R. BOIS 2000 DOFs 4000 DOFs 8000 DOFs DOFs DOFs DOFs Figure 5. Equation 13: Analytical (red) and quadratic numerical solutions (blue) on uniform meses wit u(0) = 1/2 and u(1) = 1/2. Te solution is simply u(x) = x 1 2. Note tat te exact solution belongs to te discrete finite element space and terefore, we sould expect to obtain te exact solution wen solving. However, te cange of sign of te coefficient of u (x) (te convection term) forces te development of artificial boundary layers at bot ends of te domain. Wen te number of elements is small (< 20) bot Galerkin and SUPG metods give more or less te exact solution. Te error is owever larger for te SUPG metod. Increasing te number of elements, one gets erroneous solutions suc as

14 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS DOFs 1003 DOFs Figure 6. Equation 13: Analytical (red) and quadratic numerical solutions (blue) on te adapted mes Table 3. Equation 15: L 2 -norm of te error on uniform meses u u 0 DOFs Galerkin SUPG te one presented in Figure 7 wit 32 elements (65 DOFs) obtained using te SUPG metod. Te numerical solution is essentially linear but artificial boundary layers appears at bot extremities. Bot metods (Galerkin and SUPG) ardly converge as can be seen in Table 3. Tis is terefore a case were bot Galerkin and SUPG metods fail. As explained in [9], tis beavior is due to te ill-conditioning of te corresponding linear systems. For a small number of elements, te conditioning of te Galerkin system is sligtly better but as te number of elements increases, bot systems are equally ill-conditioned and tey give similar results. Fortunately, our mes adaptation corrects te situation. Starting from te erroneous solution illustrated in Figure 7, it first removes all unnecessary nodes were te numerical solution is linear tus decreasing te number of elements and te condition number of te matrix. It ten provides te exact solution (wit only one element!) in one adaptation step. In tis specific case, mes adaptation serves as a andrail for te numerical metod.

15 422 A. FORTIN, J. URQUIZA AND R. BOIS Figure 7. Equation 15: Analytical (red) and quadratic numerical solution (blue) wit 32 quadratic elements (65 DOFs) 5.4. Singularly perturbed problem. Our next example was also proposed in [9] and takes te form (16) ɛu (x) u (x) = 2ɛ 2x, wit u(0) = u(1) = 0. Te solution is u(x) = x exp( x/ɛ) exp( 1/ɛ). 1 exp( 1/ɛ) We will consider te case were ɛ = 10 8 for wic te solution presents a very steep variation at x = 0. Tis is a classical problem wit a boundary layer and were te Galerkin formulation fails and some stabilization metod suc as SUPG is necessary. Figure 8 (top) presents a solution on a quadratically adapted mes wit 393 elements (787 DOFs) obtained using a target error e Ω = Te exact L 2 -norm of te error in tis particular case is and te L -norm of te error is To obtain a comparable accuracy, and-adapted meses suc as tose proposed in Siskin [18] wit more tan 10 5 nodes was necessary in [9]. In te boundary layer, te lengt of te elements varies from to It ten rapidly increases up to te last element wose lengt is Suc a large element is made possible by te fact tat te analytical solution is essentially parabolic apart from te boundary layer and our quadratic finite element approximation can represent it almost exactly wit a single element. We ave also illustrated in Figure 8, te first and tird adapted meses. Te first adapted mes does not capture te boundary layer correctly and presents a large number of nodes in te parabolic region. Te tird intermediate mes is already very satisfactory and afterward, te adaptation metod does not cange te mes significantly. Our adaptive metod automatically provides a better mes tan tose proposed by Siskin [18] and tis, witout any a priori knowledge of te solution or te position of te boundary layer. Te convergence rates in te different norms are presented in Figures 9 and 10 sowing optimal rates of convergence in all cases. Similar results are obtained (but not sown) for linear adaptation.

16 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 423 Final mes wit 393 elements (787 DOFs) First intermediate adapted mes Tird intermediate adapted mes Figure 8. Equation 16: Analytical (red) and quadratic numerical solutions (blue) on different meses Figure 9. Equation 16: Convergence in L -norm.

17 424 A. FORTIN, J. URQUIZA AND R. BOIS Figure 10. Equation 16: Convergence in L 2 -norm and H 1 -seminorm A discontinuous solution. We now consider a first order equation: (17) xu (x) = xe x, wit u( 1) = 0 and u(1) = 0. Te solution is caracterized by a discontinuity (a sock) at x = 0: { e u(x) = 1 e x if x < 0, e 1 e x if x > 0. Using a SUPG formulation, we ave obtained te piecewise quadratic solution illustrated on top of Fig 11 wit 691 elements (1383 DOFs). Te solution is not bad at all but small amplitude oscillations are still visible at te discontinuity. Te L 2 norm of te error is around since we used a target error e Ω = Te fact tat our mes adaptation metod does not give entirely satisfactory results sould not be a surprise. Recall tat te metod is based on a reconstruction of te derivative of te numerical solution, wic in tis particular case, does not exist at te discontinuity. Te results are neverteless surprisingly good. We now consider a sligtly perturbed problem of te form (18) ɛu (x) xu (x) = xe x, wit a very small value for ɛ (10 15 ). Te solution of tis perturbed problem is now continuous and differentiable, toug a very steep variation is present at x = 0. Te results are presented in Figure 11. Here again, tese results are very satisfactory taking into account tat computing a discontinuous solution witin a space of continuous functions is a difficult callenge. As stated in [9], a discontinuous Galerkin metod is more likely to give a better solution at te express condition tat a node is provided at te discontinuity. Tis again requires an a priori knowledge of te solution. Our adaptive metod provides a very good compromise and does not necessitate suc information Exponentially ill-conditioned problem. Our last problem is a very difficult test case: (19) ɛu (x) + xu (x) = 0, wit u( 1) = 3 and u(1) = 1. Te exact solution is u(x) = x 0 es2 /2ɛ ds 1 0 es2 /2ɛ ds.

18 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 425 ɛ = 0 ɛ = Figure 11. Equation 18: Approximation of a discontinuous solution Te simplicity of te differential equation is misleading. As can be seen, even evaluating wit sufficient accuracy te integrals appearing in te exact solution is not a trivial task. For tis reason, we will present te H 1 -seminorm of te error wic requires only te computation of te derivative u (x) = 2ex2 /2ɛ 1 0 es2 /2ɛ ds. Te difficulty of te problem was explained in O Malley and Ward [19] were it is sown tat te continuous problem as an asymptotically exponentially small leading eigenvalue beaving as 2 λ 0 πɛ e 1/2ɛ. To give an idea, for ɛ = 0.04, 0.03, 0.02 and 0.01, tis gives respectively λ , 10 7, and Obviously, te associated discrete problem inerits tis property making its numerical solution extremely ill-conditioned. We first start wit uniform meses. Te analytical solution presents boundary layers at bot x = 1 and x = 1 and uniform meses are not likely to perform well. As can be seen in Table 4, quadratic convergence is clearly obtained for ɛ = 0.04 and for ɛ = 0.03 up to 3200 DOFs. For ɛ = 0.02 we get non optimal convergence and only for small numbers of DOFs. Finally, we were not able to obtain a valid

19 426 A. FORTIN, J. URQUIZA AND R. BOIS Table 4. Equation 19: Error in different norms for uniform meses u u 1 Number of ɛ = 0.04 ɛ = 0.03 ɛ = 0.02 DOFs solution in te case ɛ = Tese results are similar to wat was observed in Sun [20]. Tis problem was really a test for our mes adaptation metod. Contrarily to all te preceeding tests, convergence was not easily obtained and was linked to te smallest eigenvalue of te problem. Indeed, te target value e Ω given as an input to our metod ad to be cosen smaller tan λ 0 to obtain convergence. For example, in te case ɛ = 0.04 (λ ), no convergence was possible unless e Ω was cosen smaller tan Optimal convergence rate was ten obtained as can be seen in Figure 12. For a target error as small as 10 12, te L 2 -norm of te error is so small on eac element tat we are scratcing te limits of macine precision available. An optimal convergence rate was also observed for ɛ = 0.03 (λ ) but for a narrower range of values of te number of DOFs using a target error between 10 8 and Finally, te situation is not very clear for ɛ = 0.02 (λ ) were we ad to use a target error between and to get a solution. We are already flirting wit te limits of double digits accuracy. Finally, despite numerous efforts, we were not able to get a correct solution for ɛ = Numerical solutions could be obtained but were far from te exact solution and were simply discarded. If our ypotesis is correct, quadruple precision would be necessary to get a correct beaviour. Tis was also te conclusion in [20]. We ave definitely reaced te limits of our adaptive metod wit respect to macine precision at our disposal. 6. Conclusions In tis work, we ave introduced a novel mes adaptive strategy in te onedimensional case. Te metod was tested on various problems to illustrate its potential but also to underline some of its limitations. In particular, te metod was sown to perform very well for boundary layer problems. Te overall iterative metod wose unknowns are te mes nodes also sows a certain robustness in te sense tat te final adapted mes does not strongly depend on te initial mes and tat, even wen starting wit a clearly inappropriate mes, we can get a satisfactory adapted mes. Te proposed adaptive metod is also implemented in two and tree dimensions (see [1, 2]) and we are currently investigating if similar results can be obtained for iger dimensional boundary layer problems.

20 MESH ADAPTATION FOR 1D BOUNDARY LAYER PROBLEMS 427 ɛ = 0.04 ɛ = 0.03 ɛ = 0.02 Figure 12. Equation 19: Convergence in H 1 -seminorm for te adapted meses. Acknowledgments Te autors wis to acknowledge te financial support of te Natural Sciences and Engineering Researc Council of Canada (NSERC). References [1] R. Bois, M. Fortin, and A. Fortin. A fully optimal anisotropic mes adaptation metod based on a ierarcical error estimator. Computer Metods in Applied Mecanics and Engineering, :12 27, february [2] R. Bois, M. Fortin, A. Fortin, and A. Couët. Hig order optimal anisotropic mes adaptation using ierarcical elements. European Journal of Computational Mecanics/Revue Européenne de Mécanique Numérique, 21(1-2):72 91, [3] F. Alauzet. Hig-order metods and mes adaptation for euler equations. International Journal for Numerical Metods in Fluids, 56(8): , [4] Y. Belamadia, A. Fortin, and É. Camberland. Anisotropic mes adaptation for te solution of te Stefan problem. Journal of Computational Pysics, 194(1): , [5] Y. Belamadia, A. Fortin, and É Camberland. Tree-dimensional anisotropic mes adaptation for pase cange problems. Journal of Computational Pysics, 201(2): , [6] W. G. Habasi, J. Dompierre, Y. Bourgault, D. Ait Ali Yaia, M. Fortin, and M.-G. Vallet. Anisotropic mes adaptation: towards user-independent, mes-independent and solverindependent CFD. Part I: General principles. International Journal for Numerical Metods in Fluids, 32: , 2000.

21 428 A. FORTIN, J. URQUIZA AND R. BOIS [7] M. Hect and B. Moammadi. Mes adaptation by metric control for multi-scale penomena and turbulence. In 35t Aerospace Sciences Meeting & Exibit, number , Reno, USA, [8] I. Babuska and W.C. Reinboldt. A-posteriori error estimates for te finite element metod. International Journal for Numerical Metods in Engineering, 12: , [9] S. Franz and H.-G. Roos. Te capriciousness of numerical metods for singular perturbations. SIAM review, 53(1): , [10] T. J. R. Huges and A. N. Brooks. A multidimensional upwind sceme wit no crosswind diffusion, volume 34 of Finite Element Metods for Convection Dominated Flows, pages Amer. Soc. of Mec. Eng., New York, [11] T. J. R. Huges and A. N. Brooks. A teoretical framework for Petrov-Galerkin metods wit discontinuous weigting functions. Applications to te streamline upwind procedure. In R. H. Gallager, editor, Finite element in fluids, volume IV. Wiley, London, [12] H.-G. Roos. Stabilized FEM for convection-diffusion problems on layer-adapted meses. Journal of Computational Matematics, 27: , [13] H.-G. Roos, M. Stynes, and L. Tobiska. Robust numerical metods for singularly perturbed differential equations, volume 24 of Springer Series in Computational Matematics. Springer- Verlag, Berlin, second edition, Convection-diffusion-reaction and flow problems. [14] G. Zou. How accurate is te streamline diffusion finite element metod? Matematics of Computation, 66(217):31 44, [15] Z. Zang and A. Naga. A new finite element gradient recovery metod: Superconvergence property. SIAM Journal for Scientific Computing, 26(4): , [16] R. E. Bank and. R. Smit. A posteriori error estimates based on ierarcical bases. SIAM Journal on Numerical Analysis, 30(4): , [17] E.F. D Azevedo and R.B. Simpson. On optimal triangular meses for minimizing te gradient error. Numerisce Matematik, 59(1): , [18] G. Siskin and L. Siskina. Difference metods for singularly perturbed problems. Capman and Hall, Boca Raton, FL, [19] R.E. O Malley and M. J. Ward. Exponential asymptotics, boundary layer resonance, and dynamic metastability. In L. P. Cook et al. and V. Roytburd & M. Tulin, editors, Matematics is for solving problems, pages SIAM publication, [20] X. Sun. Numerical analysis of an exponentially ill-conditioned boundary value problem wit applications to metastable problems. IMA Journal of Numerical Analysis, 21: , Département de matématiques et de statistique, 1045 avenue de la médecine, Université Laval, Québec, Canada, G1V 0A6, afortin@giref.ulaval.ca URL: ttp://