CONVERGENCE ANALYSIS OF FINITE ELEMENT SOLUTION OF ONE-DIMENSIONAL SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS ON EQUIDISTRIBUTING MESHES

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume, Number, Pages c 5 Institute for Scientific Computing and Information CONVERGENCE ANALYSIS OF FINITE ELEMENT SOLUTION OF ONE-DIMENSIONAL SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS ON EQUIDISTRIBUTING MESHES WEIZHANG HUANG Communicated by Zimin Zang Abstract In tis paper convergence on equidistributing meses is investigated Equidistributing meses, or more generally approximate equidistributing meses, are constructed troug te well-known equidistribution principle and a so-called adaptation or monitor function wic is defined based on estimates on interpolation error for polynomial preserving operators Detailed convergence analysis is given for finite element solution of singularly perturbed two-point boundary value problems witout turning points Illustrative numerical results are given for a convection-diffusion problem and a reaction-diffusion problem Key Words Mes adaptation, equidistribution, error analysis, finite element metod Introduction Te concept of equidistribution as been used for long for adaptive mes generation It is first used by Burcard [7] and ten by a number of researcers cf te early works [, 3, 5, 4, 7] for te error analysis of best spline approximations wit variable knots An algoritm, now known as de Boor s algoritm, is introduced by de Boor [4] for computing equidistributed meses Russell and Cristiansen [3] give an early review on mes selection strategies based on equidistribution, and one of suc strategies is implemented in te general-purpose code COLSYS by Ascer, Cristiansen, and Russell [] Te equidistribution principle as also been playing an important role in multi-dimensional adaptive mes generation Te concept can naturally be incorporated into te variational mes generation framework, and a number of metods ave been developed along tis line, eg, see [6, 8, 9, 6, 7, 9, ] Convergence analysis for te numerical solution of partial differential equations PDEs using equidistributing meses can be traced back to works in te seventies of te last century For example, Pereyra and Sewell [7] give an asymptotical bound for te truncation error wen finite differences are used for solving two-point boundary value problems on an equidistributing mes Babu ska and Reinboldt [] obtain a posteriori error estimates for finite element solutions for one dimensional Received by te editors January, 4 and, in revised form, Marc, 4 Matematics Subect Classification 65M5, 65M6, 65L5, 65L6 Tis work was supported in part by te NSF under grants DMS-744 and DMS-4545 and by te University of Kansas General Researc Fund under grant #

2 58 W HUANG problems in an asymptotic form for te size of elements going to zero Tey sow tat a mes is asymptotically optimal if all error indicators on subintervals are equal and tus te mes is equidistributing Recent progress as been made on matematically more rigorous convergence analysis Notably, Qiu and Sloan [8] and Qiu, Sloan, and Tang [9] investigate te uniform convergence of upwind finite difference approximations to a singularly perturbed problem Beckett and Mackenzie study te convergence of finite difference approximations to convection-diffusion problems witout turning points and reaction-diffusion problems in [3, 4, 3] and finite element approximations to reaction-diffusion problems in [5] Te meses considered in tese works are eiter cosen a prior or determined troug te equidistribution relation based on te explicit expression of te exact solution Te stability and convergence of te finite element solution to one-dimensional convection-dominated problems are studied by Cen and Xu [] for some a prior cosen meses Te convergence analysis to a fully discrete problem were meses are determined completely by te computed solution is recently presented by Kopteva and Stynes [] for te upwind finite difference discretization of a quasi-linear one-dimensional convection-diffusion problem witout turning points Te obective of tis paper is to study te convergence of finite element solution to one-dimensional singularly perturbed PDEs on equidistributing meses We attempt to develop a general teory for use in convergence analysis Our approac is different from tose used in te existing works Specifically, following [8, ] we first investigate interpolation error for polynomial preserving operators on a general mes Several mes quality measures are defined, and estimates for interpolation error are obtained in terms of tese mes quality measures An equididistributing mes wic satisfies te equidistribution principle, or more generally, an approximate equididistributing mes wic satisfies te equidistribution principle only approximately, is caracterized as a mes wit a bounded overall quality measure see 6 as it is refined Te interpolation error estimates are ten used to analyze te convergence of te standard finite element solution to singularly perturbed PDEs on approximate equidistributing meses Te analysis is carried out for two separate cases, te convection-diffusion case and te reaction-diffusion one To our best knowledge, tis is te first work on te convergence of standard finite element solution of one-dimensional convection-diffusion problems on equidistributing meses wile an analysis for one-dimensional reaction-diffusion problems is given by Beckett and Mackenzie [5] It is empasized tat unlike te existing approaces, our analysis does not use an a prior cosen mes nor requires te mes be given troug te equidistribution principle wit an analytical expression of te exact solution Wat we require is tat te mes satisfies te equidistribution relation approximately, ie, 6 or 39 and 4 Numerical results presented in Section 4 sow tat suc a mes can be obtained using De Boor s algoritm [4] An outline of te paper is as follows In section, we study approximation properties of polynomial preserving operators on a general mes and define several mes quality measures In Section 3, te results of Section are applied to te convergence analysis of te finite element solution to singularly perturbed boundary value problems witout turning points Convection-diffusion and reaction-diffusion equations are covered in Subsections 3 and 3, respectively Numerical results are presented in Section 4 Finally, Section 5 contains te conclusions Trougout te paper, we use C as a generic constant wic may take different values at different occurrences

3 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES 59 Error estimates for polynomial preserving operators In tis section we first study approximation properties of polynomial preserving operators on a general mes and ten define te mes quality measures according to te obtained results Our approac is similar to tat used in [8] were mes quality measures are defined for a more general situation in multi-dimensions Interpolation error estimates We start wit introducing some notation For a general open set D of R, we denote te norm and semi-norm of te Sobolev space H m D by m,d and m,d, respectively Te scaled semi-h m norm will also be used, / v m,d = v m dx v H m D, D D were D is te lengt of D Note tat v m,d represents te L average of v m Denote by P k D te space of polynomials of degree no more tan k defined on D Consider a general mes x = a < x < < x N = b on te interval Ω a, b Let = x x, = max, I = x, x For a given integer k, we consider a polynomial preserving operator Π k defined on H k+ Ω: for =,, N, Π k I v = v v P k I, were Π k I is te restriction of Π k on I We assume tat Π k is linear and continuous and as te following approximation property: for an integer m: m k +, v Π k v m,i C q+ V k+ I v H k+ Ω, were q = k + m, C is a constant independent of v and N, and V k+ is a maorizing function for v k+, viz, 3 v k+ x C V k+ x x Ω for some positive constant C Obviously, v k+ is a natural coice for V k+ But it is empasized tat our development also works for oter coices of te maorizing function Te approximation property is te key assumption to our analysis Te assumption olds for most commonly-used polynomial preserving operators For example, consider a Taylor interpolation operator defined by k Π k v I x = v i x x x i x I, =,, N i= for v H k+ Ω Using Taylor s teorem, one can readily sow tat olds for all integers m k + Anoter important example is te interpolation operator associated wit an affine family of finite elements on Ω Let Î be te reference finite element, ˆPÎ be te space of finite element approximations on Î, and s te greatest order of

4 6 W HUANG derivatives occurring in te definition of ˆPÎ It is known eg, see [] tat olds for integers k and m k + satisfying 4 H k+ Î Cs Î and P kî ˆPÎ H m Î, s were denotes te inclusion wit continuous inection and C Î is te space of functions s times continuously differentiable on Î Particularly, for an affine family of Lagrange finite elements, 4 and terefore are satisfied for all integers k and m k + Te equidistribution principle and mes quality measures We now define te quality measure for mes adaptation using te approac of [8] Te definition is based on te well-known equidistribution principle [7] wic evenly distributes a so-called adaptation or monitor function over all te subintervals between te mes nodes To be able to define a strictly positive adaptation function, we regularize te rigt-and side term of wit a to-be-determined positive parameter α wic is referred to as te intensity parameter, ie, 5 v Π k v m,i C q+ α + V k+ I = C α q+ + V k+ I α Following [8], we define te adaptation function and te intensity parameter α as 6 7 ρ = α = + b a γ V k+ I α V k+ γ I γ =,, N, for some number γ, ] As sown in [], te optimal value wic yields te smallest error bound; cf Teorem below is γ = /q + wen te error is measured in te H m semi-norm We consider a general value of γ so tat we can deal wit more complicated norms occurring in te convergence analysis of finite element approximations to differential equations; see te next section For tis adaptation function, te equidistribution principle reads as 8 ρ = σ N =,, N, were 9 σ = ρ

5 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES 6 Wit α being defined as in 7, it can be sown tat σ b a Indeed, from Jensen s inequality it follows γ σ = + α V k+ I + α γ V k+ γ I = b a + α γ = b a V k+ γ I In practice, it is more realistic to assume tat a computed mes satisfying 8 only approximately To measure te effect of tis approximation, we define te adaptation quality measure as Q adp I = N ρ =,, N σ It is not difficult to see tat max Q adp I = implies tat te mes satisfies 8 exactly Te larger max Q adp I, te farter away te mes is from satisfying 8 We now rewrite te bound on interpolation error in terms of Q adp From 5, 6,, and, we ave N v Π k v m,i Cα q ρ γ N = Cα ρ Qadp I σ ρ N q ρ = Cα σ q N q ρ Q q adp I ρ γ q Cα N q Q mes,m max ρ γ q, γ were σ b a as been used in te last step and Q mes,m, te overall mes quality measure, is defined as 3 Q mes,m = N ρ ρ Q q adp I = σ ρ Q q adp I Te letter m in te subscript is used to indicate te dependence of te definition on m It is remarked tat wen /γ q or γ /q +, we ave and reduces to ρ /γ q v Π k v m,i Cα N q Q mes,m Tese results are summarized in te following teorem Teorem Given an integer k, let Π k be a polynomial preserving operator defined on a general mes and aving te approximation property For

6 6 W HUANG any integer m: m k + and any number γ,, assume tat te adaptation function ρ, te intensity parameter α, and te overall mes quality measure Q mes,m are defined in 6, 7, and 3, respectively Ten te interpolation error is bounded by 4 v Π k v m,i Cα N q Q mes,m max ρ γ q, were q = k + m If γ γ /q + = /k + m +, 5 v Π k v m,i Cα N q Q mes,m It can be sown eg, see [] tat te optimal value for γ wic leads to a smallest error bound is γ = γ wen te error is measured in te semi-norm H m 3 Properties of approximate equidistributing meses A mes satisfying te equidistribution relation 8 is commonly called te equidistributing mes for te adaptation function 6 Naturally, we can refer to a mes satisfying a weaker constraint, 6 Q mes,m C for some constant C, as an approximate equidistrbuting mes Note tat wen C =, we ave Q mes,m = and terefore te mes is equidistributing We now study some properties of approximate equdistributing meses From and 3 it follows tat, for any =,, N, q q Q Nk ρ k N mes,mσ = k ρ k ρ q+ σ σ Tus, k= 7 ρ Q +q mes,m σ N Since σ b a and ρ, we ave 8 b aq +q Tis implies mes,m N q q +q +q =,, N 9 max as N for any approximate equididistributing mes Moreover, using 6 we can get from 7 V k+ γ I b aq +q mes,m α γ N q +q =,, N, wic proves to be useful in te convergence analysis of finite element approximations in te next section Property 9 as a significant implication tat qualities suc as α and Q mes,m converge to teir continuous counterparts as N Take α as an example It as te continuous form as α [ b a b a V k+ γ dx ] γ

7 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES 63 Recalling tat V k+ I is te L average of V k+ on I, from 7 we can bound α by te lower and upper Riemann sums, ie, γ min V γ b a x I k+ x α γ max V γ b a x I k+ x Tus, te property 9 and te Riemann integrability of V k+ imply tat α α as N Tus, by 5 and taking V k+ = v k+ we can see tat, for any function v satisfying b a vk+ γ dx <, te H m semi-norm of te interpolation error on an approximate equidistributing mes converges at a rate ON q = ON k+ m 3 Convergence analysis for finite element approximations In tis section we study te convergence of te finite element solution of singularly perturbed problems witout turning points on an approximate equidistributing mes satisfying 6 Our tools are Teorem and te mes properties 8 and We consider te general boundary value problem ɛu + bu + cu = f x,, u = u =, were < ɛ is te perturbation parameter and b, c, and f are given functions of x and ɛ We assume tat te coefficients are sufficiently smoot so tat all concerned derivatives exist and are bounded uniformly in bot ɛ and x We also assume tat 3 cx, ɛ b xx, ɛ β > x, and ɛ > For te case bx, ɛ b >, tis condition is not essential because te equation can be transformed into a differential equation satisfying 3 troug te cange of variables u = v expkx wit a proper value for K Define te bilinear operator a, in S H, = {v v H, and v = v = } as 4 au, v = ɛu, v + bu, v + cu, v, were, is te inner product of L, For te singularly perturbed problem and, it is common practice eg see [3] to use te ɛ-dependent norm 5 v ɛ ɛ v + v for convergence analysis It is easy to verify te coercive property 6 av, v β v ɛ v S, were β = min{, β } For a given integer k, we denote by S S a finite element space of degree k defined on te mes x = < x < < x N = We assume tat members of S are at least continuous on,, and piecewise polynomials of degree no more tan k form a subset of S It is known eg, see [] tat te interpolation operator associated wit S as te approximation property wit q = k m + for all integers m k + Te finite element solution u S to problem and is defined as 7 au, v = f, v v S

8 64 W HUANG Te error equation reads as 8 au u, v = v S, were u denotes te exact solution of and Let b = max bx, ɛ, c = max cx, ɛ x,ɛ x,ɛ Lemma 3 Te error in te finite element solution of degree k to problem and satisfies 9 u u ɛ C inf ɛ w u w S + b ɛ w u + c w u, were C is a constant independent of ɛ and N 3 Proof We first ave β u u ɛ by 6 au u, u u w S = au u, u w + au u, w u by 8 wit v=u w = au u, w u Te inequality ɛ u u, w u + bu u, w u + cu u, w u ab a + b will frequently be used in estimating te terms on te rigt-and side of te last inequality of 3 Let ζ be a positive number For te first term, Scwarz s inequality gives rise to ɛ u u, w u ɛζ / u u ɛ ζ / w u 3 For te second term, it follows ɛζ u u + ɛ ζ w u 3 bu u, w u ɛζ / u u Te last term can be estimated as b / w u ɛζ ɛζ u u + b ɛζ w u 33 cu u, w u ζ u u + c 4ζ w u Substituting 3, 3, and 33 into 3 gives rise to β u u ɛ ζ u u ɛ + ɛ ζ w u + b ɛζ w u + c 4ζ w u

9 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES 65 Taking ζ = β/ yields u u ɛ ɛ β w u + b ɛβ w u + c β w u Te conclusion of te lemma, 9, is obtained by taking te infimum over w S in te above inequality Tis lemma indicates tat te error in te finite element solution is dominated by te interpolation error in te H semi-norm and te L norm We recall tat on a uniform mes, te interpolation error in tese norms is bounded by u Π k u u Π k u C N k+ C N k u k+ dx, u k+ dx On te oter and, for an approximate equidistributing mes for te adaptation function 6 wit te intensity parameter given in 7 Teorem sows tat te optimal value for γ is γ = /k+ for te H semi-norm of te error m = and γ = /k + 3 for te L norm m = In te current situation, te ɛ-dependent norm ɛ involves bot H and L It is reasonable to expect tat te optimal value for γ for te ɛ-norm stays between /k + 3 and /k + Tus, we assume tat γ used in defining te adaptation function 6 is cosen between tese two values, viz, 36 By Teorem we ave k + 3 γ k + u Π k u Cα N k+ Q mes,, u Π k u Cα N k Q mes, max ρ p, were p = /γ k + and V k+ in te definitions of α, ρ, Q mes,, and Q mes, is a maorizing function for u k+ To estimate α and max ρ p, we consider two separate cases, te convectiondiffusion case wit bx, ɛ b > and te reaction-diffusion one wit bx, ɛ and cx, ɛ c > Te exact solution of problem and beaves differently in tese two cases We assume tat te approximate equidistributing mes satisfies Q mes, = 39 ρ Q k+ adp I C, σ 4 Q mes, = for some constants C and C σ ρ Q k adpi C 3 Convection-diffusion problems For tis case, te exact solution of convectiondiffusion problem and as te following property; eg see Roos, Stynes, and Tobiska [3] and O Malley [5]

10 66 W HUANG Lemma 3 Suppose tat 4 bx, ɛ b >, were b is a constant Ten, for i =,,, 4 u i C i [ + ɛ i exp b ] x, ɛ were C i is a constant dependent only on i It is trivial to get u k+ dx Cɛ k+ Hence, by Lemma 3, 34, and 35 te error in te finite element approximation on a uniform mes is bounded by 43 u u ɛ C Nɛ k + Nɛ For mes adaptation we take te singular part of u k+, ie, 44 V k+ = + ɛ k+ exp b x ɛ Note tat V k+ is monotone increasing We now derive an error bound on an approximate equidistributing mes We first estimate α Te monotonicity of V k+ and te assumption γ < from 36 give rise to 45 V k+ γ I V γ k+ x Define x by wic yields + ɛ γk+ exp γb ɛ x ɛ γk+ exp γb ɛ x =, 46 x = k + b ɛ ln ɛ Let [x, x be te interval containing x By 45, { 47 V k+ γ for x x I ɛ γk+ for x > x Te definition of α, 7, leads to b aα γ = V k+ γ I + V k+ γ I + N = + V k+ γ I Using 47 for te first and te last terms and wit m = for te second term on te rigt-and side, we get b aα γ + b aq k+3 mes, α γ N k+ k+3 + ɛ γk+ = + b a + b aq k+3 mes, α γ N k+ k+3 + ɛ γk+ x C + ɛ γk+ ln ɛ + b aq k+3 mes, α γ N k+ k+3

11 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES 67 If N is taken large enoug suc tat we ave Q k+3 mes, N k+ k+3 48 α C We now estimate te term max ρ p Using 47 and 48, it follows and ρ p Cα pγ or N 4 k+3 + ɛ γ γk+ ln ɛ γ ρ p Cα pγ k+ Q k+ mes,, in 38 By te definition of ρ, we ave α pγ + V k+ γp I ɛ γpk+ 49 α max ρ p C + ɛ k+ γk+ ln ɛ k+ ɛ γpk+ Substituting 48 and 49 into 37 and 38 and using 39 and 4 yields u Π k u C 5 + ɛ N k+ γ γk+ ln ɛ γ, u Π k u C 5 N k + ɛ k+ γk+ ln ɛ k+ ɛ γpk+ Combining tese results wit Lemma 3, we obtain te following teorem Teorem 3 Suppose tat bx, ɛ b > Let u be a finite element solution of degree k to te problem and i Wen a uniform mes is used, te error in u is bounded by 5 u u ɛ C Nɛ k + Nɛ, were C is a constant independent of N and ɛ ii Suppose tat {x } N = is an approximating equidistributing mes satisfying 39 and 4, were te adaptation function ρ, te intensity parameter α, and te maorizing function V k+ are given in 6, 7, and 44, respectively, togeter wit /k + 3 γ /k + If N 4 k+3 bounded as u u ɛ C [ N k N ɛ 53 + k+ Q k+ + ɛ γ γk+ ln ɛ γ mes,, ten te error in u is + ɛ k+ γk+ ln ɛ k+ ɛ k+ γk+], were C is a constant independent of N and ɛ It is instructive to see tat te error bound 53 becomes u u ɛ C [ ] N k ɛ k+ k+3 + N ɛ for γ = /k + 3, u u ɛ C ln ɛ k+ N k [ + ] ln ɛ N ɛ

12 68 W HUANG for γ = /k +, and u u ɛ C ln ɛ k+ N k [ + ] N ɛ for γ = /k + Te advantage of using an adaptive mes over a uniform one is clearly sown in tis teorem Indeed, 5 sows tat te error bound obtained wit a uniform mes depends strongly on ɛ, in te order of ɛ k On te oter and, te error bound wit an approximate equidistributing mes as muc weaker ɛ-dependence For example, te dependence is of order O ln ɛ k+ /N ɛ wen γ is taken as /k + 3 Reaction-diffusion problems For tis case, 54 bx, ɛ and cx, ɛ c >, and 9 reads as 55 u u ɛ C inf w S w u ɛ Te exact solution as te following property; eg see [5, 6] Lemma 33 Suppose tat 54 olds Ten, for i =,, e c, ɛ x + e 56 u i C i [ + ɛ i c, ɛ x ] It is easy to get u k+ dx Cɛ k+ Wen a uniform mes is used, combining 55 wit 34 and 35 gives rise to 57 u Π k u ɛ Cɛ 4 N + ɛ k N ɛ For mes adaptation, we take 58 V k+ = + ɛ k+ e c, ɛ x + e c, ɛ x As in te preceding subsection, we ave for N 8 k+3 k+ Q k α C + ɛ γ γk+ ln ɛ γ, mes, α max ρ p C + ɛ k+ γk+ ln ɛ k+ ɛ γpk+ Teorem 3 Suppose tat bx, ɛ and cx, ɛ c > Let u be a finite element solution of k degree to te problem and i Wen a uniform mes is used, te error in u is bounded by 6 u u ɛ Cɛ/4 N ɛ k + N ɛ, were C is a constant independent of N and ɛ ii Suppose tat {x } N = is an approximate equidistributing mes satisfying 39 and 4, were te adaptation function ρ, te intensity parameter α, and te maorizing function V k+ are given in 6, 7, and 44, respectively, togeter wit

13 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES 69 /k + 3 γ /k + If N 8 k+3 k+ Q k+ mes,, ten te error in u is bounded by u u ɛ C [ N k N + ɛ γ γk+ ln ɛ γ + + ɛ k+ γk+ ln ɛ k+ ɛ k+ γk+] 6, were C is a constant independent of N and ɛ It is instructive to see tat te error bound 6 becomes u u ɛ C [ ] N k ɛ k+3 + N for γ = /k + 3, for γ = /k +, and u u ɛ C N k [ ɛ ln ɛ k+ + u u ɛ C N k [ ɛ ln ɛ k+ + ] ln ɛ k+ N ] ln ɛ k+ N ɛ for γ = /k + Moreover, wen /k + 3 γ < /k +, u u ɛ C N k, wic sows te uniform convergence independent of ɛ as N Here we ave used te fact tat for any given positive numbers s and t, tere exists a constant Ĉ suc tat ɛ s ln ɛ t Ĉ for all ɛ, ] 4 Numerical examples We present ere some illustrative results obtained for two examples Tree metods are used Tey are briefly described below Metod I is te linear finite element metod using a uniform mes Metod II is te linear finite element metod using an adaptive mes based on te exact function 44 or 58 wit k = To be more specific, we start wit a uniform mes On an approximation to te equidistributing mes, te adaptation function is calculated troug 6 and 7, were V is computed using an analytical expression and te involved integrals are approximated by te trapezoidal quadrature De Boor s algoritm [3] is employed to find a new approximation to te equidistributing mes To improve te convergence of te iteration, te mes is updated wit relaxation: 8 x old + x new x new Te process is repeated until te maximum difference between two contiguous iterates is less tan or a maximum number of iterations is reaced Finally, te finite element solution is found on te convergent mes Metod III is te linear finite element metod using an adaptive mes based on te computed solution Tis metod is similar to Metod II, except tat te iterative process involves finding bot te mes and te finite element solution In particular, V in 6 is replaced by an approximation of te second derivative of te computed solution, wic is obtained using a derivative recovery tecnique eg, see [, 3, 33, 34] as te derivative

14 7 W HUANG of a linear least-squares fitting polynomial based on a set of te values of te first derivative at Gaussian points in a patc of elements Te patc involves tree elements for an interior element and two elements for eac boundary element Two Gaussian points are used in eac element Example 4 Te first example is a convection-diffusion problem 63 ɛu + ɛ u + ɛ u = e x 4 x, 4 4 subect to te boundary condition Te exact solution is known to be x ux = e x e ɛ e 4 x ɛ e ɛ For tis example, b / and c b x / 3/6 for all ɛ and x [, ] Figs sows te computed solution on te convergent adaptive mes and te convergence istory by using Metod II One can readily see tat bot metods lead to correct mes concentration, ie, more mes points are concentrated in te boundary layer area Te convergence istory sows tat te mes quality measures, max i Q adp I i, Q mes,, and Q mes,, quickly decrease to one in about iterations Tis indicates tat te convergent mes satisfies 39 and 4 wit C and C and tus is nearly equidistributing Fig sows te ɛ-norm of te error as function of te number of mes points N obtained wit te tree metods for two values of ɛ: and 8 It is clear tat adaptive meses lead to significantly more accurate results tan a uniform mes Tis is especially true for small values of ɛ In te case wit ɛ = 8, te convergence order of te error associated wit Metod I is less tan one in te considered range of N Moreover, te error depends severely on ɛ, confirming te teoretical prediction in Teorem 3 On te oter and, bot Metod II and III sow te first order convergence and mild dependence on ɛ Interestingly, Metod III, a truly adaptive mes metod wic utilizes approximate second order derivatives based on te computed solution during te course of adaptive mes generation, produces results comparable to tose obtained by Metod II, a metod being based on te analytical expression of V cf 44 To sow te effect of te coice of γ on mes adaptation, we depict in Fig 3 te ɛ-norm of te error wit Metod II and Metod III for tree values of γ, /k + 3, /k +, and /k + It can be seen tat te tree coices lead to nearly te same results for Metod II wereas for Metod III γ = /k + 3 yields less accurate solutions tan te oter coices γ = /k + and /k + Tis noticeable difference in solution accuracy among te tree coices of γ may be due to te nature of Metod III tat mes adaptation relies on te accuracy in approximating te second order derivatives from te computed solution and terefore on te accuracy in te computed finite element solution After all, it is empasized tat Metod III wit coices γ = /k + and /k + produces almost te same and satisfactory solutions for reasonably large N 4 Example 4 Te second problem is a reaction-diffusion problem 64 ɛu + u = ɛ x x x, subect to te boundary condition Te exact solution to tis problem is known to be ux = x x + e x ɛ e e +x ɛ + e x ɛ e x ɛ ɛ

15 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES Computed Soln Exact Soln a b u x e-6 e-8 e- max Qadp Qmes Qmes Diff iteration number Figure Example 4: Results obtained using Metod II wit ɛ = and N = 4 a: Computed and te exact solutions b: Mes quality measures max i Q adp I i, Q mes,, and Q mes,, and te maximum norm of te difference between two contiguous meses are sown against te number of iteration a epsilon = e- Uniform mes Adpt mes w Exact Adpt mes w compt b epsilon = e-8 Uniform mes Adpt mes w Exact Adpt mes w compt e _epsilon e _epsilon N N Figure Example 4: Te ɛ-norm of te error is depicted as a function of te number of mes points N for different values of ɛ γ = /k + is used in te computation of te adaptation function a Metod II for epsilon = e-8 gamma=/k+3 gamma=/k+ gamma=/k+ b Metod III for epsilon = e-8 gamma = /k+3 gamma = /k+ gamma = /k+ e _epsilon e _epsilon N N Figure 3 Example 4: Te ɛ-norm of te error in te FEM solution obtained using Metod II a and Metod III b for different values of parameter γ, /k + 3, /k +, and /k + Typical adaptive solutions and convergence istory obtained using Metod III are sown in 4 and 5 Once again, one can see tat mes points are concentrated correctly in te areas of boundary layers From te results we can make similar observations as for Example 4 except tat te coice of γ as an less significant

16 7 W HUANG effect on te solution accuracy Tis is partly because tis example as less steep boundary layers and tus mes adaptation plays a relatively less crucial role in accuracy of te numerical solution u a Computed Soln Exact Soln b max Qadp Qmes Qmes Diff - e-6 - e x e iteration number Figure 4 Example 4: Results obtained using Metod III wit ɛ = 5 and N = 4 a: Computed and te exact solutions b: Mes quality measures max i Q adp I i, Q mes,, and Q mes,, and te maximum norm of te difference between two contiguous meses are sown against te number of iteration a epsilon = e- gemma = /k+3 gamma = /k+ gamma = /k+ b epsilon = e-8 gemma = /k+3 gamma = /k+ gamma = /k+ e _epsilon e _epsilon e-5 N e-5 N Figure 5 Example 4: Te ɛ-norm of te error in te FEM solution obtained using Metod III for different values of γ and ɛ, γ = /k + 3, /k +, /k + and ɛ =, 8 5 Conclusions and Remarks In te previous sections we ave developed a convergence teory on approximate equidistributing meses for polynomial preserving operators Te adaptation or monitor function associated wit equidistribution is defined and error estimates in semi-norms of Sobolev spaces are obtained rigorously Te main results are given in Teorems As an application example, Teorem is applied to te error analysis of te finite element solution of singularly perturbed boundary value problems witout turning points Error bounds are obtained for two separate cases: convectiondiffusion problems Teorem 3 and reaction-diffusion ones Teorem 3 For te latter case, uniform convergence is obtained regardless of te size of te perturbation parameter ɛ Numerical results are presented in Section 4 for two examples to verify teoretical findings It is sown tat a truly adaptive implementation of

17 CONVERGENCE ANALYSIS ON EQUIDISTRIBUTING MESHES 73 mes adaptation tat utilizes approximations of iger derivatives second derivative in te examples based on te computed solution can produce comparable solutions to tose obtained wit an analytical expression Te analysis metod employed in tis paper does not specifically use te advantage of dimension one It is our ope tat te metod and te results can be extended to multi-dimensions Suc an investigation is currently underway References [] U Ascer, J Cristiansen, and R D Russell, A collocation solver for mixed order systems of boundary value problems, Mat Comput, [] I Babu ska and W C Reinboldt, A-posteriori error estimates for te finite element metod, Int J Numer Met Engrg, [3] G Beckett and J A Mackenzie, Convergence analysis of finite-difference approximations on equidistributed grids to a singularly perturbed boundary value problems, J Comput Appl Mat, [4] G Beckett and J A Mackenzie, On a uniformly accurate finite difference approximation of a singularly perturbed reaction-diffusion problem using grid equidistribution, J Comput Appl Mat, [5] G Beckett and J A Mackenzie, Uniformly convergent ig order finite element solutions of a singularly perturbed reaction-diffusion equation using mes equidistribution, Appl Numer Mat, [6] J U Brackbill and J S Saltzman, Adaptive zoning for singular problems in two dimensions, J Comput Pys, [7] H G Burcard, Splines wit optimal knots are better, Appl Anal, [8] W Cao, W Huang, and R D Russell, A moving mes metod based on te geometric conservation law, SIAM J Sci Comput, [9] L Cen, P Sun, and J C Xu, Optimal anisotropic meses for minimizing interpolation errors in l p -norm, Tecnical Report AM65, Department of Matematics, Te Pennsylvania State University, 4 [] L Cen and J C Xu, Stability and accuracy of adapted finite element metods for singularly perturbed problems, Tecnical Report AM7, Department of Matematics, Te Pennsylvania State University, 4 [] J Cristiansen and R D Russell, Error analysis for spline collocation metods wit applications to knot selection, Mat Comput, [] P G Ciarlet, Te Finite Element Metod for Elliptic Problems, Nort-Holland, Amsterdam, 978 [3] C de Boor, Good approximation by splines wit variable knots A Meir and A Sarma, editors, Spline Functions and Approximation Teory, pages 57 73, Basel und Stuttgart, 973 Birkäuser Verlag [4] C de Boor, Good approximation by splines wit variables knots II G A Watson, editor, Lecture Notes in Matematics 363, pages, Berlin, 974 Springer-Verlag Conference on te Numerical Solution of Differential Equations, Dundee, Scotland, 973 [5] D S Dodson, Optimal order approximation by polynomial spline functions, Purdue University, PD tesis, 97 [6] A S Dvinsky, Adaptive grid generation from armonic maps on riemannian manifolds, J Comput Pys, [7] W Huang, Variational mes adaptation: isotropy and equidistribution, J Comput Pys, [8] W Huang, Measuring mes qualities and application to variational mes adaptation, SIAM J Sci Comput, to appear [9] W Huang and R D Russell, A ig dimensional moving mes strategy, Appl Numer Mat, [] W Huang and W Sun, Variational mes adaptation II: error estimates and monitor functions, J Comput Pys, [] P M Knupp, Jacobian-weigted elliptic grid generation, SIAM J Sci Comput, [] N Kopteva and M Stynes, A robust adaptive metod for a quasi-linear one-dimensional convection-diffusion problem, SIAM J Numer Anal,

18 74 W HUANG [3] J Mackenzie, Uniform convergence analysis of an upwind finite-differnce approximation of a convection-diffusion boundary value problem on an adaptive grid, IMA J Numer Anal, [4] D E McClure, Convergence of segmented approximations of smoot functions on a bounded interval, AMS Notices, , abstract [5] R E O Malley, Singular perturbation metods for ordinary differential equations, Springer- Verlag, New York, 99 [6] E O Riordan and M Stynes, A uniformly accurate finite-element metod for a singularly perturbed one-dimensional reaction-diffustion problems, Mat Comput, [7] V Pereyra and E G Sewell, Mes selection for discrete solution of boundary problems in ordinary differential equations, Numer Mat, [8] Y Qiu and D M Sloan, Analysis of difference approximations to a singularly perturbed twopoint boundary value problem on an adaptively generated grid, J Comput Appl Mat, [9] Y Qiu, D M Sloan, and T Tang, Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: analysis of convergence, J Comput Appl Mat, 6 43 [3] H-G Roos, M Stynes, and L Tobiska, Numerical metods for singularly perturbed differential equations, Springer-Verlag, Berlin, 996 [3] R D Russell and J Cristiansen, Adaptive mes selection strategies for solving boundary value problems, SIAM J Numer Anal, [3] Z Zang and A Naga, A mesless gradient recovery metod Part I: Superconvergence property, Researc Report, Department of Matematics, Wayne State University, [33] O C Zienkiewicz and J Z Zu, Te superconvergence patc recovery and a posteriori error estimates Part : Te recovery tecnique, Int J Numer Metods Engrg, [34] O C Zienkiewicz and J Z Zu, Te superconvergence patc recovery and a posteriori error estimates Part : Error esimates and adaptivity, Int J Numer Metods Engrg, Department of Matematics, te University of Kansas, Lawrence, KS 6645, USA uang@matkuedu URL: ttp://wwwmatkuedu/ uang/