The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations

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1 The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS Abstract. We consider fully discrete schemes for linear parabolic problems discretized by the Crank Nicolson method in time and the standard finite element method in space. We study the effect of mesh modification on the stability of fully discrete approximations as well as its influence on residual-based a posteriori error estimators. We focus mainly on the qualitative, analytical and computational behavior of the schemes and the error estimators.. Introduction In recent years, a main approach towards the computation of solutions of Partial Differential Equations is based on self-adapted methods. In particular, methods utilizing self adjusted meshes have important benefits approximating PDEs with solutions that exhibit nontrivial characteristics. When appropriately chosen, they lead to efficient, accurate and robust algorithms. Adaptive algorithms are naturally related to error control. Appropriate analysis can provide guarantees on how accurate the approximate solution is through a posteriori estimates. Error control may lead to appropriate adaptive algorithms by identifying areas of large errors and adjusting the mesh accordingly. Error control and associated adaptive algorithms for time dependent problems is a challenging area, both for theory and computations. A key issue, often underestimated, is the need of spatial mesh modification (mesh movement) with time. In this paper we discuss the effect of mesh modification with time on the error control of fully discrete approximations of parabolic problems. The approximations are constructed by combining Crank Nicolson (CN) time discretization with standard finite elements for the space discretization. The finite element spaces are allowed to change in different time nodes. Roughly speaking, the main structure of an algorithm which permits mesh redistribution with time has the form: Given the approximation u n at the time step n, which belongs to a finite dimensional space V n (reflecting the space discretization method) a : choose the next space V n+, b : project u n to the new space V n+ to get ũ n, c : use ũ n as starting value to perform the evolution step in V n+ to obtain the new approximation u n+ V n+. Standard schemes involve only step (c) (uniform or nonuniform mesh). The presence of (a) and (b) are in most of the cases neglected in the analysis. It should be noted though that, on one hand, such algorithms can accumulate the nodes of the computational mesh in the areas of interest, as expected, and Date: July,. 99 Mathematics Subject Classification. 65N5. Key words and phrases. A posteriori error estimators, Crank Nicolson method, parabolic problem. The research of the second author was supported by the Chair of Applied Mathematics III of the University of Erlangen through a Research Assistantship from 7-. The third author was partially supported by the RTN-network HYKE, HPRN-CT--8, the EU Marie Curie Dev. Host Site, HPMD-CT-- and the program Pythagoras of EPEAEK II.

2 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS on the other hand (a) and (b) have fundamental influence on the qualitative behavior of the schemes. Such influence becomes evident in pressure pollution in Navier Stokes solvers. In fact, random mesh redistribution can pollute in a severe way the pressure approximation, see [5] where examples based on van Karman vortex shedding highlighting this effect are presented. On the positive side, in nonlinear hyperbolic problems geometric mesh redistribution can stabilize unstable schemes. Indeed, all stable schemes for these problems include terms inducing artificial numerical diffusion (upwinding). As it is well known, the right selection of such schemes is a nontrivial task. Recent results reported in e.g., [, ] and their references, show that when steps (a) and (b) are based on geometric information on u n then they effectively stabilize schemes even without additional terms reflecting artificial diffusion or upwinding. In the present paper, we investigate the influence of mesh change in the stability of fully discrete schemes based on Crank Nicolson time discretizations, as well as its influence on the a posteriori error estimators. These estimators are derived in detail in [6] and are the first optimal order a posteriori estimates in L (L ) for fully discrete Crank-Nicolson schemes allowing mesh modification. For completeness, we present the main ideas of the analysis here, but we focus mainly on the qualitative analytical and computational behavior of the schemes and the estimators. Our findings can be summarized as follows: () Refinement can spoil Crank Nicolson schemes. Indeed, we present examples where recursive refinement of the mesh can spoil standard Crank Nicolson schemes. This rather surprising conclusion was a consequence of our effort to understand the presence and the role in the a posteriori estimate of a term of the type ( n n )u n, here n denotes the discrete Laplacian corresponding to the space V n, see below for precise definitions. () We introduce a version of the Crank Nicolson scheme consistent with mesh redistribution. The definition of the new version of fully discrete CN scheme is motivated by the a posteriori analysis and the fact that the standard scheme is problematic when combined with mesh modification. () Refinement can influence the a posteriori error estimators. We present detailed computational experiments which show that the a posteriori estimators are of optimal order and include terms capturing separately the spatial and the temporal errors. We present a case study where refinement occurs at a given time level. In this case and when the solution is fast in the spatial variable, parts of the estimator become sensitive. This is a further indication that Crank-Nicolson fully discrete schemes should be used with great care during mesh change. () Mesh change is related to known non-smooth data effects. It is known that CN is a sensitive scheme and belongs to the border of stable time discretization methods for diffusion problems. Among its known properties is its lack of smoothing effect, see [, 7]. Smoothing is a desirable property for discretization schemes for parabolic problems and thus CN time discretization serves mostly as an interesting case study. We present computational results, as well as spectral arguments to show that this lack of smoothing of CN scheme is present and influences the behavior of the a posteriori estimators. This suggests, that nonstandard projections in the step b) of the above algorithm, might be desirable. This subtle issue requires further investigation. Our estimators are based on the methodology developed in [, ] for space discrete and fully discrete and in [, ] for time discrete schemes. The key point is the definition of an auxiliary function which we call reconstruction of the approximation U. As far as the time reconstruction is concerned we follow the approach of [] which includes the reconstructions based on approximations on one time level (two point estimators) as in [] as well as the reconstructions based on approximations on two time levels (three point estimators) as in []. The role of the elliptic reconstruction, [], is also important for the derivation of the estimators. Fully discrete a posteriori estimates for CN time discretization methods were derived previously in [, 9]. The estimators in [] are valid only without mesh change and they are of optimal order in L (H ) but not in L (L ). The estimators in [9] are not second order in time, see [5]. Compared to existing results, apart from including the possibility of mesh-change, our analysis provides optimal order estimators in L (L ) for higher-order in time fully discrete schemes. Notice also that our approach is in principle applicable to other evolution problems, not necessarily of parabolic type. A posteriori

3 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES bounds for Crank Nicolson methods applied to the linear Schrödinger equation were derived by Dörfler [7]. Alternative estimators for the discetization methods and the problem at hand based on the direct comparison of u and the numerical solution U could be derived using parabolic duality as in [9, 8]. The rest of this article is organized as follows. In the remainder of this section we introduce the problem setting and define the CN discretization. In Section we present the space time reconstruction. Section. is concerned with a posteriori error estimates. In particular we emphasize on the use of two point and three point estimators. Then in Section we discuss in detail numerical experiments highlighting in particular the staibilityof the scheme and the estimators with respect to data effects and mesh refinements. In Section we summarize our findings... The problem and its discretization. We consider the initial value problem for the heat equation: Find u L (,T; H (Ω)), with tu L (,T; L (Ω)), satisfying { ut,φ + a(u, φ) = f, φ, φ H (Ω), (.) u() = u, where f L (,T; L (Ω)) and u H (Ω). Here Ω is a bounded domain in Rd,d=,, and T>. We denote by, the duality pairing between H (Ω) and its dual H (Ω), and by a(, ) the bilinear form in H (Ω) defined as (.) a(v, w) = v, w, v, w H (Ω). For D R d bounded we denote by D the norm in L (D), by r,d and by r,d the norm and the semi-norm, respectively, in the Sobolev space H r (D),r Z +. In view of the Poincaré inequality, we consider,d to be the norm in H (D) and denote by,d the norm in H (D). In the sequel, in order to simplify the notation, we shall omit the subscript D in the notation of function spaces and norms whenever D = Ω. In order to discretize the time variable in (.), we introduce the partition =t <t < <t N = T of [,T] and we denote by I n := (t n,t n ] the subintervals, by := t n t n the time steps, and by t n / the midpoints of I n. Moreover, for given sequence {v n } N n=, we shall use the notation (.) v n := vn v n and v n v n + v n :=,n=,..., N. We shall also denote by u m (x) and f m (x) the values u(x, t m ) and f(x, t m ), respectively, throughout the rest of the paper. In addition, we shall often drop the space dependence explicitly, e.g., we shall write u m with reference to u m (,t m ). We use finite elements to discretize in space: Let {T n } N n= be a family of conforming shape-regular triangulations of the domain Ω, which corresponds to the time node t n. As emphasized earlier, we assume that the triangulations are allowed to change in time. We denote by h n the local mesh-size function of each given triangulation T n defined by (.) h n (x) := h K, K T n and x K, with h K := diam(k). For each n and for each K T n, we let E n (K) be the set of the sides of K (edges in d = or faces in d = ) and Σ n (K) E n (K) be the set of the internal sides of K. In addition, we introduce the sets E n := K Tn E n (K) and Σ n := K Tn Σ n (K). We shall also use the sets ˆΣ n := Σ n Σ n and ˇΣ n := Σ n Σ n. With e Σ n (K) we associate a unit vector n e orthogonal to e and denote by [v] e the jump of any function v across e in the direction of n e, (.5) [v] e := lim δ +[v(x + δn e) v(x δn e )], for x e. We associate with each triangulation T n the finite element spaces (.6) Ṽ n := {φ H (Ω) : K T n : φ K P l } and V n := Ṽ n H (Ω). where P l is the space of polynomials in d variables of degree at most l.

4 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS.. The fully discrete scheme. The standard Crank-Nicolson Galerkin (GCN) finite element discretization is the following usual form of the fully discrete equations: let U a given initial approximation of u and for n N, find U n V n such that (.7) U n U n This scheme can be written in the point-wise form,,φ n + a( U n + U n,φ n )= f n,φn for all φ n V n. (.8) U n P n U n + ( n )U n + ( n )U n = P n f n. Here P n : L V n is the L -projection onto V n and n is the discrete Laplacian corresponding to the finite element space V n defined by Definition.. The discrete Laplacian n : H (Ω) Vn is the operator with the property (.9) n v, φ n = a(v, φ n ) φ n V n. However, see [6], when changing the mesh, the term n U n may cause problems. If for instance T n is a refinement of T n, then the discrete Laplace operator on the finer mesh is applied to coarse grid functions leading to oscillatory behavior of the term n U n, see Fig. for a computational example in d. There, the standard Galerkin Crank Nicolson scheme (.8) was applied for time steps with global refinement each 6 time steps. Clearly the oscillatory behavior can be seen. Notice that this is particularly interesting since usually errors are not expected during refinement only, [6]. The a posteriori analysis of [6], has led to this unexpected finding. In fact, the final a posteriori estimate for the standard scheme (.8) contains a term of the type, [6], ( n n )U n which might grow without control. This suggests that a possible solution that would resolve the oscillatory behavior of the classical scheme, is to consider the following modified Crank Nicolson scheme: For n, n N find U n V n, n N, such that (.) U n Π n U n + Πn ( n )U n + ( n )U n = P n f n. Here, Π n, Π n : V n V n denote suitable projections or interpolants to be chosen. For the computations of Fig. we used Π n =Π n = P n. The reason for introducing a further operator Π n is that we would like to study schemes and corresponding estimators including several possible choices for the projection step. Fig. also displays computations with the scheme (.), which resolves the problem of this oscillatory behavior. The scheme (.) is in fact natural: One may think that the fact that the discrete Laplacian changes with time introduces an artificial time dependence of the form y t + A(t)y =, say to the space discrete ode. An application of the Trapezoidal method to this problem will yield, y n y n and the similarity to (.) is evident. + A(tn )y n + A(tn )y n =,

5 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES Figure. Comparison of the standard Galerkin Crank Nicolson scheme (.8) (blue line) and the modified method (.) (red dotted line); solutions after times steps with global refinement each 6 time steps.. A posteriori error estimates in L (L ) norm Towards error control the methodology developed in [,,, ] summarized in [] is used. It is based on the appropriate definition of an auxiliary function Û which we call reconstruction of the approximation U; here U is the piecewise linear in time interplant of {U n }. Then the error estimate relies on the separate control of u Û and Û U. A key ingredient of this approach is the fact that Û should satisfy the same PDE with the exact solution, but, perturbed with an a posteriori term which we would like to have in the final estimate (terms which are not computable but can be bounded a posteriori are also allowed). The crucial and not trivial issue is to define appropriately the reconstruction Û. For our case, following [6] we present the basic steps of this construction. As it was observed first in the time discrete case in [], U cannot lead to optimal order estimators using energy methods, see also [9,, 5]. As far as the time reconstruction is concerned we follow the approach of [] which includes the reconstructions based on approximations on one time level (two point estimators) as in [] as well as the reconstructions based on approximations on two time levels (three point estimators) as in []. We emphasize that in order to derive estimators of optimal order in L (,T; L (Ω)) we have to appropriately define Û by involving in its derivation the elliptic reconstruction operator []... The space-time reconstruction. We begin by introducing the piecewise linear approximation U : [,T] H (Ω) of u defined by linearly interpolating between the nodal values U n and U n (.) U(t) := l n (t)u n + l n (t)u n, t I n, with (.) l n (t) := tn t and l n (t) := t tn, t I n. In addition, let Θ : [,T] H (Ω) be defined as (.) Θ(t) =l n (t)π n ( n )U n + l n (t)( n )U n, t I n. To proceed with the definition of the space-time reconstructions of the fully discrete approximate solution U n, n =,..., N, defined in (.), we finally need to introduce the elliptic reconstruction

6 6 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS operator R n. Since its definition depends on the finite element space V n, the operator R n changes also with n : Definition.. (Elliptic reconstruction) For fixed v h V n, we define the elliptic reconstruction R n v h H of v h, as the solution of the following variational problem (.) a(r n v h,ψ) = ( n )v h,ψ ψ H. The elliptic reconstruction R n satisfies the Galerkin orthogonality property (.5) a(r n v h v h,χ n )=, χ n V n. To define the space time reconstruction it will be useful to rewrite our scheme (.) in the compact form (.6) where U n Π n U n + F n =, (.7) F n = Πn ( n )U n + ( n )U n P n f n. Definition.. (Space time reconstruction) We define first the piecewise linear in time function ω : [,T] H defined by linearly interpolating between the values Rn U n and R n U n (.8) ω(t) := l n (t)r n U n + l n (t)r n U n, t I n, with l n and ln defined in (.). Next, we define the space-time reconstruction, Û : [,T] H, as follows (.9) Û(t) :=R n U n + Rn Πn U n R n U n (t t n ) t R n ÎF(s) ds, t I n, t n Here ÎF( ) is a piecewise linear function such that ÎF( ) I n is a linear polynomial interpolating F n : (.) ÎF(t n )=F n = ( n )U n + Πn ( n )U n P n f n. It can be easily seen that the function Û interpolates the values Rn U n and R n U n. The first claim is obvious. Furthermore, evaluating the integral in (.9) by the mid-point rule and recalling (.), we get t n Û(t n )=R n Πn U n R n ÎF(s) ds (.) t n In addition, Û satisfies the following relation = R n { Π n U n F (t n )} = R n U n. (.) Û t (t)+r n ÎF(t) = Rn Πn U n R n U n, t I n. The analysis in [6] is based on this particular choice of Û which incorporates the effect of the mesh change in a high order in time scheme. For Backward Euler the space-time reconstruction defined in [] is different. A crucial point is the fact that the difference Û ω can be computed explicitly, and in fact is a second-order in time term, see [6].

7 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES 7 The definition of the reconstruction when no mesh change in time is present is just Û(t) :=R n{ t } U n ÎF(s) ds, t I n. t n One may view the above expression as the elliptic reconstruction operator applied to the time reconstruction constructed in the spirit of [], []. Then one can verify that Û would satisfy Ût(t),ψ + a(û(t),ψ) = Rn ÎF(t) Θ(t),ψ + a(û(t) ω(t),ψ), t I n. This simplified equation is the starting point of our analysis, since when compared to the equation for u it leads to the main error equation. Roughly speaking, the first term in the right hand side will create spatial errors and the term a(û(t) ω(t),ψ) temporal errors. When mesh modification is allowed the equation that Û satisfies is more involved since it will contain terms in the right hand side accounting for mesh change, compare to the r.h.s. of (.5) and (.8)... Main error equation. Motivated by the discussion above we could hope that ˆρ := u Û satisfies the same PDE with the exact solution but with controllable (a posteriori) r.h.s. To this end we introduce some more notation: Let ɛ := ω U be the elliptic reconstruction error, ρ and ˆρ be the parabolic errors defined by (.) ρ := u ω and ˆρ := u Û, and σ := Û ω be the time reconstruction error. Then, the error e := u U can be split as follows (.) e = u U =[u Û ]+[Û U ] =[u Û ] + [ (Û ω)+(ω U)] =ˆρ +[σ + ɛ ]. The proof of the estimate relies on two main ingredients : (a): the direct estimation of Û U via the estimate of σ and ɛ, and (b): the estimate of ˆρ using PDE stability estimates. Note that σ will account for the time discretization error, and ɛ, for the space discretization error. A crucial step in the proof is to establish the equation that ˆρ satisfies, [6]: For each ψ H, we have ˆρ t (t),ψ + a(ρ(t),ψ) = R n ÎF(t) Θ(t),ψ + l n (t) (Π n I)( n )U n,ψ (.5) kn R n Πn U n R n U n,ψ + f(t),ψ, t I n. After a rearrangement of certain terms, we conclude (.6) ˆρ t (t),ψ + a(ρ(t),ψ) = R h,ψ, ψ H, where (.7) R h :=(R n I)(ÎF(t) F n / )+(ÎF(t) Θ(t)) Π n U n U n + l n (t)(π n I)( n )U n (Rn I)U n (R n I)U n + f(t). An examination of the above equation leads to the following conclusions: (i): The error ˆρ satisfies a parabolic PDE with controllable right-hand side. Indeed, (.6) yields (.8) ˆρ t (t),ψ + a(ˆρ(t),ψ) = R h,ψ + a(σ(t),ψ), ψ H, since ˆρ(t) ρ(t) = (Û ω) = σ. We keep a(ρ(t),ψ) in the left-hand side of (.6) for technical reasons. (ii): The terms in the rhs of (.8) are either direct a posteriori terms or involve spatial error operators of the form R j I.

8 8 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS Therefore, one can prove the following result. The estimators still depend on stationary finite element errors through R j I. These terms can be estimated using residual type a posteriori estimators, although other choices on the estimators are possible. Theorem.. (Estimate in L (L ) and L (H ) for the parabolic error) Let u be the exact solution of (.), and ω and Û defined in (.8) and (.9) respectively. The following estimate holds { t (.9) max ˆρ(t) + ( ˆρ(s) t [,t m + ρ(s) ) ds } ˆρ() + J m, ] where J m,m=,..., N, are defined by m (.) J m := (Jn T +J S, with n= (.) J T n := n +J S, t n t n σ(s) ds, n +J C n +J D n ), t n (.) Jn S, := (R n I)(ÎF(t) F n / ), ˆρ(s) ds t n t n (.) Jn S, := (.) J C n := t n (Rn I)U n (R n I)U n t n, ˆρ(s) ds (Π n I)(l n (t)( n )U n U n ), ˆρ(s) ds t n t n (.5) Jn D := ÎF(s) Θ(s)+f(s), ˆρ(s) ds tn In the next paragraphs we shall focus to two special choices of space-time reconstructions... Specific Choices for the Reconstructions. To estimate of the terms appearing in the r.h.s. of the bound in Theorem., we need to specify the interpolation operator used in the definition of Û. Depending on this choice we derive estimates involving one time interval (two point estimator) or estimates involving two time intervals (two point estimators), compare to [], [].... Choice of the Interpolant: Two-point estimator. It is easily seen that the Crank Nicolson method (.) can be written as follows (.6) U n Π n U n + Θ(t n )=P n f(t n ). Let F (t) := Θ(t) P n f(t) and let Î be the piecewise linear interpolant chosen as (.7) Î(v) In P (I n ), Î(v)(t n )=v(t n ), Î(v)(t n )=v(t n ). Then, ÎF : I n V n and (.8) ÎF(t) =Î(Θ(t) P n f(t)) = Θ(t) P n ϕ(t) where ϕ(t) =Î(f(t)). Moreover, there holds (.9) ÎF(t n )) = Î(Θ(t n ) P n f(t n )) = F (t n ). We shall now calculate the terms on the right-hand side in Theorem. depending on that special choice of F and Î.

9 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES 9 Lemma.. (Calculation of ÎF(t) F n / ). We have (.) ÎF(t) F n / = (t t n / )w n, where w n is given by (.) w n := tθ(t) P n [f(tn ) f(t n )]. Proof. In view of (.8), we have (.) ÎF(t, Θ(t)) F n / = Θ(t) Θ(t n / ) P n [ϕ(t) ϕ(t n / )]. Now, in view of (.), (.) and (.), it is easily seen that Θ(t) Θ(t n / )=l n (t)π n ( n )U n + l n (t)( n )U n ( Π n ( n )U n +( n )U n) (.) = (ln (t) l n (t)) (( n )U n Π n ( n )U n ) =(t t n / ) ( ( n )U n Π n ( n )U n ) The result claimed follows by combining the last two relations with the definition of ϕ. Furthermore, we have (.) ÎF(t) Θ(t) = P n ϕ(t).... Choice of the Interpolant: Three-point estimator. Our scheme (.) can be rewritten in the form U n (.5) Π n U n + F n =. We shall need also the projected version of the same equation at the previous interval, (.6) π n U n Π n U n + π n F n =. Here π n is any projection to V n at our disposal and F n = ( n )U n + Πn ( n )U n P n f n. Then, we define the extended piecewise linear interpolant ÎF as (.7) ÎF(t) := l n / (t) F n + l n / (t) π n F n, t In, where (.8) l n / (t) := (t tn ) +, Obviously, ÎF(t) Vn for each t I n, ÎF I n l n / (tn t) (t) :=. + is a linear function of t and (.9) ÎF(t n )=F n. For the proof of the following two Lemmata we refer to [6]. Lemma.. (Calculation of ÎF(t) F n / ). We have (.) ÎF(t) F n / = (t t n / ) w n, where w n is given by (.) w n := + [( U n Π n U n ) π n( U n Π n U n )],

10 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS Lemma.. (Calculation of ÎF(t) Θ(t)). If we denote (.) ˆϕ(t) := l n / (t) P n f n + l n / (t) π n P n f n, t In, we have (.) ÎF(t) Θ(t) = ˆϕ(t) + l n / (t) [ kn ( n )U n ( Π n + Π n π n) ( n )U n + π n Π n ( n )U n ]... Error estimate based on residual estimators. Using residual-based estimators estimating the spatial finite element errors we can conclude to the final error estimate for the modified Crank Nicolson- Galerkin scheme. To this end we shall need the following definition. Definition.. (L (L ) error estimators). Let c, c i,j be appropriate constants appearing in Clément type interpolation estimates. For C E being the elliptic regularity constant we denote v C E v, v H (Ω), C j, = C E c j,. For n =,..., N, we define: The elliptic reconstruction error estimator appearing in definition of both two- and three-point estimators (.) ε n = C, h n( n )U n + C, h / n J[ U n ] Σn, Let ĥn := max(h n,h n ); the space-mesh error estimator that appears also in both two- and three-point estimators [ γ n = C, (.5) ĥ n k n ( n )U n kn ( n )U n ] + C, ĥ/ n J[ U n U n + C ] ˆΣn, ĥ/ n J[ U n U n ] ˇΣn\ˆΣ n. Let w n and w n be as in Lemma. and Lemma., respectively. We define the time reconstruction error estimator that corresponds to the two-point reconstruction by (.6) δ n := k { n wn + C, h n( n )w n + C, h / } n J[ w n ] Σn, and to the three-point reconstruction by (.7) δn := k { n wn + C, h n( n ) w n + C, h / } n J[ w n ] Σn. Further we define: The space error estimator corresponding to the two-point reconstruction (.8) η n := { C, h n( n )w n + C, h / n J[ w n ] Σn }, and to the three-point reconstruction (.9) η n := { C, h n( n ) w n + C, h / n J[ w n ] Σn }. The time error estimator in case of the two-point reconstruction (.5) θ n : = k { n c w n + c, h n ( n )w n }, and in case of three-point reconstruction (.5) θ n : = k { n c w n + c, h n ( n ) w n }.

11 The coarsening error estimator EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES (.5) β n := (Π n I)(( ) n U n + ( Π n I)kn U n, and the data approximation estimators (.5) t n ξ n, := f(s) ϕ(s) ds, t n ξ n, := c, max { h n (I P n )f n, h n (I P n )f n } and (.5) ξn, := t n t n f(s) ˆϕ(s) ds. In case of the three-point estimator the following additional estimator appears (.55) ζ n := ( + ) ( n )U n ( + Π n π n) ( n )U n + π n Π n ( n )U n. The a posteriori bounds are summarized in the following theorem. Theorem.. (Complete L (L ) a posteriori error estimates). For the reconstruction defined in Section. and for m =,..., N, the following two level estimate holds ( (.56) where max t [,t m ] u(t) U(t) u R u + (.57) E m, := ) / m θn + { Em, + Em, n= + max n m δ n + max n m ε n, m (η n + γ n + β n + ξ n, ), E m, := n= m n= k / n ξ n,. Alternatively, if we use the reconstruction defined in Section. the following three level estimate holds for m =,..., N, (.58) max t [,t m ] u(t) U(t) u R u + where (.59) Ẽ m, := ( m n= m ( η n + γ n + β n + ξ n, + ζ n ). n= } / ) / θ n + Ẽm, + max δ n + max ε n, n m n m The proof of the theorem uses Theorem.. For completeness, in the rest of the section we present the basic steps taken in [6] to prove Theorem.. We consider the case of the two point estimator only here. Thus one can associate the estimators appearing in the Theorem. (and thus in Definition.) to the bounds in Theorem.. For the complete analysis we refer to [6].

12 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS... Elliptic estimators. Using approximation properties of the Clément-type interpolants it can be proved by applying standard techniques in a posteriori error analysis for elliptic problems, cf., e.g., [8], the following estimate for the elliptic reconstruction error. Lemma.. For any ϕ V n there holds (.6) (R n I)ϕ n C, h n( n )ϕ n + C, h / n J[ ϕ n ] Σn. In particular for m =,..., N, the following estimate holds (.6) max ɛ(t) max ε n. t [,t m ] n m... Main time estimator. The main time reconstruction error is due to σ. Its estimate is based on the expression (.6) σ(t) =Û(t) ω(t) = (t tn )(t n t)r n w n, t I n. Then, (.6) max σ(t) max δ t [,t m n. ] n m The proof hinges on (.6) σ(t) (t t n )(t n t) { (R n I)w n + w n }. Next, one can bound the similar term J T n in Theorem. by first noticing (.65) σ(t) = a(σ(t),σ(t)) = (t t n )(t n t)a(r n w n,σ(t)). One can conclude in this case that (.66) J T n θ n.... Spatial error estimate. In order to estimate the term Jn S, in Theorem., which accounts for the space discretization error, we use (.6) and the expression for ÎF(t) F n / to obtain (.67) J S, n C max s [,t m ] ˆρ(s) η n.... Space estimator accounting for mesh changing. The estimate of the term Jn S, in Theorem. is based on the orthogonality properties of R n R n on V n V n. The final estimate in this case is (.68) J S, n max t [,t m ] ˆρ(t) γ n...5. Coarsening error estimate. The term J C n in Theorem. can be obviously bounded as follows (.69) J C n max t [,t m ] ˆρ(t) β n...6. Estimation of the term J D n. This term can be written as (.7) Jn D = f(s) P n ϕ(s), ˆρ(t) ds, t n and it can be estimated (.7) ( tn ) / Jn D max t t m n, + kn / ξ n, ˆρ(s). t n t n

13 . Behavior of the estimators EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES In this section we study the behavior of the error estimators. In particular, we start with the influence of the non-smooth initial data on the error indicators and compare this influence with the known non-smooth data effects of the Crank Nicolson method (Section.). Next, we study the asymptotic behavior of the error estimators of Section and compare this behaviour with the true error on four model problems, where the one of them is chosen such that the right hand-side f to satisfy non-zero boundary conditions (Section.). Finally, we investigate how refinement can influence the a posteriori error estimators. In particular we consider a computational case study where refinement occurs at a given time level and compare the behavior of the estimators (Section.). All the error estimators were implemented in a C code that uses the adaptive finite element library ALBERTA [6]. For our purpose, we consider the heat equation (.) on the unit square, Ω = [, ], and T = and the exact solution u be one of the following: case (): u(x, y, t) = sin(πt) sin(πx) sin(πy) case (): u(x, y, t) = sin(5πt) sin(πx) sin(πy) (fast in time) case (): u(x, y, t) = sin(.5πt) sin(πx) sin(πy) (fast in space). In addition we consider the following tests case (): u(x, y, t) = sin(πt)(x x + x )(y 6y +y) (u =,f on Ω) case (5): u(x, y, t) = exp( π t) sin(πx) sin(πy) (u (x, y) = sin(πx) sin(πy), f= ). The right-hand side f of each problem is calculated by applying the pde to the corresponding u. Note that in cases () and () the error is due to both space and time discretization, in cases () and (5) the error comes mainly from the time discretization and in case () mainly from the space discretization. The initial conditions vanish in the first four cases and is sin(πx) sin(πy) in the last. Finally, in cases (), (), () and (5) the right-hand side f is equal to zero on Ω and in case () f satisfies non-zero boundary conditions... Practical Implementation of the Estimators. As we mentioned, we used the finite element library ALBERTA for the computations. In ALBERTA, the sequence of the triangulations is constructed as follows: An initial triangulation of the domain is given (macro triangulation); based on an appropriate procedure which assures mesh conformity and conserves shape regularity, some simpleces are first refined by bisection and, after several refinements, some other simplexes may be coarsened. The mesh is represented as a binary tree whose nodes represent the simplexes. The children of each simplex (parent) are the two sub-simplexes obtained by bisection. During coarsening the children of a simplex are coarsened to get their parent. The leafs of the tree represent the simplexes of the current mesh. In this paragraph we shall shortly describe a practical implementation of the parts of the estimators which involve finite element functions corresponding to two successive meshes T n and T n. Let T n be the finest intermediate triangulation between T n and T n, that is (.) T n refine... refine T n We shall first describe how inner products of the form coarsen... (.) w n,w n,w n V n,w n V n coarsen T n. can be computed exactly. We note that, when the grid is only refined, that means V n V n, the expression (.) can be calculated exactly on the new grid, since w n belongs also to V n. Thus, no information during refinement is lost. On the other hand, when T n comes only from coarsening of T n, information is usually lost, since w n can not be represented exactly on the coarsest mesh T n. However, also in that case, we can calculate expression (.) exactly by working as follows: Let {ψ j } be the basis functions of V n and {φ i } be the basis function of V n. We can compute expressions of the form w n,ψ j exactly on the finest grid T n, and then, by using the representation of the basis functions {ψ j } by the

14 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS basis functions {φ i }, the data can be transformed during coarsening (from the children to their parent) such that w n,φ i is calculated also exactly. If N (.) w n = α i φ i is the representation of w n in terms of the basis {φ j }, then we have i= (.) w n,w n = i α i w n,φ i, and this computation is exact. Notice that the coefficients in coarsening are exactly the ones appearing in the representation of functions with respect to the hierarchical basis, see []. In general case when T n comes from both refinement and coarsening of T n, we may work as follows: Let { ψ j } be any basis of the finite element space Ṽ n with respect to T n. We compute w n, ψ j for all basis functions ψ j V n, on the finest grid T n, and then continue as described above in case of coarsening. We proceed with the exact computation of quantities of the form (.5) w n w n T n T n,w n V n,w n V n. By applying the Pythagorean Theorem, we obtain that (.6) w n w n T n T n = w n T n T n + w n T n T n w n,w n T n T n. The first term on the right hand-side is calculated on the intermediate grid T n and the result is transformed during coarsening in such a way that no information is lost. The second term is computed after refinement and coarsening on the new mesh T n. For the exact calculation of the last term we proceed as described above... Data effects. In this section, we shall study the behaviour of the estimators in case of non-zero initial data u. Since the Crank Nicolson method requires further regularity assumptions on the data in order to be second-order accurate, [, 7], we are interested in studying the influence of the smoothness of the data on the error estimators. Here we refer to the smoothness of the discrete approximations of u and not on the smoothness of u per se. Indeed we distinguish three cases for starting the fully discrete scheme. The first is the non-smooth choice, and the last suggested already in [] is enough for CN scheme to have the smoothing property. In particular the initial approximation U is chosen to be the interpolant I u of the initial data u ; the elliptic projection P u of u, namely the solution of the system (.7) a(p u,φ) = u,φ, φ V ; the approximation given by performing two steps of the Backward Euler with the half time-step. Following the analysis in [7] for the Crank Nicolson method applying to linear parabolic equations with non-smooth data, we shall next study the stability of w n and w n, which appear in the definitions of the error estimators, in case of uniform time meshes. In parallel we show the computational behavior of these terms for the test problem (5). Stability of two- and three-point reconstruction estimator. We consider the homogeneous heat equation with u. Let {λ j } N j= be the eigenvalues of h and {e j } N j= be a corresponding basis of orthonormal eigenvectors. Then, any function v V = V h can be written as (.8) v = N v, e j e j. j=

15 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES 5 The approximate solution U n may be written in the recursive form (.9) U n = r (k( h ))U n, where r is the rational function appearing in the Crank Nicolson method, (.) r (λ) := λ + λ. It is easily seen that (.) sup r (λ) and lim r (λ) =. λ σ( k h ) λ + According to (..), we have (.) w n = k [( h)u n ( h )U n ]= k ( h)r ( k h ) n (r ( k h ) I)U. Hence, w n can be written in the following spectral representation form, [7], (.) w n = k Similarly, we have N λ j r (kλ j ) n (r (kλ j ) ) U,e j e j. j= (.) w n = N k r (kλ j ) n (r (kλ j ) )) U,e j e j. j= The following spectral bounds for w n, w n as well as the associated numerical experiments serve the purpose of comparing the effect of the three different choices for the initial approximations. We can conclude that the discrete regularity of data, which affects the order of convergence of the Crank Nicolson method, affects also the stability of the estimators. When comparing, w n to w n it follows that the three point estimator is less sensitive to the discrete smoothness of the data. A fact which is reflected in the estimates below, in the way which the undesirable term λ max appears. At the end of the paragraph we provide an additional argument for this purpose by expressing both w n and w n in a comparable form. It turns out that w n contains an additional Backward Euler smoothing step compared to w n.... Starting with U = I u. Two-point estimator. In view of Parseval s identity and (.), we get (.5) w n = k hence, we obtain N λ j r (kλ j ) n (r (kλ j ) ) I u,e j ; j= (.6) w n k sup (kλ j )r (kλ j ) n (r (kλ j ) ) I u. λ j σ( h ) where Hence, (kλ j )r (kλ j ) n (r (kλ j ) ) kλ max. (.7) w n kλ max k I u.

16 6 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS x 6 5 x 6 5 (a) w n L (b) w n L (c) h n n w n L Figure. Test problem of case (5) (u and f = ) and U = I u. We notice here the effect of non-smooth initial data on the w n, w n, h( h )w n that appear in the definition of the two-point estimator. Three-point estimator. According to Parseval s relation and (.), it follows that (.8) w n k sup r (kλ j ) n (r (kλ j ) ) I u. λ j σ( h ) But, Hence, r (kλ j ) n (r (kλ j ) ), (.9) w n k I u. On the other hand, we have (.) h( h ) w n = h N k λ j r (kλ j ) n (r (kλ j ) ) I u,e j, j= and Thus, we have (kλ j )r (kλ j ) n (r (kλ j ) ) kλ max. (.) h( h ) w n hλ max k I u. Hence, in order the three-point space-estimator to be stable, further regularity assumptions on initial data are required.... Starting with U = P u. Next, we consider the case of the elliptic projection, namely we choose U = P u. Two-point estimator. Since ( u,ϕ) =( P u, ϕ) =( h P u,ϕ) for all ϕ V, the discrete function w n appearing in the definition of the two-point estimator may now be written as follows (.) w n = k ( h)r ( k h ) n (r ( k h ) I)( h ) P u.

17 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES (a) w n L (b) w n L (c) h n n w n L Figure. Test problem of case (5) (u and f = ) and U = I u. The non-smooth initial data influences the behaviour of h( h ) w n and w n, which are included in the definition of the three-point estimator. The Parseval relation implies Thus it follows where Hence, w n = k N r (kλ j ) n (r (kλ j ) ) P u,e j. j= w n sup r (kλ j ) n (r (kλ j ) ) P u, k λ j σ( h ) r (kλ j ) n (r (kλ j ) ). (.) w n k P u. In addition, we have Now, Therefore, h( h )w n = h k N (kλ j ) r (kλ j ) n (r (kλ j ) )(kλ j ) P u,e j. j= (kλ j )r (kλ j ) n (r (kλ j ) ) kλ max. h( h )w n hλ max k P u. Three-point estimator. In this case, in view of Parseval s identity and (.), we obtain (.) w n k with Hence, sup r (kλ j ) n (r (kλ j ) ) (kλ j ) P u, λ j σ( h ) r (kλ j ) n (r (kλ j ) ) (kλ j ). (.5) w n k P u.

18 8 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS (a) w n L (b) w n L (c) h n n w n L Figure. Test problem of case (5) (u and f = ) and U = P u : Compared to Figure, we can here observe an improvement on the results. The quantity w n decreases with respect to time as Ce πt. Moreover, the quantities w n, w n, h( h )w n show a similar behaviour compared to the quantities w n, w n, h( h ) w n in Figure. and On the other hand, we have Thus, h( h ) w n = h k N j= λ j r (kλ j ) n (r (kλ j ) ) λ j P u,e j, r (kλ j ) n (r (kλ j ) ). (.6) h( h ) w n h k P u (a) w n L (b) w n L (c) h n n w n L Figure 5. Test problem of case (5) (u and f = ) and U = P u : Compared to Figure, we can notice here an improvement on the results. The norms w n and w n decrease with respect to time as Ce πt.... Starting with two steps of Backward Euler method.

19 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES 9 Two-point estimator. In the case where we start the Crank Nicolson method by first performing two steps of Backward Euler with the half time-step, the discrete function w n is written (.7) w n =( h )r ( k h ) n (r ( k h ) I)r ( k h) I u, where r is the rational function of the Backward Euler method, (.8) r (λ) := +λ. By Parseval s relation, we obtain where In addition, we have Now Thus, w n k sup (kλ j )r (kλ j ) n (r (kλ j ) )r ( k λ j σ( h ) λ j) I u, h( h )w n = h k 6 (kλ j )r (kλ j ) n (r (kλ j ) )r ( k λ j). N (kλ j ) r (kλ j ) n (r (kλ j ) )r ( k λ j) I u,e j. j= (kλ j ) r (kλ j ) n (r (kλ j ) )r ( k λ j) 8. (.9) h( h )w n h k I u (a) w n L (b) w n L (c) h n n w n L Figure 6. Test problem of case (5) (u and f = ) and starting with Backward Euler: Compared to Figure and Figure, the norms w n, w n and h( h )w n decrease with respect to time as Ce πt.

20 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS Three-point estimator. Similarly, we get (.) w n k sup r (kλ j ) n (r (kλ j ) ) r ( kλ j λ j σ( h ) ) I u k I u, since Finally, and Therefore h( h ) w n = h k 6 r (kλ j ) n (r (kλ j ) ) r ( kλ j ). N j= kλ j r (kλ j ) n (r (kλ j ) ) r ( kλ j ) I u,e j, (kλ j )r (kλ j ) n (r (kλ j ) ) r ( kλ j ) 8. (.) h( h ) w n 8h k I u. 8 6 (a) w n L (b) w n L (c) h n n w n L Figure 7. Test problem of case (5) (u and f = ) and starting with Backward Euler: the norms w n, w n, and h( h ) w n, decrease with respect to time as Ce πt.... Direct comparison of two and three-point estimators. We will use the recursive relation U n = r (k( h ))U n, and the definition of the scheme to directly compare w n and w n in the case of constant time step and constant in time finite element mesh. Recall that (.) w n = k [( h)u n ( h )U n ] and (.) w n = k (U n U n + U n ). Using the definition of the scheme, it turns out that (.) w n = k [( h)u n ( h )U n ].

21 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES Hence, modulo a constant factor the only difference between w n and w n is the fact that w n involves the difference U n U n while w n involves the difference U n U n. It turns out that this exactly is the source of the smoother behavior of w n in certain cases. Indeed, (.5) w n = k ( h)r ( k h ) n (r ( k h ) I)U = k ( h)r ( k h ) n (r ( k h ) I)(r ( k h )+I)U = k ( h)r ( k h ) n (r ( k h ) I) r ( k h)u. Where we used the fact r (λ) + = r (λ/), where r is the rational function of the Backward Euler method r (λ) = +λ. The corresponding expression for w n is (.6) w n = k ( h)r ( k h ) n (r ( k h ) I)U. Therefore modulo a constant factor the main difference between w n and w n is that w n contains an additional Backward Euler smoothing step compared to w n... Experimental order of convergence (EOC). In this section, we study the asymptotic behaviour of the estimators. Since we are interested in understanding the asymptotic behaviour of the estimators, we conduct tests on uniform meshes with uniform time steps. Linear Lagrange elements are used for the spatial discretization. The computed quantities here and in the next sections are: The error in the L (,t m ; L (Ω)) norm max n m en := max n m u(tn ) U n and the total error which is dominated by the L (,t m ; H (Ω)) error ( ) / m e total (t m ) := max e n + e n ). n m The elliptic reconstruction and the space-mesh error estimators: m max ε n and γ n. n m The time reconstruction error estimators: The space error estimators: The time error estimators: n= n= max δ n and max δ n. n m n m m η n n= and m η n. n= ( m θn ) / and ( m ) /. θ n n= The estimator appearing only in case of the three-point reconstruction: m ζn. n= n=

22 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS The two-point estimator defined as ( / m m E m := θn) + (η n + γ n ) + max δ n + max ε n, n m n m n= n= and the three-point estimator defined as ( ) / m m Ẽ m := θ n + ( η n + γ n + ζ n ) + max δ n + max ε n. n m n m n= n= The corresponding effectivity indices defined as EI(t m ) := E m e total (t m ) and EI(t m ) := Ẽ m e total (t m ). Each curve of the plots corresponds to equal time and spatial mesh sizes. The most coarse grid corresponds to k = h =.5 (cyan color) and the finest grid corresponds to k = h =.785 (red color). The time and spatial mesh sizes are divided by two while moving from the highest to the lowest curve. On odd rows of each figure we plot the logs of the errors and the estimators and below them the corresponding EOC. Since, the finite element spaces consist of linear Lagrange elements and the Crank Nicolson method is second-order accurate, the error in L (,T; L (Ω)) norm is O(k + h ). The main conclusion of this paragraph is that all the error estimators, in both cases of time-reconstruction, decrease with optimal order with respect to time and spatial variable. Notice that also in the forth problem where the right-hand side f satisfies non-zero boundary condition, the error estimators decrease still with optimal order... Behavior of the estimators under refinement. We present here some numerical results regarding the error norms, the estimators and their EOC under mesh modification. To this end, we chose the most harmless mesh modification, namely refinement. More precisely, the following geometric refinement is performed: at time level t =.5, we mark each element K with at least one of the coordinates of the barycenter (x K,y K ) less than.5 or greater than.75 to be refined. The time mesh-size remains constant in all of the experiments of this paragraph. Each curve of the plots corresponds to runs that start with equal time and spatial mesh sizes. The most coarse grid corresponds to k = h =.5 (green color) and the finest grid corresponds to k = h =.965 (black color). The time and spatial mesh sizes are divided by two while moving from the highest to the lowest curve. We observe that, when the exact solution of the heat equation is fast in the space variable, both the two- and three-point estimators jump under the refinement procedure described above (see Figure 9). In particular, both space error estimators, n η n and n η n, both time-estimators, ( n θn and ( n θ n) /, and both time reconstruction error estimators, maxn δ n and max n δn, jump at the time level of refinement. Nevertheless the global three point estimator is only marginally affected. Notice that the previous mentioned estimators, and only them, depend on the discrete functions w n and w n. In case that the exact solution of the problem changes faster in time, the influence of the given refinement on the behaviour of the estimators is very small or non-existent (see Figure 7 and Figure 8). ) /

23 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES (a) e L (L ) (b) e total (t m ) (c) E m (d) Ẽm (e) EOC( e L (L ) ) (f) EOC(e total (t m )) (g) EI(t m ) (h) EI(t m ) Figure 8. Test problem of case (). On top we plot the logs of each quantity and below the corresponding EOC. We observe that the L (L ) error is O(h +k ), the L (H ) error is O(h+k ), and both the two-point and three-point estimators decrease with second order with respect to time and space.

24 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS 6 (a) max n δ n 6 (b) max n δn 6 (c) `P n knθ n 6 / (d) `P n kn θ n / (e) EOC() (f) EOC(max n δn) (g) EOC(`P n knθ n /) (h) EOC(`P n kn θ n /) (i) P n knηn (j) P n kn ηn (k) max n ε n (l) P n knγn (m) EOC( P n knηn) (n) EOC( P n kn ηn) (o) EOC(max n ε n) (p) EOC( P n knγn) Figure 9. Test problem of case (). On first and third row we plot the logs of each quantity and below the corresponding EOC. the elliptic reconstruction estimator max n ɛ n and the space-mesh estimator n γ n decrease with second order. Both the time reconstruction estimators max n δ n, max n δn and both the time estimators ( n θn) /, ( n θ n ) /, are of optimal order. The space estimator n η n is of second order, while n η n superconverges.

25 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES 5 6 (a) P n kn ζ n (b) EOC( P n kn ζ n) Figure. Test problem of case (). The last part of the three-point estimator, which we can see here, decreases also with optimal order with respect to both time and spatial variables. 6 (a) e L (L ) (b) e total (t m ) (c) E m (d) Ẽm 5 5 (e) EOC( e L (L ) ) (f) EOC(e total (t m )) (g) EI(t m ) (h) EI(t m ) Figure. Numerical results for the problem with exact solution the one of case (). On top we plot the logs of the errors and the estimators and below their EOC or the effectivity index. We observe that the L (L ) error is O(h + k ), the L (H ) error is O(h + k ), and both the estimators decrease with second order with respect to time and space.

26 6 E. BÄNSCH, F. KARAKATSANI, AND CH. MAKRIDAKIS (a) e L (L ) (b) e total (t m ) (c) E m (d) Ẽm (e) EOC( e L (L ) ) (f) EOC(e total (t m )) (g) EI(t m ) (h) EI(t m ) Figure. Numerical results for the problem with exact solution the one of case () (fast in space). We observe that the L (L ) error is O(h + k ), the L (H ) error is O(h + k ), and all the estimators decrease with second order with respect to time and space.

27 EFFECT OF MESH MODIFICATION ON FULLY DISCRETE SCHEMES (a) e L (L ) (b) e total (t m ) (c) E m (d) Ẽm 5 5 (e) EOC( e L (L ) ) (f) EOC(e total (t m )) (g) EI(t m ) (h) EI(t m ) Figure. Test problem of case (): On top we plot the logs of each quantity and below the corresponding EOC or the effectivity index. We observe that the L (L ) error is O(h + k ), the L (H ) error is O(h + k ), and both the two-point and three-point estimators decrease with second order with respect to time and space.