A posteriori error estimation with the p-version of the finite element method for nonlinear...

A posteriori error estimation with the p-version of thefinite element method for nonlinear parabolic

differential equations q

Javier de Frutos *, Julia Novo

Departamento de Matem�aatica Aplicada y Computaci�oon, Universidad de Valladolid, E-47005 Valladolid, Spain

Received 26 October 2001

Abstract

We analyze an a posteriori error estimator for nonlinear parabolic differential equations in several space dimensions.

The spatial discretization is carried out using the p-version of the finite element method. The error estimates are ob-

tained by solving an elliptic problem at the desired times when the estimation is wanted. Some numerical experiments

prove the efficiency of the error estimation.

� 2002 Elsevier Science B.V. All rights reserved.

Keywords: A posteriori error estimation; Parabolic equations; p-version of the finite element method

1. Introduction

In recent years spectral methods for the numerical solution of nonlinear parabolic differential equationshave received increasingly attention. The p- and h–p-version of the finite element method allow to treatcomplex geometries with spectral accuracy [12]. In this paper we focus on the p-version of the finite elementmethod also called spectral element method. The basic idea is to divide the complete domain into severalsubdomains (elements) and obtain spectral convergence by increasing the degree of the local polynomialsused in the approximation while keeping fixed the mesh. Spectral-type approximations have been proved tobe very suitable for parabolic (linear or nonlinear) equations, due to their regularization properties [22].One common approach to numerically solve parabolic equations is the method of lines. That is, apply first adiscretization of the space variables keeping time continuous to get a system of ordinary differentialequations that is finally time integrated by means of an efficient stiff ODE solver. This is the approach wefollow in the present paper.

Comput. Methods Appl. Mech. Engrg. 191 (2002) 4893–4904

www.elsevier.com/locate/cma

qPartially supported by project DGI-MCYT BFM2001-2138 partly financed by FEDER founds.* Corresponding author.

E-mail addresses: [email protected] (J. de Frutos), [email protected] (J. Novo).

0045-7825/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0045-7825 (02 )00419-X

mail to: [email protected]

In order to calculate numerical approximations with a prescribed level of accuracy in an optimal, ornearly optimal, way it is necessary to have adaptive procedures available. A posteriori error estimates are anecessary ingredient of adaptive methods for solving evolutionary partial differential equations. They areused to estimate the accuracy of computed solutions and to control adaptive refinement.

The theory of a posteriori error estimation procedures for elliptic problems has been extensively treatedin the literature, see for example [2,3,9,20,23]. Some error estimation techniques have been also developedfor the parabolic evolutionary case. We refer to [19] where the nonlinear parabolic one dimensional case isconsidered and to [1,5,6] where the authors treat the parabolic two dimensional case. In [5,6] it is proventhat the a posteriori error estimation of a linear elliptic problem yields an estimator for a parabolicproblem. In these papers the error is measured in a specific L2-norm in the space–time cylinder and theestimation of the error at a time T is based on the error estimations of elliptic problems with right-hand sidedepending on t, t 2 ð0; T Þ. In the present paper we propose a similar procedure that can be used in thenonlinear case but with the difference that the error estimation at each fixed time T �

6 T is based on theerror estimation of only one elliptic problem. Our technique can be used for estimating spatial discreti-zation errors of p-version finite element solutions for nonlinear parabolic differential equations in severalspace dimensions.

The error estimates are obtained by solving a discrete elliptic problem in a space of local polynomials ofdegree M slightly larger than N if polynomials of degree N have been used to compute the Galerkin so-lution. Our approach can be included in the class of a posteriori estimates that have been called hierarchicalbasis estimates [3,23]. The main advantage of the procedure we propose is that it allows to improve thespatial accuracy (if it is needed) without increasing the computational cost of the overall time integration,including the cost of the error estimation. The calculation of the a posteriori error estimation (and a fortiorithe correction) is far less expensive than the computation of the numerical solution and can be carried outat every time when it is needed. The technique can also be applied to a finite element space discretization.The results can be used as a basis for an adaptive numerical procedure that controls adaptive enrichmentthrough h or p refinement.

The rest of the paper is organized as follows. In Section 2 we state the model problem and introducesome notation. In Section 3 we present an a posteriori error estimation procedure. In Section 4 we showsome numerical experiments that assess the efficiency of the error estimator. The final section is devotedto some concluding remarks.

2. Preliminaries and notation

Let us consider nonlinear parabolic equations that can be written in the form

utðt; xÞ � mDuðt; xÞ þ Rðuðt; xÞÞ ¼ f ðt; xÞ; 06 t6 T ; x 2 X;

uð0; xÞ ¼ u0ðxÞ; x 2 X;

uðt; xÞ ¼ 0; x 2 oX;

ð1Þ

where X � R2 is a domain with smooth boundary, m is a positive constant, and R is either a nonlinearconvection term

RðuÞ ¼ ðu rÞu;or a reaction term

RðuÞ ¼ gðuÞ;for some smooth function g : R ! R.

4894 J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 191 (2002) 4893–4904

We will denote by ð; Þ the inner product in L2ðXÞn and by k k0 the associated norm. The value of n willbe 2 for the convection case and 1 for the reaction one. For k integer we consider the Sobolev space HkðXÞnof functions with derivatives up to order k in L2ðXÞn. If s is noninteger HsðXÞn is defined by interpolation[18]. The norm in the Sobolev space HsðXÞn will be denoted by k ks. Finally, H 1

0 ðXÞn will be the closure ofthe set of indefinitely differentiable functions with compact support respect to the H 1ðXÞn-norm.

In what follows we will assume that the forcing term, f, and the initial data u0 in (1) are such that u,ut 2 L1 ½0; T �;HsðXÞnð Þ, sP 2. We denote by

Ki ¼ sup06 t6 T

diudti

��s

; i ¼ 0; 1: ð2Þ

In H 10 ðXÞn we consider the Dirichlet bilinear form

aðu; vÞ ¼ ðru;rvÞ; u; v 2 H 10 ðXÞn:

Although the results presented here are applicable to the general case, in order to simplify the description ofthe spectral element method, we will assume that X is a rectangular domain. Let P ¼ fXigKi¼1 be a con-forming partition of X by rectangles. If u is a function defined in X ¼ [K

i¼1Xi, the restriction of u to Xi will berepresented by ui. In the sequel k ks;i will be the HsðXÞn-norm in the subdomain Xi. We set

VN ¼ fvN 2 ðC0ðXÞÞnjvNi 2 ðPN ðXiÞÞn; vNi ¼ 0 in oX \ oXi; i ¼ 1; . . . ;Kg:The following approximation result is well known, see for example [8, Theorem 3.3],

kv� pNvka 6CN a�skvks; 06 a6 1; v 2 HsðXÞn; ð3Þwhere pN is the H 1

0 ðXÞn-orthogonal projection onto VN .The Galerkin p-finite element approximation (also known as spectral element approximation) to (1)

consists in finding uN : ½0; T � ! VN such that uN ð0; Þ ¼ pNu0 and

ðuNt ;uN Þ þ maðuN ;uN Þ þ ðRðuN Þ;uN Þ ¼ ðf ;uNÞ; 8uN 2 VN : ð4ÞThe Galerkin approximation satisfies the optimal a priori bound

sup06 t6 T

kuðtÞ � uN ðtÞka 6CN�sþa; a ¼ 0; 1: ð5Þ

3. Semidiscrete error estimation

In this section we state some results about the semidiscrete error estimation of the problem (4). Let ussuppose that the error in the Galerkin solution (4) needs to be estimated at a fixed time T �

6 T . We willshow that the estimation can be carried out by means of an elliptic a posteriori error estimator. We startby defining the abstract error estimate EN ðT �Þ 2 H 1

0 ðXÞn as the solution of the following elliptic problem[5,6],

maðEN ðT �Þ;uÞ ¼ �maðuN ðT �Þ;uÞ � ðRðuN ðT �ÞÞ;uÞ � ðuNt ðT �Þ;uÞ þ ðf ðT �Þ;uÞ; 8u 2 H 10 ðXÞn: ð6Þ

Notice that the right-hand side above is just the residual of the Galerkin solution with time frozen at levelT �. Thus, to construct the abstract error estimate the only information we need is uN ðT �Þ. We do not makeuse of the values of the numerical solution at earlier times. The abstract error estimate ENðT �Þ is, of course,not directly computable from (6) and some approximation to EN ðT �Þ will be needed. However, some of theproperties of the abstract error estimator will serve us to propose practical procedures for the estimation ofthe error.

J. de Frutos, J. Novo / Comput. Methods Appl. Mech. Engrg. 191 (2002) 4893–4904 4895

Let us denote by UN ðT �Þ ¼ uN ðT �Þ þ EN ðT �Þ 2 H 10 ðXÞn. It is easy to check that UN ðT �Þ satisfies

maðUN ðT �Þ;uÞ ¼ �ðRðuN ðT �ÞÞ;uÞ � ðuNt ðT �Þ;uÞ þ ðf ðT �Þ;uÞ; 8u 2 H 10 ðXÞn: ð7Þ

Theorem 1 below proves the fact that the function UNðT �Þ is an approximation to the solution uðT �Þ of (1)of higher order than uN ðT �Þ.

Theorem 1. Let u be the solution of it Eq. (1). Fix T � > 0. Then, there exists a positive constant C ¼CðT �;K0;K1Þ, Ki being the constants in (2), such that for N big enough, the solution UN ðT �Þ of (7) satisfies

kUN ðT �Þ � uðT �Þk0 6Cm�1N�1�s: ð8Þ

kUN ðT �Þ � uðT �Þk1 6Cm�1 logðNÞN�s: ð9ÞThe proof of this theorem is readily obtained reasoning as in [14, Theorem 1], [15, Theorem 4.10]. The

key ingredient is the following superconvergence property of the Galerkin solution (see [14,15,19,22]), thatshows that uN is a better approximation (uniformly in compact time intervals) to the elliptic projection thanto the solution u itself.

sup06 t6 T

kpNuðtÞ � uN ðtÞka 6Cm�1 logðNÞaN�s�1þa; a ¼ 0; 1:

Using the previous theorem it is easy to show that the abstract error estimator defined in (6) gives anasymptotically exact estimation.

Theorem 2. Let u be the solution of Eq. (1). Let us assume that

kuðT �Þ � uN ðT �Þka PCN�sþa; a ¼ 0; 1: ð10ÞThen,

limN!1

kENðT �Þka

kuðT �Þ � uN ðT �Þka

¼ 1; a ¼ 0; 1: ð11Þ

Proof. Let us denote by eN ðT �Þ ¼ uðT �Þ � uN ðT �Þ. Using Theorem 1 we obtain

keN ðT �Þka 6 kENðT �Þka þ kuðT �Þ � UN ðT �Þka 6 kEN ðT �Þka þ C logðNÞaN�s�1þa;

kEN ðT �Þka 6 keNðT �Þka þ kuðT �Þ � UN ðT �Þka 6 keNðT �Þka þ C logðNÞaN�s�1þa; a ¼ 0; 1:

Dividing by keN ðT �Þka and taking (10) into account (11) is reached. �

We remark that with hypothesis (10) we are merely assuming that the term of order N�s (respectivelyN�sþ1) is really present in the asymptotic expansion of the error in the L2ðXÞn (respectively H 1ðXÞn) norm.

Due to the fact that UN ðT �Þ is an improved approximation to uðT �Þ the error in the Galerkin approx-imation eN ðT �Þ ¼ uðT �Þ � uNðT �Þ can be estimated by means of ENðT �Þ ¼ UN ðT �Þ � uN ðT �Þ. It is quiteobvious that uN ðT �Þ is the Galerkin approximation to UN ðT �Þ by piecewise polynomials of degree N overthe (fixed) mesh P. Then, we have reduced the original problem to a simpler one since now we only need toestimate the error in the numerical solution of the elliptic problem (7), which right-hand side is known fromthe numerically calculated uN ðT �Þ. As a consequence, any procedure to estimate the error in a linear ellipticproblem that works for the p-version of the finite element method can, in principle, be applied to estimatethe error in the numerical solution of the nonlinear parabolic problem (1). As we have pointed out before,the literature about a posteriori error estimation procedures for linear elliptic problems is exhaustive al-though most of the results deal with the h-version of the finite element method. A similar technique can also


be used for the h-version of the finite element method when combined with the method of lines to solve (1).The relevant results in this case can be found in [16,17]. The Galerkin solution can be seen as the finiteelement solution of an elliptic problem defined in an analogous manner to (7). Now, any of the numerousprocedures to a posteriori estimate the error in the finite element solution of the above mentioned ellipticproblem can be applied to estimate the error in (1).

Although the possibility of using a standard, local if possible, asymptotically exact, a posteriori errorestimator for an elliptic problem to estimate the error in the parabolic problem could prove to be a veryefficient procedure, it has several drawbacks. The main of them is that, if the estimator has detected a notenough accurate Galerkin solution at a time T �, to improve it (before continuing with the time integration)the computation must be restarted, with a refined mesh or a larger degree of the local polynomials, from thelast time where the Galerkin approximation were considered acceptable. It is clear that this process candramatically increase the required computational cost to get a solution of the nonlinear evolutionaryproblem below a prescribed level of accuracy. The a posteriori error estimator we next present try to solvethis drawback while keeping the computational cost at a reasonable level. The advantages of the method weconsider are that allow us, not only to estimate the error committed, but also to obtain an enhanced so-lution that can replace the Galerkin one if this last approximation does not verify our accuracy require-ments. Moreover, the enhanced approximation is obtained at the fixed time where the error is beingestimated without restarting the time integration from a previous time.

Let us denote by uMN ðT �Þ 2 VM , M > N , the solution of

maðuMN ðT �Þ;uMÞ ¼ �ðRðuN ðT �ÞÞ;uMÞ � ðuNt ðT �Þ;uMÞ þ ðf ðT �Þ;uMÞ; 8uM 2 VM : ð12Þ

Then, the global error estimator at time T � is defined by

EMN ðT �Þ ¼ uMN ðT �Þ � uN ðT �Þ: ð13Þ

A local error indicator is obtained by restricting EMN at each subdomain Xi

Ei;MN ¼ EM

N jXi:

Our estimator can be included in the class of estimates that have been called hierarchical basis error esti-mates (see [3,23]). A similar estimator to (13) is considered in [19] for parabolic equations in one spacedimension, this estimator is called there a linear elliptic error estimator.

The solution uMN ðT �Þ of (12) has been called the postprocessed approximation and has been used, in adifferent context, to get an improved numerical solution to (1) without increasing the computational cost.See, for example [13–15].

We remark that the semidiscrete Galerkin equation (4) are a stiff system of ordinary differential equa-tions that must be integrated by means of some implicit time stepping procedure. Thus, each time step onehas to solve a nonlinear system of equations by means of some Newton or quasi Newton iteration and,consequently, solve a discrete Helmholtz-like problem at each iteration of the nonlinear solver. As inpractice a value ofM slightly larger than N, sayM ¼ N þ 2, usually suffices for estimation purposes (see thenumerical experiments in Section 4), the cost of estimating errors by means of (13) is nearly the same as anextra single Newton (or quasi Newton) iteration. Furthermore, for a method of lines code the estimate (13)need not to be computed for each time step, but only from time to time when it is needed. This fact greatlyreduces the computational cost required in finding an error estimate.

The proof of the following theorem can be found in [14,15].

Theorem 3. Let u be the solution of Eq. (1). Fix T > 0. Then, there exists a positive constant C ¼CðT ;K0;K1Þ, Ki being the constants in (2), such that for M > N , N big enough, the improved approximationuMN ðT �Þ satisfies


kuMN ðT �Þ � uðT �Þka 6Cm�1 logðNÞaN�s�1þa þ kpMuðT �Þ � uðT �Þka 6Cðm�1 logðNÞaN�s�1þa þM�sÞ:ð14Þ

Reasoning as in Theorem 2 but using now Theorem 3 instead of Theorem 1 one can prove that ifM ¼ OðN 1þ�Þ, � > 0 arbitrary, then the error estimator (13) is asymptotically exact. Moreover, it is easy tocheck that for every value of M > N satisfying the saturation assumption

ku� uMka 6 bku� uNka; 06 b < 1; a ¼ 0; 1; ð15Þ

our estimator verifies

C1 6kEM

N ðT �Þka

kuðT �Þ � uN ðT �Þka

6C2; a ¼ 0; 1;

for some constants C1, C2. This fact will be observed in the numerical experiments of the next section. Thesaturation assumption (15) has been introduced for a posteriori error estimates based on the use of hier-archical basis functions in [7], see also [3,23]. Notice that the saturation assumption is a quite realistichypothesis in a practical setting although it fails to be true in special cases. Furthermore, it is easy to seethat (15) is satisfied forM big enough if hypothesis (10) is fulfilled. See [3, Sections 5.2, 5.7] for a discussionabout the validity of (15).

In the next section we will show that the advantage of the error estimator we propose is that allow us toboth estimate the error committed and improve the Galerkin approximation without restarting the com-putation until a previous time.

4. Numerical experiments

In this section we present two numerical experiments to assess the efficiency of the error estimator. Wefirst consider the nonlinear convection–diffusion system

ut þ ðu rÞu ¼ mDuþ g;

for x ¼ ðx; yÞ in X ¼ ½�1; 3� � ½�1; 1� subject to homogeneous Dirichlet boundary conditions. The domainX is decomposed into two subdomains X1 and X2 with X1 ¼ ½�1; 1� � ½�1; 1� and X2 ¼ ½1; 3� � ½�1; 1�. Wehave chosen a forcing term that is time independent. More precisely, g ¼ ½f ; 0�T where f ðx; yÞ ¼ f1ðxÞf2ðyÞand

f1ðxÞ ¼ x4ðx� 2Þ4 if 06 x6 2;0 otherwise;

�

f2ðyÞ ¼ 100ðy þ 0:5Þ4ðy � 0:5Þ4 if �0:56 y6 0:5;0 otherwise:

�

The initial condition was set to uð0; x; yÞ ¼ ½u1ðx; yÞ; u1ðx; yÞ�T, u1ðx; yÞ ¼ sinððxþ 1Þðp=4ÞÞ sinððy þ 1Þðp=2ÞÞ.The value of m in this experiment is m ¼ 0:1 and the errors are estimated at time T � ¼ 1. The theoretical(‘‘exact’’) solution was computed with the standard Galerkin method using a number of degrees of freedomlarge enough and a sufficiently small time-step in the time integrator. The computed solution taken as exactis reasonably far more accurate than those shown in the experiments.

For the time integration we use a variable step linearly implicit Runge–Kutta method of order 3. Thismethod was introduced in [11] where the authors show the reliability and efficiency of this code when is used


to integrate the semidiscrete equations arising after the spatial discretization of advection–diffusionequations.

We have used hierarchical basis functions based on tensor products of integrated Legendre polynomials,see, for example, [4,10,21]. With this choice the mass and stiffness matrices have a well defined blockstructure and are very sparse. The linear systems are solved by means of a conjugate gradient method withblock diagonal preconditioning.

The nonlinear terms are evaluated pseudospectrally using tensorized discrete Legendre transform.In Fig. 1 we show in a logarithmic scale the L2-norm of the global errors against the degree N of the

local polynomials used (which is the same for the two subdomains). We have represented the real errorsby circles joined by continuous line and the estimated errors by triangles joined by discontinuous line. Inthis experiment a value of M ¼ N þ 2 was found to be enough in order to have very precise error esti-mations. We can observe in the picture that the error estimate nicely fits the real error for every value ofN. Actually, for most of the values of N, the real and the estimated errors in this experiment are almostindistinguishable.

In Fig. 2 we show, with the same notation as before, the L2-norm of the local errors at every subdomain.We point out that the local estimator reproduces very precisely the size of the real errors committed at eachsubdomain. This fact is better observed in the table bellow, where we have represented the local effectivityindex in the L2ðXÞ2-norm, defined as

H0;i ¼kEM

N ðT �Þk0;ikuðT �Þ � uN ðT �Þk0;i

:

In the table we show the local effectivity index when the error estimator is computed using M ¼ N þ 2 andM ¼ N þ 4. This index approaches one as the number of degrees of freedom increases and, for every value

Fig. 1. Convection–diffusion equation; global errors. Real error: continuous line; estimated errors: discontinuous line.


of N andM, is always very near to one. As we can see a value ofM slightly larger than N is enough to havereliable error estimations and to observe the asymptotic behaviour predicted by Theorem 2.

In the second experiment the Cahn–Allen reaction–diffusion equation in the same domain X ¼½�1; 3� � ½�1; 1� is considered.

ut ¼ mDuþ u� u3 þ f ðt; xÞ:

We set m ¼ 0:005 and measured the errors at time T � ¼ 2. We chose the forcing term such that the exactsolution was uðt; x; yÞ ¼ ð2þ cosðptÞÞu1ðxÞðy2 � 1Þ, where

u1ðxÞ ¼

1

30ðxþ 1Þ 2x� 21

4

� ��16 x6 2;

1

30ð3� xÞ x� 23

4

� �26 x6 3:

8>><>>:

This time X is decomposed into four subdomains,

N M ¼ N þ 2 M ¼ N þ 4

H0;1 H0;2 H0;1 H0;2

6 1.1618 1.1919 1.2404 1.20058 0.9789 0.9793 1.0892 1.058710 0.8125 0.8520 1.0466 1.000412 0.9644 1.0039 1.0704 1.053814 1.0193 0.9473 1.0024 1.0098

Fig. 2. Convection–diffusion equation; local errors. Real error: continuous line; estimated errors: discontinuous line.


X ¼[2i;j¼1

Xij;

with Xij ¼ ½�1þ 2ði� 1Þ;�1þ 2i� � ½�1þ ðj� 1Þ;�1þ j�.The solution is periodic in time, it is smooth in the patch of subdomains X1 ¼ X11 [ X12 and it has a

singular second derivative in X2 ¼ X21 [ X22. So, it is expected that the spectral element method gives amore precise approximation in X1 than in X2.

For the time integration we have used a linearly implicit Runge–Kutta method of order 4 with variablestep size implementation that has been proved to be more efficient for reaction-diffusion equations than theorder 3 code we used in the previous experiment, see [11]. The results with other stiff time integrators, suchas a variable step variable order BDF code for example, are the same.

In this experiment we have estimated the errors using two different choices for the degreeM of the localpolynomials to be used in (12) and (13), M ¼ N þ 2 and M ¼ N 1þ�, � ¼ 0:1. In Fig. 3 we show the L2ðXiÞ-norm, i ¼ 1; 2, of the true and estimated errors. We have represented the true errors using circles joined by acontinuous line. The estimated errors have been represented using a discontinuous line and triangles orsquares for the results obtained with the first and second choice of M respectively.

Since the solution has a singularity in the second derivative in the subdomain X2 and is smooth in X1 theerrors are much larger in X2 than in X1. Clearly, the local error estimator indicates this fact even for a valueofM slightly larger than N such as M ¼ N þ 2. The discrepancy between the estimated errors and the trueerrors in X1 is due to the effect of the time integration errors. For every point that appears in the figures wehave chosen a value of the time integration tolerance such as a smaller value of the tolerance does not leadto a reduction of the global error. This is done in order to constrain the temporal errors to be smaller thanthe spatial errors that we want to plot and estimate. Then, in Fig. 3, the time integration errors are belowthe global spatial errors and then below the local spatial errors in X2 but are comparable, or even largerthan the local spatial errors in X1. A detailed study about how to estimate both the temporal and the spatial

Fig. 3. Reaction–diffusion equation; local errors. Galerkin error: continuous line; estimated errors: discontinuous line.


errors and about how to relate them to get an optimal ratio between the error achieved and its compu-tational cost will be the subject of a future research.

It can be observed, in the experiment with M ¼ N 1þ�, that the difference between the real and estimatederror decreases as the value of N increases which is an indicator of the asymptotically exact behaviour of theerror estimator. This result is in agreement with the comments made in Section 3 since the value of Mchosen in this experiment is a power of N. By the other hand, looking at the behaviour of the error estimatein the caseM ¼ N þ 2, we observe the characteristic behaviour of an efficient error estimator. The estimatederrors approach to the real errors for the first values of N and then both, real and estimated errors, lie alongparallel lines. Furthermore, in the picture it is observed that a value of M ¼ N þ 2 it is enough in order tohave valuable error estimations.

In view of the difference between the size of the estimated errors at each subdomain one could decide toincrease the number of degrees of freedom of the spectral element approximation in X2 keeping fixed thedegree of the polynomials in X1. This would lead to get a better error in this subdomain and, in conse-quence, a better global error. Besides, taken into account that, in view of Theorem 3, the postprocessedGalerkin approximation uMN ðT �Þ ¼ uN ðT �Þ þ EM

N ðT �Þ is a better approximation to the solution than uN ðT �Þ,the error estimator is a reliable tool to both estimate the error and improve the approximation if it isnecessary. The main advantage of this procedure is that the improved approximation at time T � can beobtained, directly from uN ðT �Þ, without restarting the time integration procedure from an earlier time step.Let us observe that in an adaptive code one would like to get the desired level of accuracy with a number ofdegrees of freedom as small as possible. Next, we present a possible procedure to obtain a refined solutionand simultaneously estimate the number of degrees of freedom needed for a prescribed accuracy.

In Fig. 4 we show, in a logarithmic scale, the Galerkin spectral element errors against the degree of thelocal polynomials used in the calculation which has been taken to be equal in the four elements Xi;j, i,j ¼ 1; 2. We have used circles joined by a continuous line. Once the Galerkin approximation has beencalculated we have postprocessed the numerical solution using M ¼ N in X1 and four different values ofM

Fig. 4. Reaction–diffusion equation; global errors. Galerkin error: continuous line; postprocessed Galerkin error: stars.


in X2 ranging from M ¼ N þ 2 to M ¼ N þ 8. We have used stars to represent the errors kuNþ2N ðT �Þ �

uðT �Þk; . . . ; kuNþ8N ðT �Þ � uðT �Þk of the corresponding postprocessed approximations which have been

plotted against N. We can observe that a hierarchy of approximations that give better and better errors isobtained. Furthermore, the error in the postprocessed solution with degree N in X1 and M in X2 is nearlyequal to the error by time integrating in the full interval ½0; 2� the Galerkin equations using polynomials ofdegree M in both subdomains. Then, with the postprocessed approximation we can obtain a prescribedlevel of accuracy starting from a given spectral element approximation and increasing the degree of thepolynomials only in the patch of subdomains with larger error at the cost of solving one single discreteelliptic problem. Furthermore, kuMþ2

N ðT �Þ � uMN ðT �Þk can be used as an estimator of the error

kuMN ðT �Þ � uðT �Þk � kuMðT �Þ � uðT �Þk:Then, the hierarchy of postprocessed approximations allows to estimate the number of degrees of freedomneeded for a prescribed level of accuracy. This means that if at some time T a lack of precision is detected,we can adjust the number of degrees of freedom without restarting the computation from an earlier timestep. In this way, it is possible to space out the error estimations along the time integration interval ofinterest, reducing the total number of discrete elliptic problems to be solved. It is clear that when solving anevolutionary equation, this possibility greatly reduces the cost of the error estimation.

5. Conclusions

We have presented an a posteriori error estimator for the p-version of the finite element method (spectralelement method) for nonlinear parabolic differential equations. The main idea is that the estimation of theerror in the nonlinear parabolic problem can be reduced to the estimation of the error in a particular linearelliptic problem, which right-hand side depends only on the computed numerical solution and known dataof the problem. In principle, any of the numerous procedures existing in the literature to estimate the errorin an elliptic problem can be applied to get the estimation of the error of the nonlinear parabolic problem.The estimation will be asymptotically exact if the elliptic error estimator has this quality.

The a posteriori error estimator we propose is efficient and asymptotically exact. Although the estimatorrequires to solve a global discrete elliptic problem it is not expensive to implement compared with the costof the time integration. As the semidiscrete Galerkin equations are a system of stiff ordinary differentialequations, an implicit method must be used for the time integration. Then the cost of estimating errors isonly slightly larger than the cost of a single Newton or quasi Newton iteration. Besides, the so calledpostprocessed approximation that can be used to estimate the error is an improved approximation to thesolution of the nonlinear parabolic problem at each time when it is calculated. Then, it can also be used toreplace the Galerkin approximation whenever this solution does not satisfy the prescribed accuracy re-quirements. Moreover, we have shown that a hierarchy of postprocessed approximations, each one moreaccurate than the other, at a fixed time, can be constructed with the only cost of solving one extra discreteelliptic problem per approximation. This procedure allows to estimate the number of degrees of freedomneeded for a prescribed level of accuracy. Overall, the methodology we have presented has the necessaryingredients to design an efficient adaptive procedure to time integrate nonlinear parabolic equations.

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