IEEE TRANSACTIONS ON MAGNETICS, VOL. 24, NO. I, JANUARY 1988 299

LOCAL ERROR ESTIMATES FOR ADAPTIVE MESH REFINEMENT

P. Fernandes*, P. Girdin io+, P. Molfino+, M. Repetto+

(*) Istituto per la Matematica Applicata del CNR Via L.B. Albertis 4, 16132 Genova GE (Italy)

Abstract: In this paper five methods for local error estimation in a finite element solution for adaptive meshing are analyzed. All these methods are "a posteriori", single solution, element by element error estimators. The first two of them are based on the estimation of the first neglected term in the series expansion of the solution, the third one is based on the approximate solution of a local boudary value problem with hierarchical shape functions and the last two utilize a residual evaluation. All the above methods are described and implemented in a common environment and the results obtained in a simple test case are presented and discussed.

Introduction

The evaluation of the discretization error in numerical modelling is, at present , experiencing a considerable research attention, both because of its importance for the validation of results of numerical field computation and because of its key role in adaptive mesh refinement [1-6].

In fact, in adaptive structures error estimation is a necessary step to obtain a refined mesh ensuring control of error according to predefined goals, that is to obtain "optimal" meshes.

To this aim the estimation of the discretization error has to be as "robust" as possible, to work properly with several different physical modelling and geometrical shapes of the problem but, at the same time the procedure should not be too computationally expensive.

Because of the last reasons the authors have avoided to analyze, in this paper, methods requiring dual finite element solutions of the same problem or methods requiring the solution of local error problems on "finite patches", made up with several elements, to estimate the error on each element [3].

In this paper five "a posteriori" error estimators, based on a single solution of the problem and on "element by elementll error estimation procedures, have been analyzed and compared.

Two procedures are based on approximation theory and three on the evaluation on the elements.

polynomial residual

All algorithms, most of which have already been proposed as such or in similar forms in the literature [2,6,7], are implemented in a common environment and

results of application to a standard test case are compared and discussed.

Methods based .l2!!. polynomial interpolation theory

As it is well known from the polynomial interpolation theory, the error committed by replacing an analytical function f by a n-th order polynomial approximation p, is given by a quantity related to the (n+l)th derivative of the analytical function.

Since this quantity is not known when the function f is the unknown 0.£ the problem, it can be replaced by its estimate computed by means of an higher order polynomial approximation of f.

The solution obtained by a first order finite element discretization can be considered roughly equivalent to a set of Taylor series expan sions centered in each node and truncated after the first order term.

In fact, an e stimate of the potential is provided in each node, and the first order derivatives are

(+) Dipartimento di Ingegneria Elettrica - Univ. Genova Via All'Opera Pia lla, 16145 Genova GE (Italy)

defined on the elements as the gradient of the shape function.

Therefore an estimation of the error requires an evaluation of the first neglected term in the series expansion, which is proportional to the second order spatial derivatives of the solution and should be estimated with One of several possible ways.

In this paper the first order derivatives have been obtained using two different methods. The first one computes the first order derivatives in the nodes by differentiating the shape functions of the elements and averaging, with or without weigths, their v alues on the elements surrounding each node. The second one computes the field values in the nodes by means of a five-links generalized finite difference method suitable for irregular grids [8].

One error estimation method, based on polynomial interpolation theory, has been proposed by some of the authors [7] and it is based on the evaluation of the term

g = Igradlgrad ul I (1)

where U is the computed finite element solution. As it can be seen from equation (I), the quantity

g is proportional to the second order derivative of U, which is related to the first neglected terms in a first order finite element discretization.

In order to evaluate the quantity g, the gradient of the solution is not computed by differentiating the first order shape function, but it is obtained assigning to the nodes field values calculated using the five-links method and linearly interpolating them on the elements.

This error estimation will be referred to in the following as the "gradient of field" method .

Another error estimation method, based on the same theory and proposed in [6], performs the computation of the error e on an element as the integral of the difference between the field B, obtained differentiating the shape function, and the nodal averaged field BS interpolated in the element with the same first order shape function, that is:

e = .rfi (B-Ils )dfi (2) It can be shown that , in the case of first order

elements, also this quantity is related to the first neglected term in an element centered Taylor series

expansion. This method will be referred to in the following

as the "field difference" method.

Methods based on residual evaluation

The residual evaluation is another approach to error estimation, and it is more closely connected with finite element formulations.

In fact, considering a Poisson problem defined by the differential equation

_\72u = f (3) in a region fi with boundary conditions on the boundary r = rn lJ rd

an = q on I� (4) it can be shown [3] that the approximate finite element

OOI8-9464/88/0100-0299$Ol.OO©1988 IEEE

300

solution u satisfies the equation

u = u - r d d

- V2(j = f - r - p on 0

on au - = q - r an n

(5) (6)

where the residuals r are defined as:

r = f + V2;:; on 0 U ou

rd = ud - on ld rn = q onfo an (8) while p represents the discontinuity in the normal derivatives on element sides rk not contained in r, and is defined as:

p =� [ :� lout - :� linllk Jk (9)

where 8k is the Dirac impulse function defined as non zero on fk•

Defining the error

it follows

_V2e =

e = rd on ld

e = u - u

r + p

ae an

onO

rn on fn

( 10)

(11) (12)

As it can be seen in [2] the error problem can be approximated by a Neumann problem on the i-th element Qi' that is

on (l i

� = r on ao. n r an n 1 n (13)

(14)

fn = � [ :� lout - :� lin].oi-r on 30Cf (15) Then this local problem can be solved by means of

hierarchical shape functions [2,3] obtaining directly the error estimation. Because of the special structure of hierarchical shape functions, the system to be solved is a 3x3 one for every element of the mesh.

In order to obtain an error estimation on the element it 1s necessary to calculate a norm of the error. Among the various possibilities, the most natural norm used for these problems is the energy norm [3] that is

2 , 2 Ilell =J Wei dO E � (16)

This method will be referred to in the following as the "local error problem" method.

A simplified approach is to utilize as an error estimator directly the driving function of the error equation previously defined, thus avoiding to solve it.

Under the above hypotheses this error estimate can be written as:

s = J IrldO+) +-\ au I - au I. \ d1 +l Irn Id1 (17) 0i aO.-f 2 an out an 1n aQ,nr 1 1 n

where the variables are those previously defined, and absolute values avoid that cancellation of contributions could lead to an underestimation of the error. In the following this method will be referred to as "complete residual".

A third error estimating procedure proposed by the authors is based on the evaluation of the term

R = r {f+V2 ;;)dQ (W) 'n. 1 In order to be evaluated with first order finite

element , the quantity R is transformed in a first order operator by means of Green formulas. The gradient of the solution contained in the resulting line integral is then obtained by linearly interpolating the nodal field values computed by five-links finite difference. This method does not take into account the contribution of the boundary residuals. Because of this reason the method will be referred to as "incomplete reSidual".

Example £f application

All these five methods have been implemented in a first order electromagnetic finite element package, named CEDEF, developed by a group of italian universities including that of the authors, and have been tested in a simple L-shaped magnetic case.

A problem which arises, after obt aining an error estimation on each element, is to identify the elements that have to be refined in the following step of an adaptive meshing procedure.

In order to obtain a reliable procedure in the widest possible set of cases, all methods based on assumptions on the statistical distribution of the error [9] have been discarded.

The procedure selected by the authors has been to assume as error comparison parameter of each element the ratio between the actual value of the error estimate on the element and its maximum value on the problem domain.

All five error estimation procedures have been implemented and tested for the example defined above, with various kind of simple regular meshes. Results obtained grouping the elements by 20% differences in the error comparison parameter for one of the above meshes are shown in Figs. 1 to 5.

In this simple case, an "optimal" procedure should produce error estimations greater around the upper sharp corner of the domain, and rapidly decreasing symmetrically elsewhere, so that the quality of results can be evaluated by inspection.

In table I the execution times of the various procedures as percentage of the time needed to obtain the finite element solution are shown.

However, the time comparison among the various procedures should not be considered as an absolute one, since the five error estimation routines use different data, which may or may not be directly available in the data structure depending on the particular finite element code under consideration.

TABLE I

Ratio of CPU times for the execution of the error estimation procedures.

Methods t/ttot t/ Soin (*) (+)

gradient of field 6% 1.0 field difference 10% 1.46 local error problem 22% 3.3 complete residual 12% 1.77 incomplete residual 6% 1.0

(*) ratio of the time of the current procedure with respect to the time needed for the finite element solution.

(+) ratio of the time of the current procedure with respect to the faster one.

gadient of field % of mo):

Fig. 1. Results obtained grouping the elements by 20% differences in the error comparison parameters for the "gradient of field" method.

field difference % of max

Fig. 2. Results obtained grouping the elements by 20% differences in the error comparison parameters for the "field difference" method.

local error problem % of max

,O���\. ;.; 4Q 2Q · ..

301

Fig. 3. Results obtained grouping the elements by 20% differences in the error comparison parameters for the "local error problem" method.

complete residJOI % of max

Fig. 4. Results obtained grouping the elements by 20% differences in the error comparison parameters for the "complete residual" method.

302

incomplete residual % of mox

Fig. 5. Results obtained grouping the elements by 20% diffe rences in the error comparison parameters for the Itinc omplete residual" method.

Discussion

As it can be seen in the figures of the previous section, all the five methods seem to be able to provide a reasonable definition of the area re quiring refining, in the simple case tested. The experience gained in the implementation of the methods allows a comparative consideration in the follow ing key issues:

Implementation: all the five error estimation procedures do not require input data unusual with respect to the classical methods of a finite element code, so that they should not require significant changes in the main code, except for a possible optimization of data structures.

Dependence on problem equation: only the first two methods are independent from the differential equation of the problem while the last three are based on Poisson equation. However, it should be possible in principle to adapt them to other differential equations by changing the estimation routines.

"Robustness" of the methods the "gradient of field", "difference offfeld" and "incomplete residual" methods need the field values in the mesh nodes. In general, the use of numerical derivatives of the solution reduces the "robustness" of a procedure, since those quantities are difficult to evaluate rel iably, particularly on boundaries and interfaces. Furthermore, the fields can be computed in several ways, and the experience with the three methods requiring derivatives has shown that each method works better with a specific field computation procedure.

computational load: as it can be seen from Table I, and even within the limits previously mentioned, all methods are notably faster than the solution of the problem and so they appear significantly more appealing than methods requiring a dual or complementary solution.

The ratio of the time required by the slowest to

the fastest methods, of the order of 2.5, is certainly exceeding the uncertainty range connected to the possible mismatching of data structures, and makes then appealing the fastest methods from this standpoint. However, the slowest methods are those which are theoretically more grounded as reliable error estimators, while the fastest methods seem to be related to the actual error in a much more qualitative way.

As a consequence, the fastest methods seem to be better suited for fast interactiVe adaptive meshing, requiring IIreal timen results, while the slowest methods seem to provide better error estimates for validation of finite element solutions,

Further work will be needed to asses, under more realistic conditions, the comparative reliability of the fast estimators for adaptive meshing, whereas the more complex methods for quantitative error evaluation should be also compared with estimators using "patches II

of more than one element to evaluate the error on each element and with other possible methods of error estimation, based on higher order local solutions.

References

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[2] R.E. Bank and A. Weiser, "Some a-posteriori error estimator for elliptic partial differential equations", Mathematics of computation, Vol. 44, pp. 283-301, April 1985.

[3J D.W. Kelly, J.P. De S.R. Gago and O.C. Zienkiewicz, "A-posteriori error analysis and adaptive processes in the fini te element method: part I - error analysis", Inter. Journ. for Numerical Meth. in Engineering, Vol. 19, pp. 1593-1619, 1983.

[4] A.R. P inchuk and P.P. Silvester, "Error estimation for automatic adaptive finite element mesh generation", IEEE Transaction on Magnetics, Vol. MAG-21 , pp. 2551-2554, November 1985.

[5J J. Penman and M.D. Grieve, "An appr oach to self adaptive mesh generationll, IEEE Transactions on Magnetics, Vol MAG-21, pp.--z567-2570, November 1985.

[6 J C. S. Biddlecombe, "Error analysis electromagnetic Magnetics, Vol. 1986.

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[8] G. �olinari, M.R. podesta', G. Sciutto and A. Viviani, "Finite difference method with irregular grid and transformed discretization metric", IEEE PES Winter Meeting, New York, USA, Jan. 29 - Feb. 3, 1978, paper A78, pp. 288-293.

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