CALCOLO 37, 65 – 77 (2000)

CALCOLO
© Springer-Verlag 2000

An adaptive finite element discretisation

for a simplified Signorini problem
H. Blum, F.-T. Suttmeier
Universität Dortmund, Fachbereich Mathematik, Lehrstuhl X, Vogelpothsweg 87, 44221
Dortmund, Germany
e-mail: Heribert.Blum@math.uni-dortmund.de;
Franz-Theo.Suttmeier@math.uni-dortmund.de

Received: May 1999 / Accepted: October 1999

Abstract. Adaptive mesh design based on a posteriori error control is stud-

ied for finite element discretisations for variational problems of Signorini
type. The techniques to derive residual based error estimators developed,
e.g., in ([2,10,20]) are extended to variational inequalities employing a suit-
able adaptation of the duality argument [17]. By use of this variational ar-
gument weighted a posteriori estimates for controlling arbitrary functionals
of the error are derived here for model situations for contact problems. All
arguments are based on Hilbert space methods and can be carried over to the
more general situation of linear elasticity. Numerical examples demonstrate
that this approach leads to effective strategies for designing economical
meshes and to bounds for the error which are useful in practice.

1 Introduction

A fundamental model situation for contact problems in elasticity is Sig-

norini’s problem describing the deformation of an elastic body which is
unilaterally supported by a frictionless rigid foundation. We intend to derive
efficient a posteriori error control techniques for this equation with special
emphasis on local error phenomena, e.g., the error for stresses in the contact
zone. In order to demonstrate the concept for our method for a posteriori
error estimation, we first consider the simplified case
66 H. Blum, F.-T. Suttmeier

−1u = f in  ⊂ R2 ,
u = 0 on 0D , (1.1)
u ≥ 0, ∂n u ≥ 0, u ∂n u = 0 on 0C ,

where 0C = ∂ \ 0D and ∂n u = ∇u · n.
Problem (1.1) is to be solved by the finite element Galerkin method on
adaptively optimised meshes. By variational arguments, we derive weighted
a posteriori error estimates for controlling arbitrary linear functionals of the
error. This approach leads to effective strategies for designing economical
meshes and to bounds for the error which are useful in practice. The extension
to Signorini’s problem is illustrated in the last section.
The basis for applying the finite element method to (1.1) is the formu-
lation as a variational inequality, i.e., a solution u ∈ K is sought which
satisfies

(∇u, ∇(ϕ − u)) ≥ (f, ϕ − u) ∀ϕ ∈ K, (1.2)

where we set V = {v ∈ H 1 | v = 0 on 0D } and K = {v ∈ V | v ≥ 0

on 0C }. Here, and in what follows, H m = H m () denotes the standard
Sobolev space of L2 -functions with derivatives in L2 () up to order m.
Equation (1.2) is uniquely solvable (cf. Lions and Stampacchia [14])
and, under appropriate smoothness conditions on the boundary and data,
the solution is known to satisfy the regularity result u ∈ H 2 () (see Brézis
[3]).
In the following, we apply the finite element method on decompositions
Th = {T } of  consisting of quadrilaterals T satisfying the usual condition
of shape regularity. Simplifying notation, we assume the domain  to be
polyhedral in order to ease the approximation of the boundary. More general
situations may be treated by the usual modifications. For ease of mesh re-
finement and coarsening, hanging nodes are allowed in our implementation.
The width of the mesh Th is characterised in terms of a piecewise constant
mesh size function h = h(x) > 0, where hT := h|T = diam(T ). We use
standard bilinear finite elements to construct the spaces Vh ⊂ V and assume
that Kh = K ∩ Vh .
Eventually, the finite element approximation uh of u in (1.2) is deter-
mined by

(∇uh , ∇(ϕ − uh )) ≥ (f, ϕ − uh ) ∀ϕ ∈ Kh . (1.3)

This finite dimensional problem can be shown to be uniquely solvable fol-

lowing the same line of arguments as in the continuous case. Optimal order
a priori error estimates in the energy norm have been given, for example,
An adaptive finite element discretisation for a simplified Signorini problem 67

in Falk [7] and Brezzi et al. [4]. Dobrowolski and Staib [6] show O(h)-
convergence in the energy norm without additional assumptions on the struc-
ture of the free boundary. Error estimates with respect to the L∞ -norm have
been obtained, e.g., by Nitsche [16] based on a discrete maximum principle.
Below, we shall demonstrate how functionals J (u − uh ) of the error can
be controlled in an a posteriori manner, i.e., we estimate the error in terms
of quantities at the element level containing only the discrete solution and
the data of the problem.

2 A posteriori error estimate

For elliptic variational equalities, i.e., in the case K = V , many different,

but related, approaches for a posteriori error control have been developed
in the last two decades; see, e.g., Verfürth [21] for a survey. Most estimators
are designed to control the error in the energy norm. A general concept for
estimating e = u − uh for more general error measures given in terms of
a linear functional J (·) has been proposed in Becker and Rannacher [2]
and further developed, e.g., in Kanschat [10] and Suttmeier [20]. One main
ingredient in deriving such residual based a posteriori estimates is a duality
argument known as the “Aubin–Nitsche trick” from a priori analysis. In
principle, such techniques can be carried over to variational inequalities
when, for example, penalty techniques are employed to avoid the explicit
treatment of the constraints. This again leads to variational equalities of the
form mentioned above.
In the present paper, we attack the original unpenalised problem. Since
we are mainly interested in local phenomena like the normal stress on the
contact surface, we intend to adopt local control techniques to estimate a
functional J (e).
In Natterer [17], there is described a generalisation of Nitsche’s trick
for variational inequalities, which we employ to derive an a posteriori error
estimate for the scheme (1.3). To this end, we consider the dual solution
z ∈ G of

(∇(ϕ − z), ∇z) ≥ J (ϕ − z) ∀ϕ ∈ G, (2.1)

R
where G = {v ∈ V | v ≥ 0 on Bh and 0C ∂n u(v + uh ) ≤ 0} and Bh =
{x ∈ 0C | uh (x) = 0}.
In order to show that z + u − uh ∈ G, we observe that z + u − uh ≥ 0
on Bh , since on Bh ⊂ 0C we have u ≥ 0, uh = 0. Furthermore,
Z Z Z
∂n u((z + u − uh ) + uh ) = ∂n uz ≤ − ∂n uuh ≤ 0.
0C 0C 0C
68 H. Blum, F.-T. Suttmeier

Now, we can choose ϕ = z + u − uh as a test function in (2.1) and obtain

J (e) ≤ (∇(u − uh ), ∇z).
Next, we use the solution U ∈ V of the nonrestricted problem
(∇U, ∇ϕ) = (f, ϕ) ∀ϕ ∈ V ,
to rewrite (1.2) and (1.3) in the form
(∇(U − u), ∇(ϕ − u)) ≤ 0 ∀ϕ ∈ K, (2.2)
(∇(U − uh ), ∇(ϕ − uh )) ≤ 0 ∀ϕ ∈ Kh . (2.3)
It is easily seen that uh ∈ Wh = {v ∈ V | v ≥ 0 on Bh } ∩ Vh , i.e., uh
coincides with the solution ũh ∈ Wh of the discrete variational inequality
(∇(U − ũh ), ∇(ϕ − ũh )) ≤ 0 ∀ϕ ∈ Wh . (2.4)
With zh ∈ Wh and choosing ϕ = uh + zh in (2.4) we see that the first
term on the right-hand side of the identity

(∇(u − uh ), ∇zh ) = (∇(U − uh ), ∇zh ) + (∇(u − U), ∇(zh + uh − u))

+ (∇(u − U), ∇(u − uh )) ∀zh ∈ Wh , (2.5)
is negative. So also is the last term by taking ϕ = uh in (2.2). To sum up,
we have shown the inequality
(∇(u − uh ), ∇zh ) ≤ (∇(u − U), ∇(zh + uh − u)) ∀zh ∈ Wh . (2.6)
We now proceed with estimating J (e) by
J (e) ≤ (∇(u − uh ), ∇(z − zh )) + (∇(u − uh ), ∇zh )
≤ (∇(u − uh ), ∇(z − zh )) + (∇(u − U), ∇(zh + uh − u))
= (∇(u − uh ), ∇(z − zh )) + (∇(u − U), ∇(z + uh − u))
+ (∇(u − U), ∇(zh − z)).
Due to u∂n u = 0 on 0C , we have, for z ∈ G,
Z
(∇(u − U), ∇(z + uh − u)) = ∂n u(z + uh ) ≤ 0.
0C

Eventually, we obtain the a posteriori error estimate

J (e) ≤ (∇(U − uh ), ∇(z − zh )). (2.7)
With standard techniques this can be exploited as follows. Element-wise
integration by parts yields
X
J (e) ≤ (f + 1uh , z − zh )T − 21 ([∂n uh ], z − zh )∂T , (2.8)
T ∈Th
An adaptive finite element discretisation for a simplified Signorini problem 69

where, for interior interelement boundaries, [∂n uh ] denotes the jump of the
normal derivative ∂n uh . Furthermore we set [∂n uh ] = 0 and [∂n uh ] = ∂n uh
on edges belonging to 0D and 0C respectively.
From (2.8), we deduce the a posteriori error bound
X
|J (e)| ≤ ωT ρT =: ηweight , (2.9)
T ∈Th

with the local residuals ρT and weights ωT defined by

1/2
ρT := hT kf + 1uh kT + 21 hT kn · [∇uh ]k∂T ,
n o
−1/2
ωT := max h−1T kz − z k
h T , h T kz − z k
h ∂T .

In general, the weights ωT cannot be determined analytically, but have

to be computed by solving the dual problem numerically on the available
mesh. To this end, interpreting zh as a suitable interpolant of z, one uses the
interpolation estimate
ωT ≤ Ci,T hT k∇ 2 zkT , (2.10)
for z ∈ H 2 (T ). For less regular z an estimate similar to (2.10) could be used
involving a lower power of a local mesh size, which typically corresponds to
higher values of ωT . To evaluate the right-hand side in (2.10) one may simply
take second order difference quotients of the approximate dual solution z̃h ,
ωT ≈ ω̃T := C̃i,T h2T |∇h2 z̃h (xT )|, (2.11)
where xT is the midpoint of element T . This results in approximate a pos-
teriori error bounds using
X
ηweight ≈ ω̃T ρT . (2.12)
T ∈Th

It has been demonstrated in Becker and Rannacher [2] that this approxi-
mation has only minor effects on the quality of the resulting meshes. The
interpolation constant Ci may be set equal to one for mesh designing.
In the following, we compare this weighted estimator against the tradi-
tional approach of Zienkiewicz and Zhu [22]. This error indicator for finite
element models in structural mechanics is based on the idea of higher–order
stress recovery by local averaging. The element-wise error kσ − σh kT is
thought to be well represented by the auxiliary quantity kMh σh − σh kT ,
where Mh σh is a local (super-convergent) approximation of σ . The corre-
sponding (heuristic) global error estimator reads
X 1/2
kσ − σh k ≈ ηZZ := kMh σh − σh k2T , (2.13)
T ∈Th

with σ = ∇u and σh = ∇uh .

70 H. Blum, F.-T. Suttmeier

Remark The choice of (2.1) is not uniquely determined. Other approaches

in a priori analysis in similar situations can be found, e.g., in Mosco [15].
Here separate dual problems for the negative and positive part of the error
are considered, but it seems to be difficult to exploit these techniques for a
posteriori analysis, since the data of the problem do not enter the estimate
directly.

3 Numerical results

The implementation is based on the tools of the object-oriented FE package

DEAL [1]. The solution process is simply done by an iteration of Gauss-
Seidel-type (cf. Glowinski et al. [9]). The solutions on very fine (adaptive)
meshes with about 200,000 cells are taken as reference solutions uref for
determining the relative errors

E rel := |J (uh ) − J (uref )|/|J (uref )|

on coarser meshes, while

η(uh )
Ratio :=
|J (uref ) − J (uh )|
are the overestimation factors of the error estimators.
Let an error tolerance TOL or a maximal number of cells Nmax be given.
Starting from some initial coarse mesh the refinement criteria are chosen
in terms of the local error indicators ηT := ωT ρT . Then, for the mesh
refinement, we use the following fixed fraction strategy: in each refinement
cycle, the elements are ordered according to the size of ηT and then a fixed
portion (say 30%) of the elements with largest ηT is refined which results
in about a doubling of the number N of cells. This process is repeated until
the stopping criterion η(uh ) ≤ TOL is satisfied, or Nmax is exceeded. For
the numerical tests given below, we confined ourselves to 8 adaptive cycles.
The corresponding values for Nmax can be taken from the tables below.
For determining J (uref ), we employ an adaptive algorithm based on
(2.12), where in every third adaptive step we also do a global refinement.
The approximation of the dual problem (2.1)

(∇(ϕ − z), ∇z) ≥ J (ϕ − z) ∀ϕ ∈ G,

is realised as follows. Assuming ∂n u > 0 on Bh and ∂n u = 0 on 0C \ Bh

suggests approximating G by G̃ = {v ∈ V |v = 0 on Bh }. Therefore, we
only have to solve a linear Dirichlet problem with zero boundary conditions
on 0D + Bh .
An adaptive finite element discretisation for a simplified Signorini problem 71

Examples

As a test example, we consider (1.1) on  = (0, 1)2 , 0D = {(x1 , x2 ) ∈

∂|x1 = 0} and right-hand side f = 1000 sin(2π x1 ). The contact set
B = {x ∈ 0C |u(x) = 0} in this case is determined by B = {(x1 , x2 ) ∈
0C |x1 ≥ b} with b ≈ 0.609374 taken from uref . The structure of the solution
is sketched in Fig. 1 (left).
Applying an adaptive algorithm on the basis of the indicator ηZZ yields
locally refined grids with a structure shown in Fig. 1, which can be compared
with the grids based on ηweight for the following examples (Figs. 2, 3 and 4).

Fig. 1. Isolines of the solution (left) and structure of grids produced on the basis of ηZZ
(right)

1) Point value. For the first test, we choose

J (ϕ) = ϕ(x0 ), x0 = (0.25, 0.25),

to control the point-error in x0 . The computational results are shown in Table

1. Evaluating Ratio shows the constant relation between true error and the
corresponding estimation, and consequently it is demonstrated that the pro-
posed approach to a posteriori error control gives useful error bounds. Fig. 2
(left) shows that ηweight produces a (monotonically) converging scheme with
respect to the point value in contrast to the ZZ-approach. Fig. 2 (right) shows
the structure of grids produced on the basis of ηweight .
2) Mean value. As error functional for the second test, we choose
Z
J (ϕ) = ∂n ϕ,
B
72 H. Blum, F.-T. Suttmeier

Table 1. Numerical results for the first test example: functional value J (uh ), relative error
E rel and over-estimation factor Ratio

Cells J (uh ) E rel Ratio

484 2.928820e+01 1.254902e-03 4.25
928 2.928258e+01 1.446547e-03 2.07
1720 2.929928e+01 8.770673e-04 2.64
3148 2.930866e+01 5.572038e-04 2.97
5572 2.931476e+01 3.491901e-04 3.14
9604 2.931715e+01 2.676897e-04 3.40
16468 2.931918e+01 1.984655e-04 3.96
27724 2.932013e+01 1.660699e-04 2.99

0.01
weighted
ZZ

0.001
Error

0.0001

1e-05

1e-06
100 1000 10000
Number of Elements

Fig. 2. Relative error for the first example on adaptive grids according to the weighted esti-
mate and the ZZ-indicator (left) showing that ηweight produces a (monotonically) converging
scheme with respect to the point value. Structure of grids produced on the basis of ηweight
(right)

to control the mean value of the normal derivative along the contact set B.
In this case, the treatment of J , which is determined by derivatives of u,
requires some additional care since in this case the functional is singular,
i.e., the dual solution is not properly defined on G. The remedy (cf. Becker
and Rannacher [2] and see Rannacher and Suttmeier [18] for an application
in linear elasticity) is to work with a regularised functional J ε (.). In the
present case, we set
Z
ε −1
J (ϕ) = |Bε | ∂n ϕ dx,
Bε
An adaptive finite element discretisation for a simplified Signorini problem 73

Table 2. Numerical results for the second test example: functional value J (uh ), relative
error E rel and over-estimation factor Ratio

Cells J (uh ) E rel Ratio

1840 -1.730673e+02 1.273645e-02 1.51
3256 -1.739847e+02 7.503137e-03 1.96
5980 -1.745723e+02 4.151169e-03 2.50
10528 -1.748522e+02 2.554478e-03 2.81
19204 -1.750084e+02 1.663434e-03 2.47
34540 -1.750833e+02 1.236167e-03 3.90
65212 -1.751289e+02 9.760411e-04 3.85
122284 -1.751526e+02 8.408443e-04 2.67

1
weighted
ZZ
0.1

0.01
Error

0.001

0.0001

1e-05
100 1000 10000
Number of Elements

Fig. 3. Relative error for the second example on adaptive grids according to the weighted
estimate and the ZZ-indicator (left) demonstrating ηweight to be most economical. Structure
of grids produced on the basis of ηweight (right)

where Bε := {x ∈ , dist(x, B) < ε}. For each adaptive computation, the

regularisation is done with the choice ε = 0.5ηweight (uh ), where uh is taken
from the previous step.
The numerical results are presented in Table 2. Again, it is demonstrated
that the proposed approach to a posteriori error control gives useful error
bounds. In Fig. 3 (left) the relative errors on adaptive grids according to the
weighted estimate and the ZZ-indicator are depicted, demonstrating ηweight
to be most economical. Figure 3 (right) shows the structure of grids produced
on the basis of ηweight .
74 H. Blum, F.-T. Suttmeier

3) Normal derivative. For the third test, we choose

J (ϕ) = ∂n ϕ(x0 ), x0 = (1.00, 0.25),

to control the point error of the normal derivative in x0 . This example is

chosen to indicate the applicability of the proposed techniques for our final
goal of a posteriori error estimation of contact stresses in elasticity problems.
Again the treatment of J has to be done by regularisation as in the second
example. Again the results presented in Table 3 and Fig. 4 demonstrate ηweight
to be reliable and efficient.

Table 3. Numerical results for the third test example: functional value J (uh ), relative error
E rel and over-estimation factor Ratio

Cells J (uh ) E rel Ratio

304 1.140179e+02 8.022446e-03 1.77
628 1.146645e+02 2.396903e-03 1.59
1312 1.148638e+02 6.629546e-04 2.17
2548 1.149107e+02 2.549156e-04 2.17
4912 1.149301e+02 8.613189e-05 3.27
9208 1.149339e+02 5.307117e-05 2.29
17200 1.149363e+02 3.219071e-05 2.08
31468 1.149372e+02 2.436054e-05 1.71

1
weighted
ZZ

0.1
Error

0.01

0.001
100 1000 10000
Number of Elements

Fig. 4. Relative error for the third example on adaptive grids according to the weighted
estimate and the ZZ-indicator (left) demonstrating ηweight to be most economical. Structure
of grids produced on the basis of ηweight (right)
An adaptive finite element discretisation for a simplified Signorini problem 75

4 Outlook: Application to Signorini’s problem

We now demonstrate how the above techniques might be extended for a

posteriori error control to Signorini’s problem which, in classical notation,
reads (cf. Kikuchi and Oden [11] )

− div σ = f, Aσ = ε(u) in ,
u = 0 on 0D , σ · n = t on 0N , (4.1)

σT = 0, (un − g)σn = 0
on 0C .
un − g ≤ 0, σn ≤ 0 

This idealised model describes the deformation of an elastic body occupying

the domain  ⊂ R3 , which is unilaterally supported by a frictionless rigid
foundation. The displacement u and the corresponding stress tensor σ are
caused by a body force f and a surface traction t along 0N . Along the portion
0D of the boundary the body is fixed and 0C ⊂ ∂ denotes the part which
is a candidate contact surface. We use the notation un = u · n, σn = σij ni nj
and σT = σ · n − σn n, where n is the outward normal of ∂, and g denotes
the gap between 0C and the foundation.
Further, the deformation is assumed to be small so that the strain tensor
can be written as ε(u) = 21 (∇u+∇uT ). The compliance tensor A is assumed
to be symmetric and positive definite.
The weak solution u ∈ K of (4.1) is defined by the variational formula-
tion

a(u, ϕ − u) ≥ F (ϕ − u) ∀ϕ ∈ K, (4.2)

with the definitions

V = {v ∈ H 1 × H 1 | v = 0 on 0D }, K = {v ∈ V | vn − g ≤ 0},
Z
a(v, ϕ) = A−1 ε(v)ε(ϕ) ∀v, ϕ ∈ V ,

Z Z
F (ϕ) = fϕ + tϕ ∀ϕ ∈ V .
 0N

As above, the discrete solution uh ∈ Kh = K ∩ Vh ⊂ V is determined by

a(uh , ϕ − uh ) ≥ F (ϕ − uh ) ∀ϕ ∈ Kh . (4.3)

Again, for estimating measures defined by J (.) of e = u − uh , we employ

z ∈ G given by

a(ϕ − z, z) ≥ J (ϕ − z) ∀ϕ ∈ G, (4.4)
76 H. Blum, F.-T. Suttmeier

where G = {v ∈ V | v ≥ 0 on Bh and a(U − u, v + uh − u) ≥ 0} and

Bh = {x ∈ 0C | uh (x) · n = g(x)}. In the above U denotes the solution of

a(U, ϕ) = F (ϕ) ∀ϕ ∈ V .

Eventually, the techniques used for the

Pmodel case yield an a posteriori error
estimate of the form (2.9) |J (e)| ≤ T ∈Th ωT ρT with

ρT := hT kf + div(A−1 ε(uh ))kT + 21 hT k[n · A−1 ε(uh )]k∂T ,

1/2
n o
−1/2
ωT := max h−1T kz − z h k T , h T kz − z h k ∂T .

The approximation of the dual problem (4.4) may be realised as follows.

Assuming Bh to be an appropriate approximation of B suggests replacing G
by G̃ = {v ∈ V |v = 0 on Bh } and solving a linear elasticity problem with
Dirichlet boundary conditions on 0D + Bh .
A similar situation is given for nonlinear variational equalities, where the
dependence of the dual operator on u and uh is in practice simply expressed
in terms of the computed uh alone. The experiences in the case of the sta-
tionary Navier–Stokes equations (see Becker and Rannacher [2]) and for
nonlinear elasto-plastic material behaviour (see Rannacher and Suttmeier
[19]) indicate that for these examples the perturbation of the dual problem
is not critical in stable situations. In the present case, investigations of the
influence of the approximation of (4.4) on the accuracy of the resulting a
posteriori estimate in detail have to be done and are the subject of a forth-
coming paper.

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