Local A Posteriori Error Estimates and NumericalResults for Contact Problems and Problems of FlowThrough Porous MediaC. Y. LeeJ. T. adenM. AinsworthTexas Institute for Computational MechanicsThe University of Texas at AustinAustin, Texas, U.S.A.

Summary

A new a posteriori error estimator is introduced and applied to variational inequalities oc-curring in problems of contact and flow through porous media. One issue in constructingelement-wise a posteriori error estimates is the localization of the error. In the presentwork, this is accomplished by a special mixed formulation in which continuity conditions atinterfaces are treated as constraints. An upper bou~d of the local energy norm of the erroris obtained using a duality approach. This approach leads to error indicators which providerigorous upper bounds of the element errors. A discussion of a compatibility condition forthe well-posedness of the local error analysis problem is given. Three numerical examplesare solved to check the compatibility of the local problems and convergence of the effectivityindex both in a local and a global sense, with respect to local refinements.

1 Introduction

Consider the problem of finding a function u such that

-~u 2: f : u 2: 'P in n(~u - f)(u - 'P) = 0 a.e. in n

subject to u = 9 on an, where n E JB.2 is assumed to be a bounded domain with Lipschitzboundary an and 'P is a given function. This problem is the strong form of the obstacleproblem [3]. Similar formulations arise in the modeling of fluid flow through porous media[5) or the unilateral contact of elastic bodies [4]. The major difficulty in these problems isthat there is a contact region or free boundary which is unknown a priori. This difficultyrenders these types of problems nonlinear. However, these free-boundary problems can beformulated as variational inequalities which can be solved for Uj then unknown contact regionor boundary can be calculated a posteriori.

The uniqueness and existence of the solutions are proved in Ainsworth [1] for the case ofa Signorini-Obstacle type variational inequalities, and in Kikuchi and aden [4] for the caseof Signorini type inequalities. The finite element discretization of variational inequalities isdiscussed in [4].

In the present work, adaptive finite element methods are assumed, the key component ofwhich is an a posteriori error estimator which can be used to identify the elements where the

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4. the basis from which the estimator is derived is well founded mathematically

2. property 1) holds even element-wise

I\

\

IIIIIII

\

\

\

III

\

II

Vv EK10(~u + f)(v - lp)dx ~ 0- 1o(Au + f)(u - lp)dx = 0

3. the error estimation technique is easily incorporated into existing finite element codes

2 Model Problem

Consider the following obstacle problem

Find u such that

1. the estimated error provides an upper bound to the true error and 'is very close to theexact error

approximation is poor. We attempt to develop an acceptable error estimator which satisfiesthe following criteria:

-~u ~ f j u ~ lp in n(~u + f)(u - lp) = 0 a.e. in n

Property 2) is necessary for good adaptive FEMs while 1) and 2) should be checked bynumerical experiments. In this paper, a new a posteriori error estimator for h-adaptivemethod is introduced. The mathematical proofs of our estimator are provided in [2]. Herewe emphasize properties 1) and 2), and verify the quality of the a posteriorii error estimatorwhen applied to the obstacle problem and to the flow through porous media problems, bothbeing examples of problems characterized by variational inequalities. These model problemsare taken from aden and Kikuchi [5] and Elliot and Ockenden [3]. Details of theory andproofs of properties of variational inequalities for contact problems can be also found inKikuchi and aden [4).

subject to u = 9 on an. First, we define a non-empty closed convex set of admissiblefunctions

J{ = {v E HI (n)1v 2: lp in n, v = 9 on an}where lp E C(n) and lp ::::;0 on an. Assuming Au E L2(n), we construct a weak form of theabove problem by multiplying by (v - lp) and (u - lp), and integrating over n, to obtain

673

We define a finite dimensional non-empty closed convex subset of VIIaccording to

(7)

(6)

(5)

(4)

Vv EK

V VIIE KII

VvEI<

1J == 2a(v,v) - f(v)

u E I< : J(u) ::::;J(v)

Find UIIE KII : J(UII) ::::;J(VII)

FinduEK: a(u,v-u)2:f(v-u)

We focus our attention on a general class of variational inequalities of which (6) is a typicalmember. To discretize (5), we introduce a finite dimensional subset VIIE HI(n) such that

VII= {VII E C(n)lvll E PI(nk),nk = a finite element in a partition QIIof n, VII= 9 on an}

The convergence of UIIto u as h -+ 0 is proven in [4]. In the numerical results which follow,the discrete inequalities corresponding to finite element models of (6) are solved by theprojected SOR method (5].

For simplicity, linear triangular elements are used to construct the functions in VII, since

2. linear triangular elements provide for a simple implementation of the unilateral condi-tion VII~ 'P in n (for example, when 'P vanishes identically or is convex, we need onlyto set VII~ 'P at nodal points)

Equivalently,

where

1. linear triangular meshes can be locally refined without introducing constrained nodes

with :CA a nodal point. In this case I< :::> KII when 'P vanishes or is convex. But, in general,I< 'tJ I

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3 Error Estimation

(9)

(10)

(11)

Vw E K - {UII}

L(w,µ) == J(w) - (µ,[w])

where (u,v) == L ( u(s)v(s)dxk lao.

L(w,µ) == J(w) when w E K - {UII} and [wI = 0

Find e E K - {UII} : J(e) ::::;J(w)

a(u,u-uII)::::;f(u-uII)

a(e,e) = a(u,e) -a(uII,e)::::; f(e) -a(uII,e)

1 2 If 1211ellE == 210 Ve· Ve dx = 2a(e,e) ::::;-J(e)

Let w = v - UII E J{ - {UII}' Then, since v is arbitrary, w is also arbitrary, and we arrive atthe following variational inequality for the error:

Find e E K - {UII} : a(e,w -e) ~ f(w - e) - a(uII'w - e) V wE K - {UII} (8)

where

Define K - {UII} == {wlw = v - UII V v E K} and the error e == U - UII' By replacing U withe + UII, eq. (6) becomes

a (e + UII, v - e - UII) 2: f (v - e - UII) ora (e, v - e - UII) 2: f (v - e - UII) - a (UII, v - e - UII)

and it is easy to show that (8) is equivalent to the minimization problem

1J(w) == 2a(w, w) - f(w) + a (UII, w)

From (6), with v = Uh E Kh, we have

Therefore

[wI == jump between elements which share a common edge.For each element nk, we define the elementwise admissible function set

To split the original global problem into local problems, we allow the global shape functionto have discontinuities between elements, and use these discontinuities between elements asconstraints by introducing a Lagrangian:

Clearly

(14)

(12)

inf J(w)1UEK-{u,}

675

lIell~::::; -22:: inf Jk(w) (13)k 1UEK-(u,1

Now we have an upper bound which can be calculated from the following local minimization

problem:

2: inf L(w,µ) V µ EJB.1UEK-{u,}

Integrating by parts with ~UII = 0 in the linear element case1 .

L(w,µ) = 2a(w, w) - f(w) + a (UII, w) - (µ, [w])

1 " (aUIl ). ]= 2a(w,w)-f(w)+L..k 8'w -(µ,[w)k nk 80.

~ sup inf L(w,µ)"'ER 1UEK-{u,}

inf supL(w,µ)1UEK-{u,} "'ER

From (10) and (11), we construct a saddle point problem

aUIl h . b d9 - {j; w en s E exterior oun ary

Here, we choose µ" = { ~:II} to make the coupled term vanish where { } denotes the averagebetween elements which share a common edge.

Finally, we have

where

Jk(W) == 1 1 fa2a(W,W)k - (r,w)k + 2 80. R(s)w(s)dsa(w,w)k == 1Vw· Vw dx

o.

(r'W)k == 1(f(x)+~uII)dso.

R( s) == - [~nll] when s E interior boundary

•

(15)

(17)

(16)

on interior boundary8Wk _ 1 [8uII]an - -2 {j;with the boundary condition

o on exterior boundary

-~Wk 2: f, Wk 2: -UII + 'P(-~Wk - f) (Wk + UII - 'P) = 0 in nk

But this condition is not satisfied in general and, without any modification of the local

problem, either no solution or infinitely many solutions will exist. To overcome this difficulty,we add a constant Ok to f to make the compatibility valid:

This is a Neumann type problem. As proved in [1], for a unique solution to exist to thisboundary value problem, we must have

676

The importance of (14) is that the global problem (8) has been split into a set of local

problems. But we now have to consider the well-posedness of these local problems.

Formally, the local problem is

Let

Proof: See in [1]We next apply this theorem to each elementwise local problem (14); the result is:

Then the local problems are well posed whenever Ok < O.Next, we record a duality result of some importance.

J(v) 2: J(u) = G(\7u) ~ G(p) and U E M

G(p) == -~ { p' p dx - { 'P(f + \7 . p)dx + ( 'P(g - n· p)ds210 10 180Then for any v E J{ and any p EM,

Theorem 1

and let

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zero.

(20)

(23)

(21)

4 Numerical Examples

lIell1 ~ 2:: f VWk' VWk dxk 10.

where Wk is the solution of each local problem.

Mk == {p E [L2(nk)rlV',p+f::::;0 a.e. in nkn·p 2: ~ (~:II] on ank} (18)Gk(p) == 10.p'p dx+2 10. 'P(f+ V'·p)dx+2 lao. 'P ( n . p - ~ [~II]) ds ('P == 'P - UII) (19)

With (13) and (16), we finally obtain

p = V'w

where,

Toward this end, consider the extremal condition of this dual problem

Three model problems are solved which provide typical examples of variational inequalities

to verify the assumptions and to test the accuracy of the estimator. As noted earlier, the

projected SOR method is used to solve both global and local problems, and triangular

elements are employed. For the local problems, three mesh refinements are used to check

convergence of the effectivity indices with respect to local refinement (see Fig. 1).

Note that in the calculation of local effectivity indices, zeros indicate that the exact error is

lIell~::::; 2:: Gk(p)k

In view of this result, the local errors can be estimated by identifying the p which makesthis upper bound closest to the true error.

Clearly Mk to which p belongs and K - {UII} to which Wk, the local solution, belongs,are different and it seems the two sets have nothing in common. But from the extremal

condition, the true solution in both primal and dual problems is in both sets, and that is the

common part of the two sets. Hence, even if Wk is not the common part, we expect Wk to beclose to the true solution w. So we take the p as V'Wk and check how good the assumptionis. Moreover, from the formal form of the local problem, we also expect

aWk 1 [aUII]- - - - ~ 0 on ank (22)an 2 anso we assume only the first term is dominant and neglect the others. Thus, our final estimator

is

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4.1 Obstacle Problem

This problem has an exact solution which makes it possible to calculate the exact error andeffectivity indices. The configuration and the numerical results are given in Figs. 2-7.

4.2 Dam Problem

This problem is to determine the free boundary between saturated and dry regions inside adam with given inlet and outlet water levels. With the given water levels, the discharge canbe calculated and is in effect given in the boundary conditions. The primitive form of theproblem is of mixed type but, by the use of the Baiocchi transformation, a Dirichlet type isobtained. Since no exact solution is available, a fine mesh solution (h = 5/40) is taken to bethe exact solution. This assumption was checked by computing the relative error, which isthe ratio of the L2 norm of the estimated error to the L2 norm of the numerical solution, andwe found that the relative errors were 11.37, 6.01, and 1.57 percent of the cases of h = 5/5,5/10, and 5/40, respectively. This indicates that the h = 5/40 case is relatively close to theexact solution. The configuration of the problem and the resulting global effectivity indicesfor each case are given in Figs. 8-11.

4.3 Dam With a Sheet Pile Problem

The governing equation and the boundary conditions are the same as for the previous prob-lem. In this case we do not know the discharge q a priori. This makes the problem ofnonlinear even in the boundary conditions and requires the use of an iterative procedure todetermine the discharge. Using the fact that the discharge at the tip of the sheet pile is equalto zero, we can solve for q by Newton's method. Hence, the solution scheme is: 1) first, solvefor q by Newton's method with an initial guess, 2) determine the boundary conditions, then3) use the same method as in the previous problem. However, the discharge will depend onthe mesh size, i.e., we expect that the finer the mesh, the closer the discharge is to the exactvalue. Since no exact solution is available, we again solve the problem with a comparativelyfine mesh, h = 5/40, and take this as the exact solution. This assumption was checked bycomputing the relative errors as was done in the previous problem.

This problem is solved with two different discharges. One is obtained with the coarse meshof each problem, and the other is obtained with the finest mesh which should be the bestapproximation to the exact value. Computed relative errors were 18.42, 9.49, and 2.41percent for the cases of h = 5/5, 5/10, and 5/40, respectively, with variable discharges.These were 16.83, 9.09, and 2.41 percent with the fixed discharge q = 3.8890 which wasobtained by solving the finest mesh case (h = 5/40). This indicates we could assume thath = 5/40 case is relatively close to the exact solution.

1a

679

The configuration of the problem and the resulting global effectivity indices for each caseare given in Figs. 12-16.

5 Conclusions

Our principal goal is to check the accuracy of a new a posteriori error estimator usingnumerical examples. We made the assumption that in (18), only the first term is dominant.From our numerical results, the effectivity index is increasing as the local refinement getsfiner. It is therefore likely that as we make the local mesh finer, the neglected terms becomeincreasingly negligible relative to the first term. This tendency could also be observed inlocal effectivity indices.

As to the question of how good the error estimator is in the global sense, the effectivity indicesare all less than 1.5. This level of global effectivity indices were unexpectedly low. Locally,the effectivity indices are between 1.0 and 2.0 and the distribution of the local effectivityindices become more regular as the local mesh becomes finer. As expected, indices are worstnear the free boundary.

Acknowledgement: The support of this work by the National Science Foundation under GrantNumber MSM-8805552 is gratefully acknowledge.

References

1. M. Ainsworth, "Analysis of a Signorini-Obstacle Type Variational Inequality," TICOMReport 91-1, Austin, Texas, 1991.

2. M. Ainsworth, J. T. aden, and C. Y. Lee, "Local A Posteriori Error Estimatorsand Adaptive Algorithm for Finite Element Approximation of the Obstacle Problem,"(preprint).

3. C. M. Elliot, and J. R. Ockenden, "Weak and Variational Methods for Moving Bound-ary Problems,"Pitman Pub., Boston, 1982.

4. N. Kikuchi, and J. T. aden, "Contact Problems in Elasticity: A Study of VariationalInequalities and Finite Element Methods," SIAM, Philadelphia, 1988.

5. J. T. aden, and N. Kikuchi, "Theory of Variational Inequalities, Flow Through PorousMedia," Int. J. Eng. Sci., 18, pp. 1173-1284.

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685

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---0--. SIS Mesh Size....... 5/10

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--•............•.......... -..n_.-.--- •• -0 -- - -0

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Fig. 10 Dam Problem ( h=5/5 )

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Fig.12 Configuration of Dam with a Sheet Pile

687

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Fig.17 Dam with a Sheet Pile Problem( q=3.8890. h=5/1O )

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( q=3.8890. h=5/5 )

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page1titles1 Introduction

page2titlesI I I \ I I 2 Model Problem J{ = {v E HI (n)1 v 2: lp in n, v = 9 on an}

page3titlesu E I< : J(u) ::::; J(v) Find UII E KII : J(UII) ::::; J(VII)

page4titles3 Error Estimation Vw E K - {UII} L(w,µ) == J(w) - (µ,[w]) Find e E K - {UII} : J(e) ::::; J(w) a(u,u-uII)::::;f(u-uII) 211ellE == 210 Ve· Ve dx = 2a(e,e) ::::; -J(e) Find e E K - {UII} : a(e,w -e) ~ f(w - e) - a(uII'w - e) V wE K - {UII} (8) a (e + UII, v - e - UII) 2: f (v - e - UII) or a (e, v - e - UII) 2: f (v - e - UII) - a (UII, v - e - UII) J(w) == 2a(w, w) - f(w) + a (UII, w)

page5titles(14) (12) inf J(w) lIell~::::; -22:: inf Jk(w) (13) L(w,µ) = 2a(w, w) - f(w) + a (UII, w) - (µ, [w]) 1 " (aUIl ). ] = 2a(w,w)-f(w)+L..k 8'w -(µ,[w) inf supL(w,µ)

tablestable1

page6titles• 8Wk _ 1 [8uII] 210 10 180

page7titles4 Numerical Examples lIell~::::; 2:: Gk(p) aWk 1 [aUII] - - - - ~ 0 on ank (22) an 2 an

page8titles4.1 Obstacle Problem 4.2 Dam Problem 4.3 Dam With a Sheet Pile Problem 1

page9titles5 Conclusions References

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