Galerkin least squares finite element method for the obstacle problem
We construct a consistent multiplier free method for the finite element solution of the obstacle problem. The method is based on an augmented Lagrangian formulation in which we eliminate the multiplier by use of its definition in a discrete setting. We prove existence and uniqueness of discrete solutions and optimal order a priori error estimates for smooth exact solutions. Using a saturation assumption we also prove an a posteriori error estimate. Numerical examples show the performance of the method and of an adaptive algorithm for the control of the discretization error.
keywords:Obstacle problem, augmented Lagrangian method, a priori error estimate, a posteriori error estimate, adaptive method
Our aim in this paper is to design a simple consistent penalty method for contact problems that avoids the solution of variational inequalities. We eliminate the need for Lagrange multipliers to enforce the contact conditions by using its definition in a discrete setting, following an idea of Chouly and Hild  used for elastic contact.
1.1 The model problem
We consider the obstacle problem of finding the displacement of a membrane constrained to stay above an obstacle given by (with at ):
where is a convex polygon. It is well known that this problem admits a unique solutions . This follows from the theory of Stampacchia applied to the corresponding variational inequality (see for instance ).
1.2 The finite element method
There exists a large body of literature treating finite element methods for unilateral problems in general and obstacle problems in particular, e.g., [11, 13, 6, 10, 8, 3, 16, 18, 2, 17]. Discretization of (1) is usually performed directly starting from the variational inequality or using a penalty method. The first approach however leads to some nontrivial choices in the construction of the discretization spaces in order to satisfy the nonpenetration condition and associated inf-sup conditions and until recently it has proved difficult to obtain optimal error estimates [9, 5]. The latter approach, o n the other hand leads to the usual consistency and conditioning problems of penalty methods.
An alternative is to use the augmented Lagrangian method. We introduce the Lagrange multiplier such that
under the Kuhn-Tucker side conditions
Using the standard trick of rewriting the Kuhn-Tucker conditions as
where and , cf., e.g., Chouly and Hild , we can formulate the augmented Lagrangian problem of finding that are stationary points to the functional
cf. Alart and Curnier , leading to seeking such that
For our discrete method, we assume that is a family of conforming shape regular meshes on , consisting of triangles and define as the space of –conforming piecewise polynomial functions on , satisfying the homogeneous boundary condition of .
We then formally replace element–wise by to obtain a discrete minimization problem: seek such that
The Euler–Lagrange equations corresponding to (9) take the form: Find such that
where denotes the standard -inner product, and
and, for use below,
To simplify the notation below we introduce and
We will also omit and from the argument of below, and use the notation so that
Note that the form can be interpreted as a nonlinear consistent least squares penalty term for the imposition of the contact condition. A similar method was proposed in Stenberg et al.  in the framework of variational inequalities.
We will below alternatively use the compact notation
and the associated formulation, find such that
1.3 Summary of main results and outline
In Section 2 we recall some technical results, in Section 3 we derive an existence result for the discrete solution using Brouwer’s fixed point theorem and we prove uniqueness of the solution using monotinicity of the the nonlinearity, in Section 4 we prove an a priori error estimate and using a saturation assumption we also prove an a posteriori error estimate, finally in Section 5 we present numerical results confirming our theoretical results and illustrating the performance of an adaptive algorithm based on our a posteriori error estimate.
2 Technical results
Below we will use the notation for where is a constant independent of , but not of the local mesh geometry.
First we recall the following inverse inequality,
see Thomée .
We will use the Scott-Zhang interpolant preserving boundary conditions, denoted . This operator is -stable, and the following interpolation error estimate is known to hold ,
The essential properties of the nonlinearity are collected in the following lemmas.
Let then there holds
[Proof] Developing the left hand side of the expression we have
For the proof of the second claim, this is trivially true in case both and are positive or negative. If is negative and positive then
and similarly if is negative and positive
(Continuity of ) For all , the form (11) satisfies
3 Existence of unique discrete solution
In the previous works on Nitsche’s method existence and uniqueness has been proven by using the monotonicity and hemi-continuity of the operator. Here we propose a different approach where we use the Brouwer’s fixed point theorem to establish existence and the monotonicity of the nonlinearity for uniqueness. We start by showing some positivity results and a priori bounds. Since we are interested in existence and uniqueness for a fixed mesh parameter , we do not require that the bounds in this section are uniform in .
Let and assume that
then there holds
[Proof] First observe consider the form ,
Using the monotonicity of Lemma 1 we may write
Observe that using an inverse inequality (13) we have
. We may then write
It follows that choosing and applying the Poincaré inequality
The second inequality follows by taking above and noting that then
where we used (18) in the last step. Considering the condition on and using an arithmetic-geometric inequality we may conclude.
and we conclude that and hence . Let denote the number of degrees of freedom of .
Consider the mapping defined by
where , , where and denotes the vectors of unknown associated to the basis functions of .
By the second claim of Lemma 3, there holds
we have that for any fixed the following positivity holds for sufficiently large
Assume that this positivity holds whenever . Denote by the (closed) ball in with radius and assume that there is no such that . Define the function
Then , is continuous by Lemma 2 and the assumption that for all . Hence there exists a fixed point such that
It follows that
but since , by assumption , which leads to a contradiction, since .
4 Error estimates
[Proof] Using the definition of we may write
If we may also write
It follows that
As a consequence we have the following property reminiscent of Galerkin orthogonality,
First observe that
Considering the first term in the right hand side of equation (25) we may write
The term may be bounded using the Cauchy-Schwarz inequality followed by the arithmetic geometric inequality
For the term we use the monotonicity property , with and so that
to deduce that
Collecting the above bounds and using the Poincaré inequality (20) we find,
Fixing , and fixing sufficiently small so that then there holds
and using the bound on . Assumption: (Saturation) We assume that there exists a constant such that