An augmented Lagrangian method for the Signorini boundary value problem with BEM
 Shougui Zhang^{1}Email author and
 Xiaolin Li^{1}
Received: 9 October 2015
Accepted: 5 March 2016
Published: 10 March 2016
Abstract
We analyze augmented Lagrangian and boundary element methods for the Signorini boundary value problem of Laplacian. The boundary variational formulation is presented by the boundary integral operators, and the Signorini boundary conditions are formulated as a fixed point problem. Semismooth Newton methods are applied for the numerical solution of the problem. We prove the convergence of the method and confirm the theory by some numerical experiments.
Keywords
1 Introduction
Signorini boundary value problems are of great importance in many applications, such as the electropaint process and contact problem [1–3], etc. These problems involve inequality constraints on a part of the boundary that make them nonlinear. Therefore, development of efficient numerical algorithms for Signorini boundary value problems is one of the most important branches of modern computational mathematics and mechanics. Usually Signorini boundary value problems have been considered mathematically and numerically with variational inequalities, especially by the finite element method (FEM) [1–7] and the boundary element method (BEM) [8–13]. Recently, sound and efficient algorithms for the solution of these problems were still a very active field of research (see [14–20]).
As we know, the fixed point method based on projection theory is a powerful tool to deal with complementary problems and variational inequalities in finite dimensional space [21]. The main idea of this method is to establish the equivalence between the original problem and the fixed point problem by using projection. This equivalent formulation plays a significant role in developing various iterative methods for solving an original problem. During the last 20 years, a number of projection methods have been studied extensively [4, 5, 22, 23], which are perfectly efficient for solving the problem. In these methods, the problem has been formulated by a projection algorithm, and no other inequality constraint is needed. Consequently, the method is easy to implement for the numerical solution and the proof of convergence is very simple.
In the case of Signorini boundary conditions, the unknown boundary values are the potential and its derivative on the boundary, which are considered primary variables in BEM. They can be obtained directly using boundary integral equations [24, 25]. Therefore, the method combining the projection method with the BEM is more appropriate for Signorini boundary value problems [18–20]. However, few investigations have been done on the Signorini boundary value problem by the augmented Lagrangian and fixed point methods with the BEM.
In this paper, we focus on the boundary augmented Lagrangian methods (BALM) for the solution of Signorini boundary value problems, which is inspired by classical augmented Lagrangian methods (ALM) [7] and the BEM [11]. First of all, we deduce the boundary weak formulation with SteklovPoincaré operator. Second, we use the projection technique to deal with the Signorini boundary conditions by an equality and the projection. Although the new problem is still nonlinear on the boundary, this problem no longer has the inequality constraint and can be solved by the semismooth Newton method with local superlinear convergence rate [7, 14]. Using these transformations, we propose a BALM for the Signorini boundary value problem, which only needs the iteration for boundary function and the computation of the boundary variational equation. Then we can use properties of the projection and the boundary integral operator to prove the convergence of the method. Numerical results show that our method is accurate and efficient.
The structure of the paper is as follows. In Section 2, we first describe the classical Signorini boundary value problem and use the SteklovPoincaré operator to introduce the variational formulation. Then we establish equivalent formulations between the nonlinear boundary conditions and the fixed point problem, and propose a new ALM for the problem. Section 3 is devoted to the convergence analysis of the method, which shows monotone convergence properties of the numerical solution toward the solution of the original problem. In Section 4, we present some numerical experiments to investigate the performance of our method, and finally a brief conclusion is given in Section 5.
2 The weak formulations for the Signorini boundaryvalue problem
Lemma 2.1
Lemma 2.2
Proof
Now, we obtain the boundary weak formulation (2.7) by the SteklovPoincaré operator S and the fixed point problem (2.9) for the problem (2.1)(2.4), which only involves a boundary integral operator and no inequality constraint. This alternative equivalent formulation is also useful from the numerical and theoretical analysis point of view. With the above preparations, we can now list our boundary augmented Lagrangian method (BALM) for the Signorini boundary value problem as below.
Algorithm BALM
 Step 0::

Choose \(u^{(0)}\in L^{2}(\Gamma_{S})\), \(\rho\in R^{+}\), and set \(k:=0\).
 Step 1::

Solvewith boundary condition$$ \bigl\langle Su^{(k+1)},v\bigr\rangle _{\Gamma}+\bigl\langle \lambda^{(k+1)},v\bigr\rangle _{\Gamma_{S}}=L(v) ,\quad \forall v \in H_{D}^{1/2}(\Gamma), $$(2.10)and obtain \(u^{(k+1)}\) and \(\lambda^{(k+1)}\) on \(\Gamma_{S}\).$$ u^{(k+1)}g\bigl[u^{(k)}g\rho\bigl( \lambda^{(k+1)} f\bigr)\bigr]_{+}=0 \quad \mbox{on } \Gamma_{S}, $$(2.11)
 Step 2::

Update (2.11) by \(k:=k+1\) and return to Step 1.
For the nonlinear problem (2.11), we can apply the semismooth Newton method for its solution [7, 14]. In the next section the convergence of the algorithm is analyzed.
3 Convergence of the algorithm
Lemma 3.1
Proof
Theorem 3.1
Let \(\{(u^{(k)},\lambda^{(k)})\}\) be the sequence generated by the BALM, for all k then \(u^{(k)}\) converges to \(u^{*}\) in \(H^{1/2}(\Gamma )\) and \(\lambda^{(k)}\) converges to \(\lambda^{*}\) in \(L^{2}(\Gamma_{S})\) as \(k\to\infty\).
Proof
According to (3.7), the sequence \(\{u^{(k)}\}\) is bounded and the sequence \(\{\\delta_{u}^{(k)}\_{\Gamma_{S}}\}\) is monotonically decreasing. Furthermore, we can see that the larger value of the parameter ρ results in faster convergence to the algorithm.
4 Numerical experiments
To test the numerical verification of the theory, the algorithm above has been implemented and applied to some examples of Signorini problems in this section. An analytic solution is known for the first example, and the exact solution for the two other examples is not known. For the sake of simplicity, we apply a constant BEM to the problem (2.10) with iterations. Here, we choose \(\u^{(k+1)}u^{(k)}\_{\infty,\Gamma_{S}}\le 10^{10}\u^{(k+1)}\_{\infty,\Gamma_{S}}\) as a stopping criterion.
4.1 DirichletSignorini problem
Number of iterations with different values of N and ρ
N = 40  N = 80  N = 160  N = 320  N = 640  

ρ = 10^{0}  9  9  10  10  12 
ρ = 10^{1}  6  6  7  8  8 
ρ = 10^{2}  5  5  6  7  8 
ρ = 10^{3}  4  4  5  6  7 
ρ = 10^{4}  3  4  5  6  7 
ρ = 10^{5}  3  4  5  6  7 
4.2 DirichletSignorini problem
Number of iterations for \(\pmb{\varepsilon=0.4}\) with different N and ρ
N = 40  N = 80  N = 160  N = 320  N = 640  

ρ = 10^{0}  48  48  47  47  46 
ρ = 10^{1}  15  15  16  16  16 
ρ = 10^{2}  9  9  10  11  11 
ρ = 10^{3}  8  8  9  9  9 
ρ = 10^{4}  7  7  8  9  9 
ρ = 10^{5}  6  6  7  8  8 
Number of iterations for \(\pmb{\varepsilon=0.5}\) with different N and ρ
N = 40  N = 80  N = 160  N = 320  N = 640  

ρ = 10^{0}  22  25  24  25  24 
ρ = 10^{1}  11  11  12  13  13 
ρ = 10^{2}  8  9  9  11  11 
ρ = 10^{3}  7  8  9  10  10 
ρ = 10^{4}  6  7  8  10  10 
ρ = 10^{5}  6  7  8  9  9 
Number of iterations for \(\pmb{\varepsilon=0.55}\) with different N and ρ
N = 40  N = 80  N = 160  N = 320  N = 640  

ρ = 10^{0}  22  20  21  21  21 
ρ = 10^{1}  10  11  11  12  12 
ρ = 10^{2}  8  8  9  10  10 
ρ = 10^{3}  7  7  8  9  10 
ρ = 10^{4}  6  6  8  9  10 
ρ = 10^{5}  6  6  8  9  9 
Number of iterations for \(\pmb{\varepsilon=0.7}\) with different N and ρ
N = 40  N = 80  N = 160  N = 320  N = 640  

ρ = 10^{0}  17  17  16  16  16 
ρ = 10^{1}  9  10  10  10  11 
ρ = 10^{2}  7  8  8  9  10 
ρ = 10^{3}  6  7  7  8  9 
ρ = 10^{4}  5  7  7  8  9 
ρ = 10^{5}  5  7  7  8  9 
4.3 Signorini problem
Number of iterations with different N and ρ
N = 40  N = 80  N = 160  N = 320  N = 640  

ρ = 10^{0}  43  46  46  45  46 
ρ = 10^{1}  15  16  17  17  18 
ρ = 10^{2}  10  11  13  13  14 
ρ = 10^{3}  8  10  12  12  13 
ρ = 10^{4}  8  9  11  11  13 
ρ = 10^{5}  7  9  11  11  13 
5 Conclusion
In this paper, we have proposed a new ALM for the solution of Signorini boundary value problems and proven its convergence. Using the BEM and the fixed point method, we can easily apply this algorithm to the Signorini boundary value problems defined in domains of arbitrary shape. Each iteration only needs to solve an elliptic variational problem and the semismooth Newton method is used to find the solution. The examples tested show the feasibility and effectiveness of the algorithm.
Declarations
Acknowledgements
This work was funded by China Scholarship Council, the Natural Science Foundation Project of CQ CSTC of China (Grant Nos. cstc2013jcyjA30001 and cstc2013jcyjA10049) and Fundamental Research Funds of Chongqing Normal University of China (Grant No. 13XLB001), the National Natural Science Foundation of China (Grant Nos. 11471063 and 11301575).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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