A modified modulus-based multigrid method for linear complementarity problems arising from free boundary problems

This work is dedicated to Professor Hua DAI on the occasion of his 60th birthday
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Abstract

The linear complementarity problem arising from a free boundary problem can be equivalently reformulated as a fixed-point equation. We present a modified modulus-based multigrid method to solve this fixed-point equation. This modified method is a full approximation scheme using the modulus-based splitting iteration method as the smoother and avoids the transformation between the auxiliary and the original functions which was necessary in the existing modulus-based multigrid method. We predict its asymptotic convergence factor by applying local Fourier analysis to the corresponding two-grid case. Numerical results show that the W-cycle possesses an h-independent convergence rate and a linear elapsed CPU time, and the convergence rate of the V-cycle can be improved by increasing the smoothing steps. Compared with the existing modulus-based multigrid method, the modified method is more straightforward and is a standard full approximation scheme, which makes it more convenient and efficient in practical applications.

Introduction

Free boundary problems, which arise in flow through porous dams, journal bearing lubrication, elastic-plastic torsion and American option pricing, can be reformulated into continuous linear complementarity problems; see [9], [10], [16], [37], [28] and the references therein. Letting L to be a Laplace operator, we consider the continuous linear complementarity problem (LCP): find a nonnegative function u(x) on DR2 such that{Lu(x)+f(x)0,u(x)0andu(x)(Lu(x)+f(x))=0,inD,u(x)=g(x),onD, where f(x) and g(x) are two given functions. By using finite differences on a prescribed grid Dh where h is the grid size, the discrete LCP of (1) is denoted byrh(x):=Lhuh(x)+bh(x)0,uh(x)0anduh(x)rh(x)=0,xDh, where Lh and uh(x) are the discrete analogs of L and u(x), and bh(x) is the discrete analog of f(x) where the known values uh(x) on Dh are eliminated. The discrete LCP (2) is abbreviated as LCP(bh,Lh).

For large sparse linear complementarity problems, the projected iteration methods [11], [10], [1] and the modulus iteration methods [7], [14], [17], [2] are two kinds of basic iteration methods. The modulus-based iteration method presented in [2] is practical and effective in actual implementations. Further studies and generalizations were presented in recent years. The corresponding parallel algorithms were proposed and studied in [3], [4], [33], [12], [34], [13], [35] to improve the computing efficiency. Acceleration and preconditioning techniques were presented and studied in [38], [39], [25], [36] to improve the convergence rate. The modulus-based iteration method was also generalized to solve nonlinear complementarity problems [32], [20], [26], [24], [31].

It is known that, if the discrete grid Dh is refined, the convergence rates of projected and modulus-based iteration methods will slow down inevitably. To overcome the reduced convergence rates on the finer grid, corresponding projected multigrid and modulus-based multigrid methods were presented and studied [9], [27], [5]. For further development and generalizations of the multigrid methods for variational inequalities, we refer to previous studies [19], [21], [22] and the references therein. Numerical experiments showed that the convergence rates of projected multigrid and modulus-based multigrid methods with W-cycles are h-independent. However, with respect to the computing efficiency, the modulus-based multigrid method is superior to the projected multigrid method, due to the efficient basic linear algebra subroutines in the former and the inefficient projection operation in the latter.

As linear complementarity problems are essentially nonlinear problems, the modulus-based multigrid method in [5] is a complicated full approximation scheme, in which the smoothing operator and the transfer operator act on the auxiliary function zh(x) and the original function uh(x), respectively. Therefore, the transformation between zh(x) and uh(x) is necessary in the modulus-based multigrid method, which makes the corresponding multigrid cycle become rather involved. In this paper, we will improve the algorithm to avoid the transformation between the auxiliary and the original functions. By analyzing the relationship of the auxiliary function zh(x) on the fine and the coarse grids, we present a modified modulus-based multigrid method in which both the smoothing operator and the transfer operator act on the auxiliary function zh(x).

The rest of this paper is organized as follows. In Section 2, a modified modulus-based multigrid method is presented to solve the linear complementarity problems arising from free boundary problems. In Section 3, we give the two-grid convergence analysis to predict the asymptotic convergence factor of this modified modulus-based multigrid method. In Section 4, we use two examples to verify the two-grid convergence analysis and show the effectiveness of our new multigrid method. Finally, we give a few concluding remarks in Section 5.

Section snippets

Modified modulus-based multigrid method

We start this section with the introduction of the modulus-based splitting iteration method [2], [18], [23], [38], [15]. For simplicity, the grid functions uh(x), rh(x), and bh(x) in (2) are written as uh, rh, and bh. We introduce the auxiliary grid function zh and let uh=1γ(|zh|+zh), rh=ωhγ(|zh|zh) with ωh=ωh2, where ω and γ are positive numbers. Note that this substitution guarantees that uh and rh are nonnegative on Dh, and the linear equation rh=Lhuh+bh can be reformulated into the

Two-grid convergence analysis

In this section, we will give the two-grid convergence analysis of the modified modulus-based multigrid method by using the local Fourier analysis (LFA); see [8], [29], [30] for notations, definitions, and basic facts of the LFA. Here, we consider the linear complementarity problem arising from the free boundary problem, in which the domains z(x)0 and z(x)<0 are separated by the free boundary. Note that both the fixed-point equation (3) and the iteration formula (4) are different on Dh+ and Dh

Numerical results

In this section, we will use two examples to verify whether the two-grid convergence analysis can predict the convergence factor of the modified MFAS (Method 2.1) and show the effectiveness of the modified MFAS in actual applications.

Example 4.1

[9] Consider the linear complementarity problem arising from the porous flow free boundary problem: find u on the domain [0,16]×[0,24] such that in the domainu0,uxx+uyy1andu(uxx+uyy1)=0, and on the boundaryu={12(24y)2,forx=0,0<y24,12(4y)2,forx=16,0<y4,132(242

Concluding remarks

We have presented a modified modulus-based multigrid method for solving large sparse linear complementarity problems, which avoids the transformation between the auxiliary and the original functions in the coarse grid correction. Compared with the existing modulus-based multigrid method in [5], the modified one is more straightforward and is a standard full approximation scheme, which makes it more convenient and efficient in practical applications.

The asymptotic convergence factor can be

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    This work is supported by the National Natural Science Foundation (Nos. 11301141 and 11771467), NG Teng Fong/Sino Outstanding Youth Fund of HUEL, the Key Research Project of Henan Higher Education Institutions (No. 21A110003), and the Disciplinary Funding of Central University of Finance and Economics, P.R. China.

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