A modified modulus-based multigrid method for linear complementarity problems arising from free boundary problems☆
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 to be a Laplace operator, we consider the continuous linear complementarity problem (LCP): find a nonnegative function on such that where and are two given functions. By using finite differences on a prescribed grid where h is the grid size, the discrete LCP of (1) is denoted by where and are the discrete analogs of and , and is the discrete analog of where the known values on are eliminated. The discrete LCP (2) is abbreviated as LCP.
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 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 and the original function , respectively. Therefore, the transformation between and 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 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 .
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 , , and in (2) are written as , , and . We introduce the auxiliary grid function and let , with , where ω and γ are positive numbers. Note that this substitution guarantees that and are nonnegative on , and the linear equation 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 and are separated by the free boundary. Note that both the fixed-point equation (3) and the iteration formula (4) are different on and
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 such that in the domain and on the boundary
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.