Local and parallel multigrid method for semilinear elliptic equations

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Abstract

This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations on the coarsest mesh. Furthermore, the local and parallel strategy is used to solve the involved linear elliptic equations. Since solving large-scale semilinear elliptic equations in fine space, which can be fairly time-consuming, is avoided, the proposed local and parallel multigrid scheme will significantly improve the solving efficiency for the semilinear elliptic equations. Besides, compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only requires the Lipschitz continuation property of the nonlinear term. We make a rigorous theoretical analysis of the presented local and parallel multigrid scheme, and propose some numerical experiments to support the theory.

Introduction

Many physical phenomena in nature can be described by a variety of mathematical models presenting linear and nonlinear partial differential equations of elliptic type. How to efficiently derive the approximate solutions of such equations is thus an important and challenging scientific task. In this study, we will present a novel local and parallel method to solve semilinear elliptic equations based on multigrid discretization.

The local and parallel finite element algorithm was first proposed by Xu and Zhou in [34] for linear elliptic equations; it was based on the combination of the local defect correction technique and the two-grid finite element discretization scheme. As we know, for the finite element approximate solution, the low-frequency components describe the corresponding global behavior and the high-frequency components describe the corresponding local behavior (more detailed description can be found in [4], [23], [27], etc). Thus, the local and parallel algorithm proposed in [34] first used a coarse mesh to approximate the low-frequency components, and then used a fine mesh to correct the approximate solution which is mainly composed of the high-frequency components. Since then, the local and parallel scheme has been widely used to design algorithms for other types of partial differential equations, for instance [2], [3], [7], [8], [9], [10], [12], [19], [20], [21], [24], [25], [26], [34], [37], [39], [40], [41] and the literatures that they referred to. In [8], [13], [35], the local and parallel algorithm is also designed for solving nonlinear elliptic equations. In these papers, two grid method and Newton iteration technique are adopted to linearize the nonlinear elliptic equations.

In this study, we aim to design a novel local and parallel multigrid algorithm to solve semilinear elliptic equations through combining the aforementioned local and parallel techniques and multigrid method and utilizing the recent advances made in the multilevel correction scheme [5], [15], [16], [17], [18], [28], [29], [30], [31], [32], [33], [38]. Through the novel local and parallel multigrid scheme, we just need to solve a linearized elliptic equation in each level of the multigrid space sequence by the local and parallel technique, and then, solve a low-dimensional semilinear elliptic equation in a specially designed correction space whose dimension will be fixed during the entire solving process. Thus, with the refinement of the mesh, the main computational work will be spent on the linear elliptic equations. This means through the novel local and parallel multigrid scheme, solving semilinear elliptic equation will cost a similar computational work as the classical local and parallel method for linear elliptic equations. Moreover, compared with the developed multigrid schemes and local and parallel schemes for semilinear elliptic equations which need the bounded second order derivatives of the nonlinear term (see e.g. [8], [13], [14], [23], [35]), the proposed method only requires the Lipschitz continuation property of the nonlinear term.

An outline of the paper is organized as follows. In Section 2, we introduce some basic finite element theories regarding the local error estimates. Section 3 is devoted to introducing the finite element method for the concerned semilinear elliptic equations and meanwhile, presenting the a priori error estimates. The novel local and parallel multigrid algorithm to solve concerned semilinear elliptic equations and the corresponding theoretical analysis are presented in Section 4. In Section 5, the numerical experiments demonstrating the efficiency of the novel algorithm are presented. Finally, some concluding remarks are presented in the last section.

Section snippets

Preliminaries of finite element method

First, some notations used for describing the subsequent analysis are detailed. Next, the a priori error estimates for the finite element solution of the linear elliptic equations are presented. Sobolev spaces Ws,p(Ω) and some standard notations, including Hs(Ω)=Ws,2(Ω), and H01(Ω)={vH1(Ω):v|Ω=0}, are used. More detailed definitions can be found in [1].

Besides, for three nested domains GDΩ, the notation GD denotes dist(DΩ,GΩ)>0 (see Fig. 1). Further, because any wH01(Ω0) can be

Finite element method for semilinear elliptic equation

Now, the finite element method and the corresponding theory for semilinear elliptic equation are proposed.

In this paper, we investigate the semilinear elliptic equation as follows: Find uH01(Ω) such that{(Au)+b(x,u)=f,inΩ,u=0,onΩ, where b(x,u) is a nonlinear term with respect to the second variable u.

The variational form of the semilinear elliptic equation (9) is defined as: Find uH01(Ω) such thata(u,v)+(b(x,u),v)=(f,v),vH01(Ω).

Using the standard finite element method, the following

Local and parallel multigrid method for semilinear elliptic equations

This section introduces a new type of local and parallel algorithm based on multigrid discretization. Different from the classical local and parallel scheme for nonlinear elliptic equations, the proposed approach is defined on a multigrid mesh sequence. When adopting the new approach, only a linear elliptic equation on each level of the multigrid mesh sequence needs to be solved by local and parallel schemes, and the accuracy of the derived approximation can then be improved by solving a

Numerical result

In this section, three numerical experiments are used to show the efficiency of Algorithm 4.2. The numerical experiments are carried out on a computing cluster with the Linux system. The adopted computing node has two 18-core Intel Xeon Gold 6140 processors at 2.3 GHz and 192 GB memory.

Concluding remarks

In this paper, we design a new type of local and parallel multigrid method to solve the semilinear elliptic equations based on local and parallel scheme, multigrid discretization and our recent advances in multilevel correction method. With the novel local and parallel multigrid scheme, we just need to solve a series of linear elliptic equations by local and parallel technique in a multigrid mesh sequence, and then correct the approximate solution in each mesh level by solving a small-scale

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    This work is supported in part by the National Science Foundation of China (Grant Nos. 11801021, 11971047), Soft Science Foundation of Science and Technology Department of Guangdong, China (Grant No. 2019A101002019).

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