1. Introduction
In 1962–1964, Fedorenko [18, 19] proposed a multigrid idea for solving the Poisson’s equation on a unit square. A more complicated case of variable coefficients was later considered by Bakhvalov [4]. The actual efficiency of multigrid methods was recognized by Brandt [11, 12]. To analyze the convergence of multigrid methods, Hackbusch [20, 21] developed some fundamental elements for multigrid analysis. For other representative work on the early development of multigrid methods, we refer to [23, 41, 16, 32] and the references therein. Since the early 1980s, multigrid has been well developed and widely applied in scientific computing (see, e.g., [32, 34]).
The general convergence proofs (e.g., [22, 7]) for multigrid methods required regularity properties of the boundary value problem and quasiuniformness of the underlying finite element or finite difference meshes. These requirements led to the further development of hierarchical basis methods [5, 1, 40, 6]. Convergence theories based on some algebraic approximation assumptions appeared in [26, 31, 25]. For secondorder elliptic boundary value problems without full elliptic regularity, the convergence theory of the Vcycle multigrid with the Richardsontype smoother was studied by Brenner [15]. A unified treatment for multigrid convergence is via the method of subspace corrections [9, 10, 35, 8]. Under such a framework, multigrid convergence can be established without the regularity and quasiuniformness assumptions. An exact characterization for the convergence factor of the method of subspace corrections (as well as the method of alternating projections) in a Hilbert space setting was established by Xu and Zikatanov [36], which is the socalled XZidentity. In 2010, Napov and Notay [27] presented a systematic comparison of the convergence bounds for the Vcycle multigrid methods. It is not possible to review all relevant literature on multigrid convergence here; for more details, we refer to the survey papers [41, 24, 30, 37] and the references therein.
Algebraic multigrid [14, 13, 31] constructs the coarsening process in a purely algebraic manner that requires no explicit knowledge of geometric properties, which has been widely applied in scientific and engineering computing, especially in the situations associated with complex domains, unstructured grids, problems with jump coefficients, etc (see, e.g., [34, 37]
). It is feasible to show optimal convergence properties (e.g., independent of the mesh size) of multigrid methods via the convergence theories mentioned above, e.g., the theory of subspace correction methods. However, the convergence estimates of such approaches do not, in general, give satisfactory predictions of actual multigrid convergence speed
[32, Page 96]. Moreover, some required assumptions may be difficult to check for algebraic multigrid methods. Unlike the mature convergence theory developed for geometric multigrid methods, twogrid analysis is still a main strategy for motivating and analyzing algebraic ones [24, 30].A common wisdom on multigrid convergence can be stated as follows: If exact twogrid methods converge sufficiently well (i.e., the convergence speed is fast enough), then the corresponding multigrid method with cycle index (see Algorithm 2) has the similar convergence properties as twogrid ones [32, Page 77]. In 2007, Notay [29, Theorem 3.1] proved that, if the convergence factor of exact twogrid methods is uniformly bounded by , then the convergence factor of the corresponding Wcycle multigrid method (corresponding to ) is bounded by . This analysis (as well as the standard one in [32, Theorem 3.2.1]) fails to deliver a levelindependent upper bound for the convergence factor of the Vcycle multigrid (corresponding to ). In 2010, Napov and Notay [28] further investigated the connection between twogrid convergence and Vcycle multigrid convergence, and showed that, besides the uniform twogrid convergence, additional conditions are required to derive a levelindependent upper bound for the convergence factor of the Vcycle multigrid.
Since for , Notay’s result [29, Theorem 3.1] is only applicable for . In practice, it is observed that the Wcycle multigrid methods may have the similar convergence properties as twogrid ones even if . For example, we apply the classical algebraic multigrid method [31] to solve the 2D Poisson’s equation with homogeneous Dirichlet boundary condition on a unit square (using the Pfinite element on a quasiuniform grid with one million interior vertices). From Table 1, we observe that the Wcycle multigrid methods behave similarly to the corresponding twogrid ones even when the twogrid methods converge slowly.
Smoother type  Cycle type  Convergence factor 

Gauss–Seidel with CFordering  TG  0.458 
W  0.552  
V  0.899  
Weighted Jacobi with weight 0.5  TG  0.536 
W  0.640  
V  0.931  
Weighted Jacobi with weight 0.7  TG  0.859 
W  0.866  
V  0.941 
Motivated by these observations, we revisit the convergence analysis of multigrid methods and establish a new convergence theory for multigrid methods based on the inexact twogrid theory developed in [39]. Our main results can be divided into two parts.

The first part includes three types of convergence estimates: (3.12), (3.18), and (3.20). These results are valid for any cycle index , from which one can readily get the convergence estimates for the Vcycle and Wcycle multigrid methods. The upper bounds in (3.12) and (3.18) are strictly decreasing with respect to , both of which tend to the maximum of twogrid convergence factors over all levels as . In particular, Notay’s result [29, Theorem 3.1] can be directly deduced from the estimate (3.18). The third estimate (3.20) involves the level index . The upper bound in (3.20) is strictly increasing with respect to , which tends to the bound in (3.18) as .

The second part is concerned with two alternating combinations of the Vcycle and Wcycle multigrid methods, which can be viewed as multigrid methods with a fractional cycle index . This part contains two types of convergence estimates. The first type consists of (4.4) and (4.6), which depend on the parity of level index and the extreme quantities defined by (3.8) and (3.10). As a corollary, if the convergence factor of exact twogrid methods is uniformly bounded by , then the convergence factor of the alternating multigrid methods is bounded by or (which depends on the parity of level index). The second type includes (4.19) and (4.20), which involve the level index . If , then the upper bounds in (4.19) and (4.20) tend to the bounds in (4.4) and (4.6), respectively.
It is worth mentioning that our estimates do not require the coarsestgrid problem to be solved exactly.
The rest of this paper is organized as follows. In Section 2, we first introduce the convergence estimates for inexact twogrid methods, and then give some basic assumptions and properties on multigrid methods. In Section 3, we present a new convergence analysis of standard multigrid methods, which contains three types of estimates. In Section 4, we establish a convergence theory for the alternating combinations of the Vcycle and Wcycle multigrid methods. In Section 5, we give some concluding remarks.
2. Preliminaries
In this section, we introduce two important convergence estimates for inexact twogrid methods, and give some general assumptions involved in the convergence analysis of multigrid methods. For convenience, we first list some basic notation used in the subsequent discussions.

denotes the identity matrix (or when its size is clear from context).

and
denote the minimum and maximum eigenvalues of a matrix, respectively.

denotes the spectrum of a matrix.

denotes the energy norm induced by a symmetric and positive definite (SPD) matrix . That is, for any , ; for any , .
2.1. Twogrid methods
Consider solving the linear system
(2.1) 
where is SPD, , and . To describe twogrid methods, we need the following assumptions.

is a nonsingular smoother, and is SPD.

is a prolongation matrix of rank , where is the number of coarse variables.

is the Galerkin coarsegrid matrix.

is a general SPD coarsegrid matrix.
Given an initial guess , the standard twogrid scheme for solving (2.1) can be described as Algorithm 1.
The iteration matrix of Algorithm 1 is
(2.2) 
which satisfies
In particular, if , then Algorithm 1 is called an exact twogrid method. In this case, the iteration matrix is denoted by
(2.3) 
Define
Then, the convergence factor can be characterized as
(2.4) 
where
(2.5) 
The identity (2.4) is often called the twolevel XZidentity [17, Theorem 4.3] (see also [36, 42]).
The following theorem presents more general estimates for the convergence factor of twogrid methods [39, Corollaries 3.10 and 3.18], from which one can readily get the identity (2.4).
Theorem 2.1.
Let and . For Algorithm 1, if the coarsegrid matrix satisfies that
(2.6) 
then
(2.7) 
Alternatively, if the coarsegrid matrix satisfies that
(2.8) 
then
(2.9) 
2.2. Multigrid methods
The fundamental module of multigrid methods is the twogrid procedure described by Algorithm 1. To design a well converged twogrid method, it is not necessary to solve the coarsegrid problem exactly, especially when the problem size is still large. Instead, without essential loss of convergence speed, one can solve the coarse problem approximately. A natural idea (i.e., multigrid idea) is to apply the twogrid scheme recursively. This validates that multigrid can be regarded as an inexact twogrid scheme, which enables us to analyze multigrid convergence via inexact twogrid theory.
By recursively applying Algorithm 1 in the coarsegrid correction steps, one can obtain a multigrid algorithm. To describe the algorithm concisely, we give some notation and assumptions.

The algorithm involves levels with indices , where and correspond to the coarsestlevel and the finestlevel, respectively.

For each , is the number of coarse variables at level , and .

For each , denotes a prolongation matrix from level to level , and .

Let . For each , denotes the Galerkin coarsegrid matrix at level .

is an SPD approximation to , and is a symmetric and positive semidefinite (SPSD) matrix.

For each , denotes a nonsingular smoother at level , and is SPD.

At level , the number of presmoothing is equal to that of postsmoothing, which is denoted by .

The cycle index involved in the coarsegrid correction steps is denoted by , which is a positive integer.
Given an initial guess , the standard multigrid method for solving the linear system can be described as Algorithm 2. The symbols and in Algorithm 2 mean that the corresponding schemes will be carried out and iterations, respectively. In particular, and correspond to the V and Wcycles, respectively (see Figure 1).
The iteration matrix of Algorithm 2 is
(2.10) 
which satisfies
For brevity, we define an equivalent smoother by the relation
(2.11) 
Due to is SPD, it follows that
which, together with (2.11), yields
This implies that is also SPD. In addition, from (2.11), we have
Thus,
(2.12) 
with
In view of (2.12), we have
where
Applying mathematical induction, we can deduce that is symmetric and
which leads to
As a result, can be written as
(2.13) 
where is SPD and is SPSD. Combining (2.12) and (2.13), we obtain the recursive relation
where
(2.14) 
Interchanging the roles of and in (2.14) yields another symmetrized smoother
(2.15) 
It is easy to check that both and are SPSD matrices.
3. Convergence analysis: Standard cycles
Comparing (2.2) with (2.12), we see that Algorithm 2 is essentially an inexact twogrid method with , , , and
(3.1) 
Remark 3.1.
Define
(3.2)  
(3.3) 
The quantities and are referred to as the convergence factors of the exact twogrid method and the multigrid method at level , respectively. According to the lower bound in (2.7) (or (2.9)), we deduce that
(3.4) 
which reveals that a fast exact twogrid method is necessary for good convergence of the corresponding multigrid method.
Based on Theorem 2.1, we can derive the following upper bounds for .
Lemma 3.2.
For any , it holds that
(3.5) 
and
(3.6) 
Proof.
In what follows, we establish three types of convergence estimates for Algorithm 2 based on Lemma 3.2. For convenience, we define
(3.8)  
(3.9)  
(3.10) 
3.1. Estimate of the first kind
The definitions (3.8)–(3.10) imply that
which, together with the positive semidefiniteness of , lead to the assumptions in the following lemma.
Lemma 3.3.
Assume that , , and . Then, there exists a strictly decreasing sequence with the limit such that is a root of the equation
Proof.
Define
Obviously, is a continuous function in . Direct computations yield
Hence, has at least one root in .
Let be a root of . Note that is a strictly increasing function with respect to . We then have
Since , there exists an such that . Repeating this process, one can get a strictly decreasing sequence .
Due to , it follows that
which yields
This completes the proof. ∎
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