Clearly, the same inclusion property is inherited by the corresponding Finite Element functional spaces and hence we find .
Thus, the matrix representing the prolongation operator is formed, column by column, by representing each function of the basis of
as linear combination of the basis of
, the coefficients being the values of the functions
on the fine mesh grid points, i.e.,
4.1. Case
Firstly, let us consider the case of
Finite Elements, where, as is well known, the stiffness matrix is the scalar Toeplitz matrix generated by
, and, for the sake of simplicity, let us consider the case of
partitioning with five equispaced points (three internal points) and
partitioning with nine equispaced points (seven internal points) obtained from
by considering the midpoint of each subinterval. In the standard geometric multigrid, the prolongation operator matrix is defined as
Indeed, the basis functions with respect to the reference interval
are
,
, and, according to Equation (
12), the
coefficients are
giving the columns of the matrix in Equation (
13). However, we can think the prolongation matrix above as the product of the Toeplitz matrix generated by the polynomial
and a suitable cutting matrix (see [
15] for the terminology and the related notation) defined as
i.e.,
.
Two-grid/Multigrid convergence with the above defined restriction/prolongation operators and a simple smoother (for instance, Gauss–Seidel iteration) is a classical result, both from the point of view of the literature of approximated differential operators [
17] and from the point of view of the literature of structured matrices [
10,
15].
In the first panel of
Table 1, we report the number of iterations needed for achieving the predefined tolerance
, when increasing the matrix size in the setting of the current subsection. Indeed, we use
and its transpose as restriction and prolongation operators and Gauss–Seidel as a smoother. We highlight that only one iteration of pre-smoothing and only one iteration of post-smoothing are employed in the current numerics. Therefore, considering the results of Remark 1 and the subsequent explanation, there is no surprise in observing that the number of iterations needed for the two-grid, V-cycle, and W-cycle convergence remains almost constant when we increase the matrix size, numerically confirming the predicted optimality of the methods in this scalar setting.
4.2. Case
Let us consider the case of
Finite Elements, where we have that the basis functions with respect to the reference interval
are
For the sake of simplicity, let us consider the case of partitioning with five equispaced points (three internal points) and partitioning with nine equispaced points (seven internal points) obtained from by considering the midpoint of each subinterval.
Thus, with respect to Equation (
12), the
coefficients are
while the
coefficients are
and so on again as for that first couple of basis functions. Notice also that, to evaluate the coefficients, for the sake of simplicity, we are referring to the basis functions on the reference interval, as depicted in
Figure 1. To sum up, the obtained prolongation matrix is as follows
Hereafter, we are interested in setting such a geometrical multigrid strategy, proposed in [
17,
19,
20], in the framework of the more general algebraic multigrid theory and in particular in the one driven by the matrix symbol analysis. To this end, we represent the prolongation operator quoted above as the product of a Toeplitz matrix generated by a polynomial
and a suitable cutting matrix. We recall that the Finite Element stiffness matrix could be thought as a principal submatrix of a Toeplitz matrix generated by the matrix-valued symbol that, from Equation (
8), has the compact form
Then, it is quite natural to look for a matrix-valued symbol for the polynomial
as well. In addition, the cutting matrix is also formed through the Kronecker product of the scalar cutting matrix in Equation (
14) and the identity matrix of order 2, so that
Taking into account the action of the cutting matrix
, we can easily identify from Equation (
15) the generating polynomial as
where
that is
A very preliminary analysis, just by computing the determinant of
shows there is a zero of third order in the mirror point
, being
Moreover, the analysis can be more detailed, as highlighted in
Section 2.
We highlight that our choices are in agreement with the mathematical conditions set in Items
(A) and
(B). Condition
(C) is violated and we discuss it in
Section 5 and Remark 2. Nevertheless, it is possible to derive the following TGM convergence and optimality sufficient conditions that should be verified by
f and
, exploiting the idea in the proof of the main result of [
18]:
with
where
,
is the
null matrix,
is a constant independent on
n, and we denote by
(respectively,
) the positive definiteness (respectively, non-positive definiteness) of the matrix
. The condition in Equation (19) requires the matrix-valued function
to be uniformly bounded in the spectral norm. These conditions are obtained from the proof of the main convergence result in [
18], where, after several numerical derivations, it was concluded that the above conditions are the final requirements needed.
To this end, we have explicitly formed the matrices involved in the conditions in Equations (
18) and (19) and computed their eigenvalues for
. The results are reported in
Figure 2 and are in perfect agreement with the theoretical requirements.
In the second panel of
Table 1, we report the number of iterations needed for achieving the predefined tolerance
, when increasing the matrix size in the setting of the current subsection. Indeed, we use
and its transpose as restriction and prolongation operators and Gauss–Seidel as a smoother. Again, we remind that only one iteration of pre-smoothing and only one iteration of post-smoothing are employed in our numerical setting.
As expected, we observe that the number of iterations needed for the two-grid convergence remains constant when we increase the matrix size, numerically confirming the optimality of the method.
Moreover, we notice that also the V-cycle and W-cycle methods possess optimal convergence properties. Although this behavior is expected from the point of view of differential approximated operators, it is interesting in the setting of algebraic multigrid methods. Indeed, constructing an optimal V-cycle method for matrices in this block setting might require a specific analysis of the spectral properties of the restricted operators (see [
18]).
4.3. Case
Hereafter, we briefly summarize the case of
Finite Elements, following the very same path we already considered in the previous section for
Finite Elements. The basis functions with respect to the reference interval
are
For the sake of simplicity, let us consider the case of partitioning with seven equispaced points (five internal points) and partitioning with 13 equispaced points (11 internal points) obtained from by considering the midpoint of each subinterval.
Thus, with respect to Equation (
12) (see also
Figure 3), the
coefficients are
while, the
coefficients are
and the
coefficients are
Thus, the obtained prolongation matrix is as follows:
Thus, taking into consideration that the stiffness matrix is a principal submatrix of the Toeplitz matrix generated by the matrix-valued function
we are looking for the matrix-valued symbol
as well. By defining
it is easy to identify the generating polynomial as
where
that is
A trivial computation shows again shows there is a zero of fourth order in the mirror point
, being
However, the main goal is to verify the conditions in Equations (
18) and (19): we have explicitly formed the matrices involved and computed their eigenvalues for
. The results are in perfect agreement with the theoretical requirements (see
Figure 4). This analysis links the geometric approach proposed in [
17,
19,
20] to the novel algebraic multigrid methods for block-Toeplitz matrices.
In the third panel of
Table 1, we report the number of iterations needed for achieving the predefined tolerance
, when increasing the matrix size in the setting of the current subsection. Indeed, we use
and its transpose as restriction and prolongation operators and Gauss–Seidel as a smoother (one iteration of pre-smoothing and one iteration of post-smoothing).
As expected, we observe that the number of iterations needed for the two-grid convergence remains constant when we increase the matrix size, numerically confirming the optimality of the method. As in the case, we also notice that the V-cycle and W-cycle methods possess the same optimal convergence properties.
Comparing the three panels in
Table 1, we also notice a mild dependency of the number of iterations on the polynomial degree
k. In addition, we can see in
Table 2 and
Table 3 that the optimal behavior of the two-grid, V-cycle, and W-cycle methods for
remains unchanged if we test different tolerance values.
Remark 2. In the cases analyzed in this section, we notice that, even though and
do not commute, the two-grid method is still convergent and optimal. The latter commutation property, along with Conditions (A) and (B) reported in Section 2, is sufficient to have optimal convergence of the two-grid method. This analysis reveals that commutativity is not a necessary property. Indeed, in our examples, we show that the operator is uniformly bounded in the spectral norm. However, we notice that in all cases the commutatorcomputed in 0 is a singular matrix. In particular, computing our commutator matrixin, we obtain:which are indeed singular matrices. Remark 3. It is worth stressing that the results hold also in dimension. In fact, interestingly, we observe that the dimensionality d does not affect the efficiency of the proposed method, as well shown inTable 4for the case. We finally remind that the tensor structure of the resulting matrices highly facilitates the generalization and extension of the numerical code to the case of. Indeed, the prolongation operators in the multilevel setting are constructed by a proper tensorization of those in 1D. Furthermore, we highlight that the presented analysis for
can be easily extended to the case on non-constant coefficients
in 1D and
in 2D, since, following a geometric approach, the prolongation operators for the general variable coefficients remain unchanged. In
Table 5 and
Table 6, we show the number of iterations needed for the convergence of the two-grid, V-cycle, and W-cycle methods for
in one and two dimensions for different values of
.