Intergrid transfer operators

In standard nested multigrid algorithms, the intergrid transfer operators are straightforward. A popular approach is to take injection for the prolongation and its adjoint for the restriction. For a skeletal system care must be taken in the con- struction of these operators to ensure the convergence of the multigrid algorithm under consideration. In the following we propose “physics-based energy-preserving” operators using the fine scale DtN maps.

4.3.2.1 Prolongation

The prolongation operator, Ik: Mk−1 −→ Mk, transfers the error from the grid

level k − 1 to the finer grid level k. Note that standard prolongation using injection would set the values on the interior edges to zero and thus does not work. For our multigrid algorithm it is defined as a function of the solution order p using DtN maps. In particular,

Ik:=  −A−1 k,IIAk,IBJk Jk  for k = 2, . . . , N, (4.11) • and for p > 1, Ik :=      JN for k = N, "

−A−1_k,IIAk,IBJk

Jk

#

for k = 2, . . . , N − 1. (4.12)

Here, Jk : Mk−1 −→ Mk is the injection (interpolation). To understand the idea

behind the prolongation operator, let us consider p = 1 in (4.11). First, through Jk,

we interpolate the error from the grid level k − 1 to obtain error on the boundary edges of the finer grid level k. Then we solve (via the first block in the definition of Ik) for the error on the interior edges as a function of the interpolated error on the

boundary. For p > 1 the prolongation is defined in (4.12), namely, we use the same prolongation as in the case of p = 1 for all levels k ≤ N − 1 and interpolate error from piecewise linear polynomials at level N − 1 to piecewise pth-order polynomials at level N .

4.3.2.2 Restriction

The restriction operator, Qk−1: Mk −→ Mk−1, restricts the residual from level

k to coarser level k − 1. The popular idea is to define the restriction operator as the adjoint (with respect to the L2_{-inner products h·, ·i}

Ek and h·, ·iEk−1 in Mk and Mk−1)

of the prolongation operator, i.e., Qk−1 = Ik∗, and from our numerical studies (not

shown here) it still works well. However, this purely algebraic procedure, though convenient, is not our desire. Here, we construct the restriction operator Qk−1 such

that the coarse grid problem, via the Galerkin approximation, is exactly a discretized DtN problem on level k − 1. To that end, let us define Qk−1 as

• for p = 1, Qk−1 := −Jk∗Ak,BIA−1k,II J ∗ k  for k = 2, . . . , N, (4.13) • for p > 1, Qk−1:= (J∗ N for k = N, h −J∗ kAk,BIA−1k,II J ∗ k i for k = 2, . . . , N − 1. (4.14) Note that if A is symmetric, then our definition of the restriction operator Qk−1 is

indeed the adjoint of the prolongation operator Ik.

Given the prolongation and restriction operators, we obtain our coarse grid equation using the discrete Galerkin approximation [145]

Qk−1AkIkλk−1 = Qk−1gk, (4.15)

where the coarse grid operator

Ak−1 := Qk−1AkIk, (4.16)

for either p = 1 and k ≤ N or p > 1 and k ≤ N − 1, reads

Ak−1 = −Jk∗Ak,BIA−1k,II J ∗ k   Ak,II Ak,IB Ak,BI Ak,BB   −A−1 k,IIAk,IBJk Jk 

= J_k∗ Ak,BB − Ak,BIA−1k,IIAk,IB
Jk, (4.17)

and

AN −1= JN∗ANJN, (4.18)

for p > 1 and k = N .

Energy preservation. In order for the multigrid algorithms to converge the intergrid operators should be constructed in such a way that the “energy” does not in- crease when transferring information between a level to a finer one [18, 74]. Note that

“energy” here need not necessarily be associated with some physical energy. Indeed, if we associate Akwith a bilinear form ak(., .) such that ak(κ, µ) = hAkκ, µi_Ek∀κ, µ ∈

Mk, where again h., .i_Ek represents the L2−inner product on Ek, then we call ak(κ, κ)

the energy on level k associated with κ. Nonincreasing energy means

ak(Ikλ, Ikλ) ≤ ak−1(λ, λ) ∀λ ∈ Mk−1, ∀k = 2, 3, . . . , N.

Proposition 1 (Energy preservation). The proposed multigrid approach preserves the energy in the following sense: ∀k = 2, 3, . . . , N ,

ak(Ikλ, Ikλ) = ak−1(λ, λ) ∀λ ∈ Mk−1. (4.19)

Proof. We proceed first with p = 1. From the definition of Ak in (4.10) and the

definition of the prolongation operator Ik in (4.11) we have

ak(Ikλ, Ikλ) =   −A−1 k,IIAk,IBJkλ, Jkλ  ,  Ak,II Ak,IB Ak,BI Ak,BB   −A−1 k,IIAk,IBJkλ Jkλ  Ek

= Ak,BB − Ak,BIA−1k,IIAk,IB
Jkλ, Jkλ

Ek

=J_k∗ Ak,BB− Ak,BIA−1k,IIAk,IB
Jkλ, λ

Ek−1 = ak−1(λ, λ) ,

where the last equality comes from the definition of the coarse grid operator Ak−1 in

(4.17). For p > 1, we need to prove (4.19) only for k = N , but this is straightforward since from (4.12), (4.14), and (4.18) we have

aN(INλ, INλ) = hANJNλ, JNλi_Ek = hJN∗ANJNλ, λi_Ek−1 = ak−1(λ, λ) .

We now prove that the coarse grid operator (4.16) is also a discretized DtN map on every level.

Proposition 2. At every level k = 1, . . . , N , the Galerkin coarse grid operator (4.16) is a discretized DtN map on that level.

Proof. The proof is a straightforward induction using the proposed coarsening strat- egy and the definition of the intergrid transfer operators. First, consider the case of p = 1. The fact that AN = A in (4.9) is a discretized DtN map on the finest level is

clear by the definition of the hybridized methods. Assume at level k the operator Ak

in (4.10) is a discretized DtN map. Taking λk,B = Jkλk−1 and condensing λk,I out

yield

Ak,BB − Ak,BIA−1_k,IIAk,IB
Jkλk−1 = gk,B− Ak,BIA−1_k,IIgk,I

which, after restricting on the coarse space Mk−1 using Jk∗, becomes

J_k∗ Ak,BB − Ak,BIA−1_k,IIAk,IB
Jkλk−1= Jk∗ gk,B − Ak,BIA−1_k,IIgk,I
,

which is exactly the coarse grid equation (4.15). That is, Ak−1 is the Schur comple-

ment obtained by eliminating all the trace unknowns inside all the macro-elements on level k − 1. By definition, the coarse grid operator Ak−1 on level (k − 1) is also a

discrete DtN map.

For the case of p > 1, it is sufficient to show that AN −1 is a discrete DtN map,

but this is clear by: (1) taking λ = JNλN −1 in (4.9), (2) restricting both sides to

MN −1 using JN∗, (3) recalling that AN is a discretized DtN map, and 4) observing

that the resulting equation coincides with the coarse grid equation (4.18).