We now summarize some previous approaches to the algebraic construction of a multi- grid hierarchy for saddle point systems.
We start with an AMG method for constrained systems, especially contact problems, [Ada04] of the form
A BT B 0 u λ =f g
where A is obtained from the discretization of a PDE and B comprises the constraint equations (contacts). An AMG method for the A part is assumed to be available, that is, we have a hierarchy of interpolation operators {Pl}L−1l=1 , coarse grid operators {Al}Ll=1
and smoothers denoted by Ml, l = 1, . . . , L. It remains to construct interpolation
operators { ¯Pl}Ll=1 for the Lagrangian multipliers. Then, the overall coarse operator can
be constructed as PT l 0 0 P¯T l Al BlT Bl 0 Pl 0 0 P¯l .
The main idea is to compute a symmetric auxiliary matrix Gl for the Lagrangian mul-
tiplier space and to derive the coarse grid and the interpolation operator by using plain unsmoothed aggregation AMG techniques (see Section 2.11.2) applied to Gl. Within
the context of contact problems, the ansatz
Gl:= BlPlPlTBlT
is preferred, as this allows to maximize the angles between the coarse level constraints Bl+1 = ¯PlTBlPl, see [Ada04] for details. The constraint interpolation ¯Pl is constant per
aggregate and scaled such that ¯PlTP¯l is the identity.
Smoothing is performed by either symmetric inexact Uzawa smoothing 4.3.2 or a Schwarz subdomain smoother, where each subdomain consists of a subset of the constraints as well as the primal variables involved.
A special case is the saddle point AMG proposed in [LOS04]. Here the constraints describe the Dirichlet boundary conditions for a reproducing kernel particle method (RKPM) applied to an elliptic partial differential equation. More precisely, the entries of B are computed by bij := Z ΓD ˜ ψitr ψj
where the ansatz functions ψj are defined throughout the domain Ω and are used for
the discretization of the PDE, while the basis functions ˜ψi are defined on the boundary
ΓD ⊂ ∂Ω. Here, tr denotes the trace operator on the boundary ΓD.
The coarsening process for the constraints employs a matrix C defined by
Cij :=
Z
ΓD
˜ ψiψ˜j.
Then, a smoothed aggregation algorithm 2.11.2 is applied to both A and C to obtain interpolation operators P and ¯P for the ansatz and the constraint space, respectively. the overall interpolation operator is assembled as
P =P 0
0 P¯
and the coarse grid operator is computed by the Galerkin product PTKP.
The smoother employed here takes the form of a Braess-Sarazin method (4.54)–(4.56),
λit+1 ← BωD−1
BT−1B uit+ ωD−1(f − Auit) − g uit+1 ← uit+ ωD−1
(f − Auit) − ωD−1BTλit+1,
that is a ω-damped Jacobi iteration for the ansatz space, while a direct solver is used to compute BωD−1BT−1
.
In [Wab03], [Wab04], [Wab06], a monolithic semi-algebraic AMG approach to the so- lution of the Navier-Stokes equations is described. To this end, the incompressible Navier-Stokes equations
∂
∂tu − ν∇ · (∇u) + (u · ∇) u + ∇p = f (4.57)
−∇ · u = 0 (4.58)
given on a domain Ω ⊂ Rd, d = 2, 3, are linearized to obtain the Oseen equations, ∂
∂tu − ν∇ · (∇u) + (w · ∇) u + ∇p = f (4.59)
−∇ · u = 0. (4.60)
Here, (assuming that the fluid is Newtonian, i.e. has a constant shear to stress ratio,) the kinematic velocity ν := µρ is defined as the quotient of the dynamic viscosity µ and the density ρ, which both depend on the material. As in the case of Stokes’ equations, we additionally introduce boundary conditions on ∂Ω = ΓD∪Γ˙ N,
u(x) = r(x) for all x ∈ ΓD, (4.61)
ν∂u
∂n(x) − np(x) = s(x) for all x ∈ ΓN, (4.62) and (as we consider an instationary problem), initial conditions for t = 0,
u|t=0 = u0, p|t=0= p0. (4.63)
For a detailed introduction to these equations, we refer to [GS98] and [ESW05].
The linear Oseen equations (4.59)–(4.60) can be used inside a fixed point iteration for (4.57)–(4.58), where the newly introduced vector field w is set to the previous approx- imation of u. Now, using a finite element discretization for the spatial variables and a time-stepping scheme, we end up with a linear system of equations of the form
K(w)x =A(w) B T B −C u p =f g . Here, A(w) = c1M + AD + c2AC(w) + c3AS(w) + c4AR(w) (4.64)
consists of a mass matrix M which stems from the time discretization, a diffusion matrix (Laplacian) AD, non-symmetric convection and reaction matrices AC(w) and AR(w) and
a symmetric positive definite convection stabilization matrix AS(w). Depending on the
discretization, C is zero or a pressure stabilization matrix, see Section 4.2.2.
Depending on the element chosen, the coarsening is either performed element-wise (i.e. AMGe, cf. Section 2.11.1) for the Crouzeix-Raviart element (4.40), or point-wise (clas- sical Ruge-St¨uben) by coarsening the nodes of the finite element mesh. In case of a P1 − P1-stab discretization (4.41), the same coarse mesh is used for all physical un-
knowns u and p, while in case of the P1isoP2 − P 1-element (4.38), the pressure mesh
is coarsened first and the velocity mesh at each level l ≥ 2 is identical to the pressure mesh at level l − 11. In all cases, the interpolation is unknown-wise, i.e.
P = PV(1) 0 0 0 PV(2) 0 0 0 PW or P = PV(1) 0 0 0 0 PV(2) 0 0 0 0 PV(3) 0 0 0 0 PW
where each PV(i), i = 1, . . . , d is a scalar interpolation scheme. The coarse level system is
computed by the Galerkin product with two slight modifications: the coarse convection stabilization term AS,l+1 part needs to be rescaled,
AS,l+1 := d
r nl
nl+1
PVTlAS,lPVl
to prevent oscillations, and, in the case of the P1-P1-stab element, the matrix Cl+1needs
to be computed as Cl+1 := λmax(Dl−1Ml) h2 P T WlClPWl,
where Ml := PWTl· · · PWT1M PW1· · · PWl is the Galerkin projection of the fine level (pres-
sure) mass matrix M to level l, Dl is the diagonal of one of the component blocks of
AD,l, and h is the mesh width on the finest level.
The stability of the coarse level systems, i.e. the existence of a discrete inf-sup condition on all levels can only be shown with additional geometric information. In the case of P1isoP2-elements, a rigorous analysis is not known and stability can only be motivated heuristically. We refer to [Wab03], [Wab04], and [Wab06] for details. We now give the stability lemma for the P1-P1-stab element, which will be the basis of our more general stability results presented in Sections 4.8–4.10.
Lemma 4.2 ([Wab03], Lemma 4.3) Assume that for all elements τ ∈ Th the diameter
hτ fulfills
αh ≤ hτ ≤ αh,
1For stability reasons, it might be necessary to use the pressure mesh at level l − 2 instead for l ≥ 3
with positive constants α and α and the discretization parameter h, and assume further that AD,l is symmetric and of essentially positive type (2.28) and that for all vl ∈ Vl we
can find Πlvl ∈ Vl+1 such that
kvl− PVlΠlk
2
Dl ≤ βkvlk
2
AD,l, (4.65)
with some constant β. Then for all levels l ∈ {1, . . . , L} there exist positive cl and dl
such that sup 06=v∈Vl vBT l p kvkAD,l ≥ clkpkMl− dl p TC lp 12 for all p ∈ Wl.
The approximation property 4.65 is fulfilled for many of the classical AMG interpolation operators, see Section 2.8.
In the remainder of this chapter, we describe the components of our algebraic multigrid approach to saddle point problems. Unlike the ansatzes introduced above, we are not restricted to specific discretizations or geometric information. We only need to know the decomposition of the saddle point matrix K,
K =A B
T
B −C
.
In many applications, the vector u describes discretization of two or three physical quantities (e.g. the velocity components in different spatial directions). In consequence, A itself is a discretization of a system of elliptic partial differential equations. We have already presented three approaches to system AMG methods in Section 2.12: First, one can ignore the decomposition into the physical unknowns and just apply AMG to A (VAMG, Section 2.12.1). The second possibility is to split A by the physical components (velocity directions) and to apply scalar AMG component-wise (UAMG, Section 2.12.2). The third ansatz is to utilize additional information to sort the matrix A by (discretization) points (PAMG, Section 2.12.3). All of them can be combined with the techniques described below to obtain an algebraic multigrid method for saddle point systems.