We present a unified approach in analyzing Uzawa iterative algorithms for saddle point problems. We study the classical Uzawa method, the augmented Lagrangian method, and two versions of inexact Uzawa algorithms. The target application is the Stokes system, but other saddle point systems, e.g., arising from mortar methods or Lagrange multipliers methods, can benefit from our study. We prove convergence of Uzawa algorithms and find optimal rates of convergence in an abstract setting on finite-or infinite-dimensional Hilbert spaces. The results can be used to design multilevel or adaptive algorithms for solving saddle point problems. The discrete spaces do not have to satisfy the LBB stability condition. Introduction.In this paper, we provide a unified approach for Uzawa methods for linear saddle point systems. Such systems arise in solving various partial differential equations (PDEs) or systems of PDEs at the continuous level or at the discrete level. Typical examples of such PDEs are second-order elliptic problems, Stokes equations, and elasticity problems. We analyze the classical Uzawa Method (UM) [1], the augmented Lagrangian Uzawa method (ALUM) [14], the inexact Uzawa method (IUM) [7,13], and a modified (or multilevel) inexact Uzawa method (MIUM) under a general approach on abstract Hilbert spaces. The motivation for considering abstract versions of Uzawa algorithms on infinite-dimensional Hilbert spaces is that the analysis at the continuous level of an algorithm for solving a PDE gives the right strategy for discretizing the PDE. In addition, the convergence factors of certain multilevel or adaptive algorithms for solving saddle point systems depend on the stability parameters of the continuous problem, and in many cases the discrete LBB stability condition is not required to be satisfied (see [4,12] or section 6). Next, we formulate the general framework of the saddle point problem to be studied in this paper and indicate the way the paper is organized.We let V and P be two Hilbert spaces with inner products a(·, ·) and (·, ·), with the corresponding induced norms | · | V = | · | = a(·, ·) 1/2 and · P = · = (·, ·) 1/2 . The dual parings on V * × V and P * × P are denoted by ·, · and (·, ·), respectively. Here, V * and P * denote the dual of V and P , respectively. We identify P * and P as Hilbert spaces so that (·, ·) represents both the inner product on P and the duality between P * and P . In applications to Stokes systems, V = (H 1 0 ) d (d = 2, 3, . . . ), P is a subspace of L 2 of codimension one and (·, ·) is the standard inner product on L 2 . Next, we consider that b(·, ·) is a continuous bilinear form on V × P , satisfying the

Let k be a sequence of triangulations of a polyhedron ⊂ n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u k ∈ S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet-Neumann boundary conditions on a polygon.

We construct a sequence of meshes k that provides quasi-optimal rates of convergence for the solution of the Poisson equation on a bounded polyhedral domain with right-hand side in H m−1 , m ≥ 2. More precisely, let ⊂ 3 be a bounded polyhedral domain and let. Also, let S k be the finite element space of continuous, piecewise polynomials of degree m ≥ 2 on k and let u k ∈ S k be the finite element approximation of u, then u − u k H 1 ( ) ≤ C dim(S k ) −m/3 f H m−1 ( ) , with C independent of k and f . Our method relies on the a priori estimate u ≤ C f H m−1 ( ) in certain anisotropic weighted Sobolev spaces = m+1 a+1 ( ), with a > 0 small, determined only by . The weight is the distance to the set of singular boundary points (i.e., edges). The main feature of our mesh refinement is that a segment AB in k will be divided into two segments AC and CB in k+1 as follows: |AC | = |CB| if A and B are equally singular and |AC |= |AB| if A is more singular than B. We can choose ≤ 2 −m/a . This allows us to use a uniform refinement of the tetrahedra that are away from the edges to construct k .

We review the definition of a Lie manifold (M, V) and the construction of the algebra Ψ ∞ V (M ) of pseudodifferential operators on a Lie manifold (M, V). We give some concrete Fredholmness conditions for pseudodifferential operators in Ψ ∞ V (M ) for a large class of Lie manifolds (M, V). These Fredholm conditions have applications to boundary value problems on polyhedral domains and to non-linear PDEs on non-compact manifolds. As an application, we determine the spectrum of the Dirac operator on a manifold with multi-cylindrical ends.

MSC:Keywords: Inexact Uzawa algorithms Saddle point system Multilevel methods Adaptive methods a b s t r a c t For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility Ladyshenskaya-Babušca-Brezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In light of the new spectral results for the Schur complements, we review the classical Babušca-Brezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions.