On a posteriori error bounds for approximations of the generalized Stokes problem generated by the Uzawa algorithm

Abstract In this paper, we derive computable a posteriori error bounds for approximations computed by the Uzawa algorithm for the generalized Stokes problem. We show that for each Uzawa iteration both the velocity error and the pressure error are bounded from above by a constant multiplied by the L2-norm of the divergence of the velocity. The derivation of the estimates essentially uses a posteriori estimates of the functional type for the Stokes problem.


Introduction
Let Ω ∈ R n be a bounded connected domain with a Lipschitz continuous boundary ∂ Ω. Henceforth, we use the space of vector valued functions where M n×n is the space of symmetric n × n-matrices (tensors). The scalar product of tensors is denoted by two dots (:), and the L 2 norm of Σ is denoted by · Σ . The L 2 norm of scalar and vector valued functions is denoted by · .
ByS(Ω) we denote the closure of smooth solenoidal functions w with compact supports in Ω with respect to the norm ∇w Σ . Let V 0 (Ω, R n ) denote the subspace 322 I. Anjam, M. Nokka, and S. I. Repin of V (Ω, R n ) that consists of functions with zero traces on ∂ Ω. The space of scalar valued square summable functions with zero mean is denoted by L 2 (Ω, R).
The classical statement of the generalized Stokes problem consists of finding a velocity field u ∈S(Ω) + u D and pressure p ∈ L 2 (Ω) which satisfy the relations −Div (ν∇u) where f ∈ L 2 (Ω, R n ), and ∂ Ω u D · n dx = 0.
The generalized solution of (1. The quantity in L A is called the augmented Lagrangian (in which λ ∈ R + ). We have From the right-hand side inequalities we see that Ω (p − q)div u dx = 0 for all q ∈ L 2 , from which we conclude that div u = 0. From the left-hand side inequalities it follows that for any solenoidal v we have On a posteriori error bounds 323 Indeed, the exact solution of the problems is (u, p). For a detailed exposition of this subject, we refer to [4]. Finding approximations of (u, p) can be performed by the Uzawa algorithm presented below.
For the Lagrangian L, we have the problem: Find u k ∈ V 0 + u D such that: For the Lagrangian L A , we have the problem: Find u k ∈ V 0 + u D such that: (1.7) 4: Set k = k + 1 and go to step 2.
Our goal is to deduce computable bounds of the difference between u k and the exact solution u in terms of the energy norms provided that 0 < ρ < 2 min(ν, µ) (1.8) and p 0 ∈ L 2 (Ω). If µ ≡ 0, the condition is 0 < ρ < 2ν. These conditions are the same for both (1.5) and (1.6).
Proof. The proof is based on well known arguments (see, e.g., [13]). However, for the convenience of the reader, we present the proof for the generalized Stokes problem, in the case of (1.5).
The exact solution of the generalized Stokes problem satisfies the relation We set w = u k − u and subtract (1.9) from (1.5), which gives Let v k := u k − u and q k := p k − p. Then we rewrite this relation in the form On the other hand, (1.7) is equivalent to .
By combining (1.10) and (1.11), we obtain On a posteriori error bounds 325 where δ ∈ (0, 1). Note that and, therefore, (1.12) implies the estimates Now, we sum inequalities (1.13) for k = 0, . . . , N and find that (1.14) Because of condition (1.8), there exists a δ * ∈ (0, 1) such that We set δ = δ * in (1.14), and see that Also, we see that q k = p k − p is bounded in L 2 (Ω), so p k is bounded in L 2 (Ω). We also observe from (1.14), that so we can extract from p k a subsequence p k ′ , which converges to some element p * weakly in L 2 (Ω). The equation (1.5) gives in the limit and by comparison to (1.9) we find that In other words, the sequence p k ′ converges weakly to p in L 2 (Ω) However, if p 0 ∈ L 2 , then it is easy to see from (1.7) that p k ∈ L 2 with all k. From this we make the conclusion that the sequence p k ′ converges weakly to p in L 2 (Ω).

Error estimates for exact solutions generated by the Uzawa algorithm
In this section, we show that the errors of approximations generated by the Uzawa algorithm are controlled by the L 2 -norm of the divergence of the velocity. First, we compare approximations computed on two consequent iterations and establish the following result.
In addition, for (1.6) we also have Proof. The equation for pressure (2.2) follows directly from (1.7). By subtracting the kth equation (1.5) from the (k + 1)th equation, we obtain we can estimate the right-hand side with Henceforth, we will use functional a posteriori error estimates for the Stokes problem derived in [11,12]. For a consequent exposition of the theory of functional a posteriori error estimates we refer the reader to [8,10].
For some simple domains the constant C LBB , or the bounds for it, are known (see, e.g., [3,5,9]). Lemma 2.1 implies an important corollary. Let v ∈ V 0 , and div v = g. Then there exists a function v g ∈ V 0 such that div(v − v g ) = 0, and

This means that there exists a solenoidal field
Thus, we can find a function w 0 ∈S(Ω) + u D such that With the help of (2.4) we can now derive our main results. We show that the errors of u k and p k generated on the iteration k of the Uzawa algorithm are both estimated from above by the L 2 -norm of the divergence of u k multiplied by a constant depending on C LBB . The proofs are based on the derivation of functional a posteriori error estimates for the generalized Stokes problem as they are presented in [12].
Here C F is the constant in the Friedrichs inequality w C F ∇w Σ and C LBB is the constant in the LBB-condition.
Proof. Let u 0 ∈S(Ω) + u D be such that, by using (2.4), we have Let the pair (u k , p k ) be an approximation of the saddle point computed on the iteration k. We can now write First, we estimate from above the first term on the right-hand side of (2.8). Let w ∈S. By subtracting the integral Ω (ν∇u 0 : ∇w + µu 0 · w) dx from both sides of (1.4) we obtain It is easy to see that Ω (Div τ · w + τ : ∇w)dx = 0 ∀τ ∈ Σ(Div, Ω), w ∈ V 0 (Ω) (2.10) and By adding (2.10) and (2.11) to the right-hand side of (2.9), we rewrite it in the form which is equivalent to Let us choose τ = ν∇u k and q = p k . In view of (1.5), we see that that the first integral of (2.13) vanishes. Indeed, Since w is a function fromS, the same conclusion is also true if u k has been calculated by (1.6). We combine (2.9) with (2.12)-(2.14), and arrive at the relation The right-hand side of (2.15) can be estimated from above as follows: where we have used the Cauchy-Schwarz inequality. We set w = u − u 0 , and find that Note that for all w ∈ V we have We substitute (2.17) into (2.8), and use (2.18) with w = u − u 0 , and obtain In order to prove a similar estimate for the pressure, we also need Lemma 2.1. Let q ∈ L 2 be an approximation of the exact pressure p. Then (p − q) ∈ L 2 and there exists a function w ∈ V 0 such that div(w) = p − q (2.20) and where C = 2C 2 for (1.5), and C = 2C 2 + λ for (1.6).
Proof. We use (2.20) for q = p k and obtain Multiplying (1.1) by w and integrating over Ω, we obtain In view of this relation, we have We use (2.10) with w = w, and arrive at the relation As before, we choose τ = ν∇u k , and observe that the first integral is zero. By estimating the latter integral with the help of the same arguments as in (2.16), we find that p − p k 2 ||| u − u k ||| ||| w ||| .
(2.24) By (2.18) and (2.21), we obtain where C is defined in (2.6). Substituting (2.25) into (2.24) results in the estimate In the case of (1.6), we add Ω λ div(u k − u k )div w dx = 0 to (2.23) and obtain Again, we choose τ = ν∇u k , and see from (1.6) that the first and second integrals are zero. By estimating the latter integral with same arguments as in (2.16), we obtain Recall that div w = p − p k . Now, (2.25) and (2.26) imply the estimate Applying Theorem 2.2 results in (2.22).
By Theorems 2.2 and 2.3, we easily conclude the following statement.
Remark 2.1. The classical Stokes problem corresponds to the case where µ ≡ 0 and ν is a constant. Let (u k , p k ) be the exact solution computed on the iteration k of the Uzawa algorithm, for the Stokes problem. Then, for velocity we have (for both cases (1.5) and (1.6)) For the pressure we have p − p k C div u k whereC = 2C −2 LBB ν for (1.5) andC = 2C −2 LBB ν + λ for (1.6).

Computable error estimates for approximations generated by the Uzawa algorithm
Let T h be a mesh having the characteristic size h, and let the spaces V 0h (Ω, R n ) and Q h (Ω) be finite dimensional subspaces of V 0 (Ω, R n ) and L 2 (Ω), respectively. We assume that for all v h ∈ V 0h + u D it holds that div v h ∈ Q h . We also assume that the spaces are constructed so that they satisfy the discrete LBB-condition, i.e, for any q h ∈ Q h with zero mean, there exists v h ∈ V 0h such that where the positive constant c does not depend on h.
Let u k h ∈ V 0h + u D be an approximation of u k calculated on the mesh T h . We need to combine the error of the pure Uzawa algorithm with the approximation error. Below we present the corresponding results, where we set p k = p k h ∈ Q h on the iteration k, and understand u k as satisfying (1.5), or (1.6), with the chosen p k h . Then, the pair (u k , p k h ) can be viewed as the exact pair associated with the Uzawa algorithm on iteration k.
Our first goal is to derive fully computable error majorants M k ⊕ and M k,λ ⊕ for approximate solutions (e.g., u k h ) of the problems generated at the first step of Uzawa algorithm by the Lagrangians L and L A , respectively. In order to make the quality of the majorants robust with respect to small or large values of the material functions ν or µ, we apply the same method that was suggested in [12] for the generalized Stokes problem.
Later we combine these estimates with the estimates of the difference between u and u k and obtain estimates applicable for approximate solutions computed within the framework of finite dimensional approximations.
First, we prove the following result for the problem generated by the Lagrangian L.

4)
Here I denotes the unit tensor.

Proof. By equation (1.5) we have
We subtract the integral Ω ν∇u k h : ∇w + µu k h · w dx from both sides of the above equation, and obtain By adding (2.10) to the right-hand side of (3.5) we have where we have used the notation (3.3) and (3.4). Note that where 0 α(x) 1. Also, we have By (3.7) and (3.8) the right-hand side of (3.6) becomes We set w = u k − u k h , use (3.6) and (3.9), and obtain It is easy to see that the optimal value of α is defined by the relation so that (3.10) implies the estimate where we have used the notation (3.1) and (3.2).

Remark 3.1.
It is easy to see that the upper bound M k ⊕ is sharp. Indeed, by setting τ = ν∇u k − Ip k h , and letting β tend to infinity, we get the exact error in the energy norm ||| · |||.
Proof. Indeed, from (3.15) and (3.16) we find that Applying the error bounds presented in Theorems 3.1 and 3.2 completes the proof.
This paper is focused on theoretical analysis of a posteriori error bounds for approximations computed by the Uzawa algorithm. However, it is worth adding some comments on the practical applications of the above derived error majorants. The majorants contain the function τ ∈ H(Div, Ω) and a positive parameter β , which in general can be taken arbitrary. Getting sharp estimates requires a proper selection of them. Finding an optimal β leads to a one-dimensional optimization problem which is easy solvable. The reconstruction of the stress tensor τ based upon computed functions u k h and p k h provides a reasonable first guess. A better selection can be performed by methods that have been developed and tested for various elliptic problems (see, e.g., [8,10,14] and the references cited therein). A systematical study of computational questions in the context of above derived estimates will be exposed in a separate paper, which is now in preparation.