{'fraction_non_alphanumeric': 0.04742393201305107, 'fraction_numerical': 0.01836254647545337, 'mean_word_length': 4.04594180704441, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A posteriori error estimates of finite element methods by preconditioning

We present a framework that relates preconditioning with a posteriori error estimates in finite element methods. In particular, we use standard tools in subspace correction methods to obtain reliable and efficient error estimators. As a simple example, we recover the classical residual error estimators for the second order elliptic equations.


Introduction
Adaptive finite element methods (AFEMs) have been an active research area since the pioneering work [2]. In contrast to finite elements based on quasi-uniform meshes, AFEMs produce a sequence of locally refined grids that is able to resolve the singularity arising from irregular data in the underlying boundary value problems. Readers are referred to e.g., [3,31,36] for a thorough introduction. Among the key concepts in AFEMs, a posteriori error estimates are the building block for comparing errors on different elements and marking elements with large errors for refinement. For details on various AFEM error estimation techniques we refer to works on: explicit residual estimators [36]; implicit estimators based on local problems [2,4,14,29]; recovery-based estimators; [42,13,7,8,41]; hierarchical basis estimators [5,6,24,23]; functional estimators [32]; and equilibrated estimators [1,26,10,19].
On the other hand, parallel with the development of AFEMs, there are also substantial research efforts in studying efficient preconditioning, which is a technique for approximating the inverse of a differential operator. Usually, such approximations are aimed at accelerating Krylov subspace iterative methods for solving linear systems resulting from discretized partial differential equations. Popular techniques used for preconditioning include e.g., multigrid [35,11,21,22,40,39] and domain decomposition/subspace correction methods [18,34,37]. In practice, subspace correction methods provide an efficient way of reducing the condition number of a large-scale but finite-dimensional linear system. However, the analysis of uniform convergence rate of those methods often benefits from the general setting of infinite-dimensional Hilbert spaces (see, for example, [30,20,38]).
The work of Zikatanov was supported in part by NSF grants DMS-1720114 and DMS-1819157.
In this paper we present a general framework relating abstract operator preconditioning [37,20,38,25,27] to a posteriori error estimates. In particular, we shall show that such standard techniques for developing preconditioners also yield reliable and efficient error estimators. Here, for clarity of presentation, we focus on the symmetric and positive-definite problems although extensions to more general cases are definitely within reach. As a simple example, with this framework, we are able to recover the classical residual error estimators for elliptic equations in primal form.
The rest of this paper is organized as follows. In section 2, we set up the model variational problem and define the operator notation which is convenient when constructing preconditioners. In section 3, we develop the main theory on posteriori error estimates via preconditioning. Section 4 is devoted to the example of second order elliptic equation that illustrates the aforementioned abstract theory. Concluding remarks are found in Section 5.

Preliminaries
Let V be a Hilbert space and V ′ denote the dual space of V. Let a : V × V → R be a continuous bilinear form and f ∈ V ′ . We consider the following variational problem: Find u ∈ V such that for all v ∈ V Here ·, · is the duality pairing between V ′ and V . Let · V denote the norm on V and · V ′ the dual norm of V ′ . For simplicity, we assume that the bilinear form a(·, ·) is symmetric and positive-definite (SPD). The continuity and positive-definiteness of a(·, ·) imply for all v, w ∈ V, where α, α > 0 are absolute constants. Such a bilinear form naturally defines a bounded isomophism A : V → V ′ for which we have Av, w := a(v, w), ∀v, w ∈ V.
Hence (2.1) is equivalent to the operator equation (2.2a) and (2.2b) imply that A induces the inner product A·, · on V. For all v ∈ V, the A-norm on V is defined as v A := Av, v 1 2 , which is equivalent to the V -norm.
2.1. Approximation from a subspace. Let us consider a general case where we approximate the solution to (2.1) by restricting it to a subspace V h ⊂ V , namely: Find u h ∈ V h such that Note that the subspace V h does not even have to be finite dimensional, although it usually is in applications. It follows from (2.2a), (2.2b) and the well-known Lax-Milgram theorem that (2.4) admits a unique solution.
For such a subspace V h ⊂ V , we consider the natural inclusion I h : V h ֒→ V and its adjoint Q h : We introduce the operator In this way, the discrete problem (2.4) reads

A posteriori error estimates by preconditioning
A posteriori error estimates are of the form where C 1 , C 2 are absolute positive constants and η h is computed from u h . In AFEMs, η h is the sum of error indicators on all elements. The local error indicators can be used to compare errors on different elements and those elements with large errors will be refined. In this way, the errors estimated by η h are equidistributed over all elements in the mesh. The optimal computational complexity of AFEMs is often attributed to the aforementioned equidistribution of errors. Rigorous analysis of convergence and optimality of AFEMs can be found in e.g., [17,28,9,33,15].

Links with operator preconditioning. Let
Clearly, from our discusion above, it follows that constructing a posteriori error estimators is equivalent to estimating a norm of the error e = A −1 r by computable bounds. We note, however, that a direct computation of the norm of A −1 r will be, in general, impossible or too expensive, since one needs to compute the action of A −1 on r. As we pointed out in the introduction, approximating such action has been also studied for several decades and is known as as preconditioning. Following this simple observation we now borrow some simple ideas from this field and apply them in constructing a posteriori error estimators.
First, we need a bounded isomorphism (the preconditioner ) B : V ′ → V , whose particular form will be given later. For the time being we only assume that B is bounded and SPD, i.e., ·, B· is an inner product on V ′ . Let S : V ′ → V be a SPD operator, which we will refer to as "the smoother " and is such that its range approximates well the high frequency part of the range of A −1 , i.e., the result of the action Sr provides a good approximation to the high frequency components of the error. Now, a simple choice for B is which is known as additive Schwarz preconditioner. Just to simplify the presentation, we will not consider the multiplicative preconditioner in this paper although following the abstract framework developed in [20,38] similar results can also be obtained in the multiplicative case as well. Let β, β be two positive absolute constants. We say that B is a preconditioner for A provided there exist constants β > 0 and β < ∞, such that The inequality (3.1) is known as spectral equivalence, or norm equivalence, and is a common ingredient in the analysis of convergence of iterative methods for large-scale linear systems.

3.2.
Estimating the residual. We now show that the norm (spectral) equivalence (3.1) naturally yields a two-sided estimate on e A . This is the central result in this paper.
i.e., Q h r = 0. Hence Throughout the rest, r, Sr will serve as a (nearly) computable a posteriori error estimator that is proved to be both an upper and lower bound of the error e A . In order to derive an error estimator within our framework, the key step is to suitably select the smoother S such that the spectral equivalence (3.1) holds.
3.3. Additive Schwarz smoother. In this subsection, we construct a particular S using the additive Schwarz method. For such a smoother, we present a lemma that serves as a criterion for verifying (3.1).
For n ∈ N, 1 ≤ k ≤ n, let V k ⊂ V be subspaces providing a decomposition of V , namely, Let I k : V k ֒→ V be the natural inclusion and Q k : V ′ → V ′ k denote its adjoint. We further set A k := Q k AI k . Next, let S k : V ′ k → V k be spectrally equivalent to A −1 k . More precisely, for 1 ≤ k ≤ n and v k ∈ V k , we assume that By the definition of B, we obtain The norm of B can be estimated using the following lemma, which can be found in e.g., [20,38,34,12].
Lemma 3.2. We have the following identity where the infimum is taken over v h ∈ V h and v k ∈ V k for 1 ≤ k ≤ n.
The proof that B is a good preconditioner for A is standard. We include it here for completeness and we follow the proof in [38].
The definition of M(k) and M implies n j,k=1 Combining the previous estimate with (3.9) and (3.7) gives Taking the infimum with respect to all decompositions and using Lemma 3.2, we obtain the upper bound For the lower bound in (3.1), v k be the decomposition that satisfies (3.8). It then follows from (3.7) and (3.8) that Using the previous estimate and Lemma 3.2, we obtain The proof is complete.

Examples
In this section, we consider the typical example of a scalar elliptic equa- For a given f ∈ L 2 (Ω) and K ∈ [W 1 ∞ (Ω)] d×d , the bilinear and linear forms in equation (2.1) are: We assume K is uniformly elliptic, i.e., Hence (2.2a) and (2.2b) holds. Let T h be a conforming and shape-regular simplicial partition of Ω. Let P p (T ) denote the set of polynomials of degree at most p. The subspace Let {x k } n k=1 denote the set of interior vertices in T h . For each x k , let φ k denote the continuous piecewise linear function that takes the value 1 at x k and 0 at other vertices. We denote for 1 ≤ k ≤ n, Ω k := supp φ k . Obviously we have We note that the framework also works for other local patches. For instance, Ω k can be chosen as the union of the two elements around each interior face.
For a fixed k, the set M(k) defined in Lemma 3.3 translates into In this case, M = max 1≤k≤n #M(k) is an absolute constant by the shaperegularity of T h . Throughout the rest, we adopt the notation C 1 C 2 provided C 1 ≤ C 3 C 2 with C 3 being a generic constant dependent only on K and M. We say C 1 C 2 provided C 1 C 2 and C 2 C 1 . We set S k := A −1 k and thus γ = γ = 1 in (3.7). The corresponding smoother S yields an error estimator. In order to show the reliability and efficiency, we need to verify (3.8) in Lemma 3.3.
Corollary 4.1. We have the following estimate Proof. To verify (3.8), we take v h = Π h v ∈ V h , where Π h is a H 1 -stable interpolation which also enjoys standard approximation properties: A simple choice for Π h is the Clément interpolation [16]. We now set For ϕ ∈ V k , we have Hence computing η k := A −1 k Q k r ∈ V k amounts to solving the variational problem: Taking ϕ = η k in (4.4) implies that  is expected to be an accurate upper and lower bound of e A provided the degree of polynomials in V k is sufficiently high.
Taking ϕ = η k in (4.6) and (4.4), we obtain Combining the previous estimate with (4.5) provides the following computable lower bound for the error To derive a computable upper bound for e A , let us first write the action of the residual on V k = H 1 0 (Ω k ). We denote the set of all (d − 1)-dimensional faces in the triangulation T h by E h . Clearly, h denotes the set of all boundary faces and E o the interior faces. We further denote T h | Ω k by T k and E h |Ω k by E k , respectively. Note that E k does not include the edges on ∂Ω k . For each T ∈ T h , let Further, for each e ∈ E o h , let T 1 , T 2 ∈ T h be the two elements sharing e, n 1 (resp. n 2 ) the outward unit normal to ∂K 1 (resp. ∂K 2 ), and It then follows from (4.4) and integration by parts that Given a Lipschitz domain D, let h D denote the diameter of D. The shaperegularity of T h implies that h k h T h e and we will use these notions interchangeably. Now, let us introduce the computable quantity , which is the standard explicit residual error estimator. We take ϕ = η k and use (4.7) and the Cauchy-Schwarz inequality to obtain that Finally, combining the previous inequality with the trace inequality and the Poincaré inequality η k L 2 (Ω k ) h k ∇η k L 2 (Ω k ) yields Hence we obtain the following computable upper bound for the error

Concluding remarks
For SPD problems, we have shown how preconditioning can be used to derive a posteriori error estimates. Extensions of this abstract theoretical framework and its application to derive estimators for indefinite, nonconforming, and discontinuous Galerkin methods are ongoing. A close inspection of the arguments shows that not only preconditioning can give a unified way to derive a posteriori error estimators. This is a two-way street: the a posteriori error estimators may provide efficient smoothers for multilevel methods. For example, the operator S we have introduced in our framework is a clear analogue of smoothing (relaxation) operator. We hope that some of the error indicators and estimators may give efficient smoothers in case of non-symmetric and or indefinite problems which are, in general, hard to precondition.