Adaptive finite element methods (AFEMs) have been an active research area since the pioneering work . 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 ; 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 ; 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]).
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.
Let be a Hilbert space and denote the dual space of Let be a continuous bilinear form and . We consider the following variational problem: Find such that for all
Here is the duality pairing between and . Let denote the norm on and the dual norm of For simplicity, we assume that the bilinear form is symmetric and positive-definite (SPD). The continuity and positive-definiteness of imply
for all where are absolute constants. Such a bilinear form naturally defines a bounded isomophism for which we have
Hence (2.1) is equivalent to the operator equation
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 , namely: Find such that
Note that the subspace 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 , we consider the natural inclusion and its adjoint defined as
We introduce the operator which approximates on . In this way, the discrete problem (2.4) reads
3. A posteriori error estimates by preconditioning
A posteriori error estimates are of the form
where are absolute positive constants and is computed from . In AFEMs, 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 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].
3.1. Links with operator preconditioning
Clearly, from our discusion above, it follows that constructing a posteriori error estimators is equivalent to estimating a norm of the error by computable bounds. We note, however, that a direct computation of the norm of will be, in general, impossible or too expensive, since one needs to compute the action of on . 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) , whose particular form will be given later. For the time being we only assume that is bounded and SPD, i.e., is an inner product on . Let 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 , i.e., the result of the action provides a good approximation to the high frequency components of the error. Now, a simple choice for 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 be two positive absolute constants. We say that is a preconditioner for provided there exist constants and , 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 . This is the central result in this paper.
Let (3.1) hold. Then we have the following two sided bound
Since is SPD, we use the Cauchy–Schwarz inequality to obtain
The inequality (3.1) implies
where we used in the first equality. The upper bound
can be shown in a similar fashion. In summary, we have
On the other hand, for any , (2.4) implies
Throughout the rest, will serve as a (nearly) computable a posteriori error estimator that is proved to be both an upper and lower bound of the error . In order to derive an error estimator within our framework, the key step is to suitably select the smoother such that the spectral equivalence (3.1) holds.
3.3. Additive Schwarz smoother
In this subsection, we construct a particular using the additive Schwarz method. For such a smoother, we present a lemma that serves as a criterion for verifying (3.1).
For , , let be subspaces providing a decomposition of , namely,
Let be the natural inclusion and denote its adjoint. We further set . Next, let be spectrally equivalent to . More precisely, for and , we assume that
where are positive absolute constants. The smoother (additive Schwarz method) is then defined to be
By the definition of , we obtain
We have the following identity
where the infimum is taken over and for
The proof that is a good preconditioner for is standard. We include it here for completeness and we follow the proof in .
For each , let
and . In addition, assume that for all , there exist with satisfying
Then (3.1) holds with constants ,
For , assume the decomposition with , . Direct calculation shows that
The definition of and implies
Taking the infimum with respect to all decompositions and using Lemma 3.2, we obtain the upper bound
In this section, we consider the typical example of a scalar elliptic equation. Let where is a Lipschitz polytope. For a given and , the bilinear and linear forms in equation (2.1) are:
We assume is uniformly elliptic, i.e.,
Let be a conforming and shape-regular simplicial partition of . Let denote the set of polynomials of degree at most . The subspace is
where is an integer.
Let denote the set of interior vertices in For each , let denote the continuous piecewise linear function that takes the value 1 at and 0 at other vertices. We denote for . Obviously we have
4.1. A posteriori error estimates for Lagrange elements
Now, let which is a subspace of by zero extension. The partition of unity (4.1) implies
We note that the framework also works for other local patches. For instance, can be chosen as the union of the two elements around each interior face.
For a fixed , the set defined in Lemma 3.3 translates into
In this case, is an absolute constant by the shape-regularity of . Throughout the rest, we adopt the notation provided with being a generic constant dependent only on and We say provided and .
We have the following estimate
To verify (3.8), we take , where is a
-stable interpolation which also enjoys standard approximation properties:
4.2. Computable error estimator
where is a subspace of polynomials. Generally speaking, the approximate estimator is expected to be an accurate upper and lower bound of provided the degree of polynomials in is sufficiently high.
Combining the previous estimate with (4.5) provides the following computable lower bound for the error
To derive a computable upper bound for , let us first write the action of the residual on . We denote the set of all -dimensional faces in the triangulation by . Clearly, where denotes the set of all boundary faces and the interior faces. We further denote by and by , respectively. Note that does not include the edges on . For each , let
Further, for each , let be the two elements sharing , (resp. ) the outward unit normal to (resp. ), and
It then follows from (4.4) and integration by parts that
Given a Lipschitz domain , let denote the diameter of The shape-regularity of implies that and we will use these notions interchangeably. Now, let us introduce the computable quantity
which is the standard explicit residual error estimator. We take 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 yields
Hence we obtain the following computable upper bound for the error
5. 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 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.
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