On the Convergence of
Adaptive Iterative Linearized Galerkin Methods
A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work  that covers some prominent procedures (including the Zarantonello, Kačanov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main purpose of this paper is the derivation of an abstract convergence theory for the unified ILG approach (based on general adaptive Galerkin discretization methods) proposed in . The theoretical results will be tested and compared for the aforementioned three iterative linearization schemes in the context of adaptive finite element discretizations of strongly monotone stationary conservation laws.
Key words and phrases:Numerical solution methods for quasilinear elliptic PDE, monotone problems, fixed point iterations, linearization schemes, Kačanov method, Newton method, Galerkin discretizations, adaptive mesh refinement, convergence of adaptive finite element methods
2010 Mathematics Subject Classification:35J62, 47J25, 47H05, 47H10, 49M15, 65J15, 65N12, 65N30, 65N50
In this paper we analyze the convergence of adaptive iterative linearized Galerkin (ILG) methods for nonlinear problems with strongly monotone operators. To set the stage, we consider a real Hilbert space with inner product and induced norm denoted by . Then, given a nonlinear operator , we focus on the equation
where denotes the dual space of . In weak form, this problem reads
with signifying the duality pairing in . For the purpose of this work, we suppose that satisfies the following conditions:
The operator is Lipschitz continuous, i.e. it exists a constant such that
for all .
The operator is strongly monotone, i.e. there is a constant such that
for all .
The existence of a solution to the nonlinear equation (1) can be established in a constructive way. This can be accomplished, for instance, by transforming (1) into an appropriate fixed-point form, which, in turn, induces a potentially convergent fixed-point iteration scheme. To this end, following our approach in , for some given , we consider a linear and invertible preconditioning operator . Then, applying to (1) leads to the fixed-point equation
For any suitable initial guess , the above identity motivates the iteration scheme
Equivalently, we have
we may write
In order to discuss the weak form of (5), for a prescribed , we introduce the bilinear form
Then, based on , the solution of (5) can be obtained from the weak formulation
Throughout this paper, for any , we assume that the bilinear form is uniformly coercive and bounded. The latter two assumptions refer to the fact that there are two constants independent of , such that
respectively. In particular, owing to the Lax-Milgram Theorem, these properties imply the well-posedness of the solution of the linear equation (7), for any given .
Let us briefly review some prominent procedures that can be cast into the framework of the linearized fixed-point iteration (7): For instance, we point to the Zarantonello iteration given by
in the special case that is independent of . Finally, we mention the (damped) Newton method which is defined by
for a damping parameter . Here signifies the Gâteaux derivative of (provided that it exists). For any of the above three iterative procedures, we emphasize that convergence to the unique solution of (1) can be guaranteed under suitable conditions; see our previous work  for details.
The ILG approach
Consider a finite dimensional subspace . Then, the Galerkin approximation of (2) in reads as follows:
We note that (13) has a unique solution since the restriction still satisfies the conditions (F1) and (F2) above. The iterative linearized Galerkin (ILG) approach is based on discretizing the iteration scheme (7). Specifically, a Galerkin approximation of , based on a prescribed initial guess , is obtained by solving iteratively the linear discrete problem
for . For the resulting sequence of discrete solutions it is possible, based on elliptic reconstruction techniques (cf., e.g., [13, 14]), to obtain general (abstract) a posteriori estimates for the difference to the exact solution, , of (1), i.e. for , , see [12, §3]. Based on such a posteriori error estimators, an adaptive ILG algorithm that exploits an efficient interplay of the iterative linearization scheme (14) and automatic Galerkin space enrichments was proposed in [12, §4]; see also . We refer to some related works in the context of (inexact) Newton schemes [1, 2, 9, 8], or of the Kačanov iteration [4, 11].
Goal of this paper
The convergence of an adaptive Kačanov algorithm, which is based on a finite element discretization, for the numerical solution of quasi-linear elliptic partial differential equations has been studied in . Furthermore, more recently, the authors of  have proposed and analyzed an adaptive algorithm for the numerical solution of (1) within the specific context of a finite element discretization of the Zarantonello iteration (10). The latter paper includes an analysis of the convergence rate which is related to the work  on optimal convergence for adaptive finite element methods within a more general abstract framework. The purpose of the current paper is to generalize the adaptive ILG algorithm from  to the framework of the unified iterative linearization scheme (5); furthermore, arbitrary (conforming) Galerkin discretizations will be considered. In order to provide a convergence analysis for the ILG scheme (14) within this general abstract setting, we will follow along the lines of , however, we emphasize that some significant modifications in the analysis are required. Indeed, whilst the theory in  relies on a contraction argument for the Zarantonello iteration, this favourable property is not available for the general iterative linearization scheme (5). To address this difficulty, we derive a contraction-like property instead. This observation will then suffice to establish the convergence of the adaptive ILG scheme, and to (uniformly) bound the number of linearization steps on each (fixed) Galerkin space similar to ; we note that the latter property constitutes a crucial ingredient with regards to the (linear) computational complexity of adaptive iterative linearized finite element schemes.
Section 2 contains a convergence analysis of the unified iteration scheme (5). On that account we will encounter a contraction-like property, which is key for the subsequent analysis of the convergence rate of the adaptive ILG algorithm in Section 3. Here, in addition, a (uniform) bound of the iterative linearization steps on each discrete space will be shown. In Section 4, we will test our ILG algorithm in the context of finite element discretizations of stationary conservation laws. Finally, we add a few concluding remarks in Section 5.
2. Iterative linearization
In this first section we will address the convergence of the linearized iteration (5). We begin with the following a posteriori error estimate.
In addition to (F1) and (F2), let us make a further assumption on the (nonlinear) operator from (1), which will play an essential role in the ensuing analysis.
The operator possesses a potential, i.e. it exists a Gâteaux differentiable functional such that .
We notice the following relation between the norm and the potential .
Suppose that the operator satisfies (F1)–(F3), and denote by the unique solution of (1). Then, we have the estimate
In particular, takes its minimum at .
For fixed , define the real-valued function , for . Taking the derivative leads to
By invoking the fundamental theorem of calculus and implementing (1), this yields
Applying the assumptions (F1) and (F2) we can bound the integrand from above and below, respectively. Indeed, the strong monotonicity (F2) implies that
Likewise, by invoking (F1) instead of (F2), we find that
Combining the above bounds leads to (16). ∎
There is a constant such that the sequence defined by (5) fulfils the bound
where is the potential of introduced in (F3).
Before turning to the proof of the above proposition, we establish an auxiliary result.
Consider a sequence which satisfies the estimate
for some constant . Then, it holds the bound , for any .
Let us define the sequence , . Using (21), we note that
for all . By induction, this implies that for any . Therefore, we infer that
Rearranging terms completes the proof. ∎
Proof of Proposition 2.4.
Let be arbitrary. Then, we note the telescope sum
Thus, by virtue of (17), we infer that
We aim to bound the left-hand side. To this end, we employ Lemma 2.3, which implies that
This, together with Lemma 2.1, leads to
From (20) we immediately obtain the following result.
Under the assumptions of Proposition 2.4, it follows that is a null sequence as .
We are now ready to state and prove the main result of this section.
In the proof of Theorem 2.7 the application of Lemma 2.1 can be replaced by using [12, Proposition 2.1] instead. We note that the latter result does not require property (F1) to hold. Indeed, assume that (F2), (8) and (9) are satisfied, and that and are continuous mappings from into its dual space with respect to the weak topology on . Then, if the sequence defined by (5) satisfies as , it converges to the unique solution of (1).
2.4. Some remarks on condition (F4)
Suppose that the assumptions (F1)–(F3) are satisfied, and consider the sequence generated by the iteration (5). Analogously as in the proof of Lemma 2.3, for fixed , we define the real-valued function , for . Then, it holds the identity
Consequently, if the bilinear form , for any given , is uniformly coercive with constant , cf. (8), where refers to the Lipschitz constant occurring in (F1), then we obtain that
i.e. (17) is satisfied with .
which, upon using Proposition 2.9, shows that (F4) is satisfied for any . Under suitable assumptions, a similar observation can be made for the Newton method (12) provided that the damping parameter is chosen sufficiently small; cf. [12, Theorem 2.6].
The above Proposition 2.9 delivers a sufficient condition for (F4). We note, however, that it is not necessary. In particular, if the coercivity constant in (8) is much smaller than the Lipschitz constant from (F1), then the bound on in Proposition 2.9 is violated. Nonetheless, in that case, we can still satisfy (17) by imposing alternative assumptions; cf., e.g., (K2) in .
3. Adaptive ILG Discretizations
In this section, following the recent approach , we will present an adaptive ILG algorithm that exploits an interplay of the unified iterative linearization procedure (5) and abstract adaptive Galerkin discretizations thereof, cf. (14). Moreover, we will establish the (linear) convergence of the resulting sequence of approximations to the unique solution of (1), and comment on the uniform boundedness of the iterative linearization steps on each discrete space. We proceed along the ideas of [10, §4 and §5], and generalize those results to the abstract framework considered in the current paper. Throughout this section, we will assume that any iterative linearization is of the form (7), with (8) and (9) being satisfied.
3.1. Abstract error estimators
We generalize the assumptions on the finite element refinement indicator from [10, §4]. Let us consider a sequence of hierarchical finite dimensional Galerkin subspaces , i.e.
Suppose that, for any , there is a computable error estimator
which satisfies the following two properties:
For all it holds that
Here, are two constants.
The following result shows that the two estimators for and are equivalent once the linearization error is small enough.
Suppose that satisfies (F1)–(F2), and that the a posteriori estimator fulfils (A1). Furthermore, for some , assume that
with a constant , where
Then, we have that
Moreover, the two error estimators and are equivalent in the sense that
Invoking the Lipschitz continuity (A1), we obtain
Since we have that . Hence, manipulating the above inequality yields
Similarly, employing (33), it follows that
This completes the argument. ∎
3.2. Adaptive ILG algorithm
We focus on the adaptive algorithm from , which was studied in the context of finite element discretizations of the Zarantonello iteration (10). It is closely related to the general adaptive ILG scheme in . The key idea is the same in both algorithms: On a given Galerkin space, we iterate the linearization scheme (14) as long as the linearization error dominates. Once the ratio of the linearization error and the a posteriori error bound is sufficiently small, we enrich the Galerkin space in a suitable way.
|Let , and enrich the Galerkin space appropriately based on the error estimator in order to obtain .|
We emphasize that we do not know (a priori) if the while loop of Algorithm 1 always terminates after finitely many steps. Moreover, it may happen that , for some , i.e. the algorithm terminates. Let us provide two comments on this issue:
Suppose that there is an enrichment of generated by the above Algorithm 1 such that for all ; in this situation, the while loop will never end. Given the assumptions of Proposition 2.4, it follows from Corollary 2.6 that as . In addition, by virtue of Theorem 2.7 (applied to the discrete setting (13) and (14)), we have that as . Then, invoking the reliability (A2) and the continuity (A1), we conclude that
3.3. Perturbed contractivity
We will now turn to the proof of the convergence of Algorithm 1. More precisely, we will show that the sequence generated by the above ILG procedure converges, under certain assumptions, to the exact solution of (1). In view of Remark 3.2 we may assume that the while loop always terminates after finitely many steps with for all .
We begin with the following result, which corresponds to [10, Proposition 4.10]. Since we consider general Galerkin discretizations, an additional assumption, cf. the perturbed contraction (34) below, is imposed.
Let (F1)–(F2) and (A1) be satisfied, and be given. Moreover, for each , assume that the while loop of Algorithm 1 terminates after finitely many steps, thereby yielding an output , with . Furthermore, suppose that there are constants and such that it holds the perturbed contraction bound
where is the unique solution of (13). Then, we have that as .
Set , and denote by the solution of the weak formulation
For any , Galerkin orthogonality reads for all . Thus, by the Lipschitz continuity (F1) and strong monotonicity (F2) we find that
for any . This results in the Céa type estimate
Recalling the nestedness (24) of the Galerkin spaces, and exploiting the definition of , the above bound (35) directly implies that for . Consequently, we deduce that for . Hence, by (34), the estimator , , is contractive up to a non-negative perturbation which tends to 0. This implies that as , see, e.g., [3, Lemma 2.3]. Since satisfies by construction of Algorithm 1, Lemma 3.1 yields the equivalence of and . Hence, we conclude that as . ∎
3.4. Linear convergence
In this section we show the linear convergence of the output sequence generated by Algorithm 1. Our analysis follows closely the work [10, Theorem 5.3]. Again, we formulate and prove the result within a more general setting, and, for this purpose, under the additional assumption (34) as before.
with from (F2), we introduce the quantity
for any .
Let satisfy (F1)–(F3), and assume (A1)–(A2). Furthermore, suppose that there are constants and such that (34) holds true. Then, upon setting
with from (38), and with , the following contraction property holds: If the while loop of Algorithm 1 terminates after finitely many steps with , for all , then we have the (linear) contraction property
Moreover, there exists a constant such that
i.e. the error estimators decay at a linear rate.
Given any integers , and corresponding Galerkin subspaces . Then, using Lemma 2.3, with being replaced by , we have that