An Inexact Newton Method with Inner Preconditioned CG for Non-Uniformly Monotone Elliptic Problems

. The present paper introduces an inexact Newton method, coupled with a preconditioned conjugate gradient method in inner iterations, for elliptic operators with non-uniformly monotone upper and lower bounds. Convergence is proved in Banach space level. The results cover real-life classes of elliptic problems. Numerical experiments reinforce the convergence results.

The use of similar preconditioners for elliptic problems can be found in [13], where the authors introduce its applicability for quasi-Newton methods. This has also been extended recently for elliptic operators with non-uniformly monotone lower and upper bounds, see [8,9].
This paper provides an inexact Newton method, coupled with preconditioned conjugate gradient method in inner iterations, for non-uniformly elliptic problems based on the setting of [8,9,13]. The preconditioners are based on spectrally equivalent operators. Additionally, the results of a numerical experiment for a subsonic flow model (see [5]) are provided as an example.
Section 2 contains the convergence result, Section 3 presents models that fall under our assumptions, while Section 4 shows results of the numerical experiment.

Abstract inner-outer iteration in Banach spaces
The theorems below show convergence results for inner-outer iteration in Banach space.

Convergence of the inexact Newton's method
We make the following assumptions. Assumption 1. (i) Let X be a real Banach space with norm · , and X its dual, with usual notation v, u := vu (where v ∈ X , u ∈ X). The norm in X is also denoted by · .
(ii) We study operator equation where F : X → X is a nonlinear operator with bihemicontinuous Gâteaux derivative. The latter is denoted by F (u) at given u ∈ X. The unique solution of equation (2.1) is denoted by u * .
(iii) For any u ∈ X the operator F (u) is symmetric.
(iv) There exists a continuous nonincreasing function λ : (v) There exists a continuous nondecreasing function L : R + → R + such that Remark 1. If the function λ in (2.2) can be chosen constant, then the operator is uniformly monotone. However, we allow inf t∈R + λ(t) = 0, which means nonuniform monotonicity. Algorithm 1. For arbitrary u 0 ∈ X let (u n ) ⊂ X be the sequence defined by u n+1 =u n + p n , n ∈ N, where the energy norm · n is defined below in (2.6) and Theorem 1. Let Assumption 1 (i)-(v) be satisfied. Then the sequence defined by Algorithm 1 converges locally to u * with order (1 + γ), namely, there exists a neighbourhood U of u * that for a given u 0 ∈ U there exists constants C > 0 and 0 < Q < 1 such that Some lemmas and definitions in [9] that are needed for the proof are repeated below, mainly for the sake of convenience. The proofs shown in the reference for the lemmas apply here, if not shown otherwise below.
We define the following energy norms in X : v u := v, F (u) −1 v 1/2 (for given u ∈ X), · * := · u * , · n := · un (for given n ∈ N), (2.6) and strictly increasing function Λ : R + → R + , t → L(t)t + F (0) . For fixed u ∈ X, the norms · u and · are equivalent, namely: Specifically:λ There exists a strictly increasing function R * : R + → R + , such that The investigation of norms of elements of X in certain segments leads to the following observation.

Lemma 6.
There exists a strictly increasing function Φ * : Proof. The following result can be readily obtained from [9]. There exists a strictly increasing function G * : Let us define p * n := −F (u n ) −1 F (u n ), which is the Newton step, and write expansion u n+1 = (u n + p * n ) + (p n − p * n ), (2.13) where, due to (2.7), (2.10) and (2.12), the following estimation holds for the first term On the other hand, one can write using (2.4), (2.5), (2.8), Lemma 3 and (2.10) results in the following estimation of the second term of (2.13) the result follows.
Lemma 7. The following estimate holds for all u, v ∈ X: in particular: (2.14) Proof of Theorem 1. For given n ∈ N, one can write expansion

Inner-outer iteration
In what follows, the applied inner iteration is specified, i. e., for given n ∈ N, the method of obtaining approximate solution p n ∈ X to auxiliary equation Let us introduce the energy inner product on X as x, y B = Bx, y .
Algorithm 2. For fixed n ∈ N, since F (u n ) is a uniformly positive bounded linear symmetric operator, we can apply the preconditioned conjugate gradient method to obtain the p n , namely, let B n be a uniformly positive bounded linear symmetric operator, for which holds (M n ≥ m n > 0). The resulting sequence is p n + F (u n ), respectively. The step is defined as follows, where s n denotes the conjugate directions: Bn , ∀h ∈ X, therefore, the conjugate gradient method can be applied [3] in the energy space corresponding to operator B n .
We choose outer iteration step p n := p (k) n , for some k ≥ k n,min , where k n,min = ln(δ n /2) ln(Q n ) (2.20) is the minimum number of iterations, and Proof.
For the conjugate gradient method in the energy space corresponding to operator B n , the following is known on the other hand Combining these and using r
In accordance with [9], the definition of operators B n can be given with scalar coefficients Let us define a by replacing min with max in (3.7). If the function β satisfies a(x, r 2 ) ≤ β(x, r 2 ) ≤ a(x, r 2 ), e.g. β(x, r 2 ) := 1 2 a(x, r 2 ) + a(x, r 2 ) , then, for all n, spectral equivalence of B n and F (u n ) can be obtained readily, in other words, (2.19) holds. The Newton equation must be discretized which adds to the inexactness. This can be based on a combination of a coarse and a fine mesh which can save computer time (see [2,4,21], etc.). Future work in the topic might include corresponding investigations.

Subsonic flow example
The following boundary value problem describing potential flow in a wind tunnel section Ω ⊂ R 2 has been presented in [5,Chap. 9]. For the geometry, see [5,Fig. 5]. Let us consider: is the air density at infinity, and u is the velocity potential. M ∞ > 0 denotes the Mach number at infinity, v ∞ stands for the constant velocity potential on Dirichlet boundary Γ D , while Γ N := ∂Ω \ Γ D is the Neumann boundary. The range of γ is {0,ṽ ∞ }, whereṽ ∞ > 0 is a parameter describing outflow velocity. We only deal with the case v ∞ = 0 without the loss of generality, since u is a potential (one may observe that (4.1) only contains derivatives of u except for the constant Dirichlet boundary condition).
By involving problem (4.1), our goal is to test that our method may work even beyond the limitations posed by our previous theoretical assumptions. Namely, the condition for ellipticity is that |∇u| is pointwise below the subsonic limit, hence the operator cannot be defined on a whole function space. Therefore, the above subsonic flow problem described by (4.1) is not precisely contained in the elliptic model (3.6) with assumptions (3.7)-(3.8). However, one can expect that the method of present paper converges properly while the solution and the utilized part of the iterative sequence satisfy the subsonic limit condition.
We apply the results of Section 2, and use the finite element method (FEM) for the discretization of the problem, namely, Courant elements. Hence the above Banach space X is a finite dimensional space consisting of Courant elements for which u |Γ D = 0. Therefore, all norms are equivalent.
Thus one can define the operator describing (4.1) in weak form as Consequently, the FEM problem becomes the task of finding u ∈ V h , such that F (u), v = 0, ∀v ∈ V h . This can be written shortly in the form of (2.1) as The Gâteaux derivative of the operator can be obtained readily in weak form The applied preconditioner in the n-th outer step is It provides a substantial simplification of the Gâteaux derivative of the operator.

Numerical results
The results of the above experiment with five different meshes and two different v ∞ values are presented below.
Let symbol DoF stand for the degrees of freedom of the FEM model. Denote n 1 , n 2 the number of outer iteration steps necessary to achieve smaller relative residual error than 10 −4 and 10 −6 , respectively. Let k denote the number of inner iteration steps required for relative residual error to be smaller than 10 −4 for an individual outer iteration step. The numerical results are summarized in Table 1, where the outer step number, which the given value of k corresponds to, can be identified in the header.
As a conclusion, we state the following observations. Firstly, we readily find robustness of the method developed here for the subsonic flow example. A similar robustness result for a quasi-Newton method with the same preconditioner can be found in [9].
Secondly, (2.20) states for the subsonic model that at most 8 inner iterations are sufficient to reach relative tolerance 10 −4 . Comparing this to Table 1 shows that in fact far less iterations can be sufficient as well.