Adaptive inexact Newton methods for discretizations of nonlinear diffusion PDEs. II. Applications∗

Adaptive inexact Newton methods for discretizations of nonlinear diffusion PDEs. II. Applications ^∗

Alexandre Ern

Martin Vohral´ık

March 21, 2012

Abstract

We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations of diffusion type. In order to solve them, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. In Part I of this work, we have developed a general abstract framework hinging on equilibrated flux reconstructions to derive stopping criteria for both iterative solvers and to control the size and distribution of the overall approximation error. In this Part II, we apply this framework to various discretization schemes like finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, and lowest-order mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers.

This leads to new guaranteed and robust a posteriori error estimates for nonlinear diffusion problems in the presence of linearization and algebraic errors. Moreover, for many discretization schemes, we improve on, or derive new, flux equilibration techniques.

Key words: nonlinear diffusion PDE, nonlinear algebraic system, adaptive linearization, adaptive algebraic solution, a posteriori error estimate, finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, mixed finite elements

1 Introduction

In Part I of this work, we considered the following nonlinear diffusion problem: find a scalar-valued function u, termed the potential, such that

−∇·σ(x, u(x),∇u(x)) =f in Ω, (1.1a)

u= 0 on∂Ω, (1.1b)

where Ω ⊂R^d, d≥2, is a polygonal (polyhedral) domain, σ : Ω×R×R^d →R^d, and f : Ω→ Ris the source term. Given a potential u, the vector-valued function−σ(·, u,∇u) : Ω→R^d is termed theflux. To simplify, we omit the dependence on the space variablexand simply writeσ(u,∇u). Assuming that there is a real number p∈(1,∞) such thatf ∈L^q(Ω) withq:= _p−1^p , the model problem (1.1) is formulated as follows: findu∈V :=W₀^1,p(Ω) such that

(σ(u,∇u),∇v) = (f, v) ∀v∈V, (1.2)

where, for w∈L^q(Ω), v∈L^p(Ω), (w, v) stands for R

Ωw(x)v(x) dx. In what follows, we assume that there exists a unique weak solution of (1.2).

∗This work was partly supported by the Groupement MoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN) and by the ERT project “Enhanced oil recovery and geological sequestration of CO2: mesh adaptivity, a posteriori error control, and other advanced techniques” (LJLL/IFPEN).

∗Universit´e Paris-Est, CERMICS, Ecole des Ponts ParisTech, 77455 Marne la Vall´ee cedex 2, France (ern@cermics.enpc.fr).

†UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France & CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France (vohralik@ann.jussieu.fr).

Problem (1.2) covers two important examples. Firstly, thequasi-linear diffusionproblem whereσ(v,ξ) = A(v)ξ with the assumption that the (tensor-valued) function A is bounded and takes symmetric values with minimal eigenvalue uniformly bounded away from zero. In this case,σ depends linearly onξ, and the functional setting corresponds to the choice p= 2. Secondly, the Leray–Lions problem where σ depends nonlinearly on ξ, but is independent of v, in the form σ(ξ) = A(ξ)ξ with suitable assumptions on the (tensor-valued) functionA, see Part I. A typical example is thep-Laplacian where A(ξ) =|ξ|^p−2IandIis the identity tensor.

A posteriori estimates of discretization errors for problems of type (1.2) have been derived in various spe- cific situations. Verf¨urth [36] developed a general framework for reliable and efficient a posteriori estimates in the finite element setting. Pousin and Rappaz [33] considered such ideas for general Petrov–Galerkin approximations. Repin [35] derived guaranteed a posteriori error estimates for arbitrary conforming dis- cretizations; approximate solution of a global problem is, however, often necessary. Quite tight guaranteed upper bounds have been obtained by Carstensen and Klose [10] for the p-Laplacian. Other discretization schemes were also studied. Creus´eet al. [13] derived a posteriori error estimates for mixed finite elements in the p-Laplacian case, whereas Houston et al.[26] considered the discontinuous Galerkin method in the quasi-linear diffusion setting. Kim [28] derived guaranteed estimates in the quasi-linear diffusion setting for locally conservative methods.

Recently, there has been an interest in setting up unified frameworks for a posteriori analysis. We mention, in particular, the work of Carstensen [9], Kim [28], and Ainsworth [1]. The last two references fall into the flux equilibration approach, which can be traced back to Prager and Synge [34], cf. also Luce and Wohlmuth [30], Braess and Sch¨oberl [5], and the references therein. In [21], a unified analysis framework is developed whose application to a given discretization consists in the definition of flux reconstructions and the verification of a couple of assumptions on these fluxes.

The discretization of problem (1.2) leads to a system of nonlinear algebraic equations. In practice, this system is solved iteratively using a nonlinear solver which, at each step, employs an iterative linear solver. It is therefore important to distinguish and estimate separately the threeerror components, namely discretization, linearization, and algebraic errors. We have achieved this goal in Part I of this work, where we have proposed a posteriori error estimates for these components and balanced them throughstopping criteria for the iterative nonlinear and linear solvers, leading to anadaptive inexact Newton method. Moreover, the obtained error estimates areguaranteedandrobustwith respect to the size of the nonlinearity owing to the chosen error measure, see (2.1) below. These developments are presented in a unified abstract framework.

They extend the ideas of Chaillou and Suri [11,12] and [18] concerning linearization errors, and those of [27]

concerning algebraic errors. To our knowledge, this is the first time that the three error components are analyzed simultaneously. Alternative approaches include that of Han [24] for linearization errors and that of Beckeret al.[4] and Arioliet al.[3] for algebraic errors, see also the references therein.

The aim of this Part II is to apply the framework of Part I to a wide class of discretization schemes, namely nonconforming finite elements, conforming finite elements, discontinuous Galerkin, finite volumes, and lowest-order mixed finite elements. In Section2, we first synthesize the unified framework of Part I. The presentation somewhat differs from Part I so as to emphasize the key algorithmic aspects of the adaptive inexact Newton method and to identify the ingredients needed for its implementation together with the two key assumptions to be verified by the flux reconstructions. This verification is undertaken in Section 3–

Section 7 for the various discretization schemes. For each scheme, we exemplify two iterative nonlinear solvers, namely fixed point and Newton, while the iterative linear solver can be arbitrary. It turns out that in many cases, we improve on, or develop new, flux equilibration techniques that are of independent interest.

A brief discussion is included in the section dedicated to each discretization scheme. Finally, we draw some conclusions in Section8.

2 The adaptive inexact Newton method

This section presents the basic setting and synthesizes the main results derived in Part I.

2.1 Basic notation

LetThbe a simplicial mesh of Ω. For simplicity, we suppose thatThdoes not contain hanging nodes, so that, for two distinct elements of Th, their intersection is either an empty set or a commonl-dimensional face, 0≤l≤d−1. The (d−1)-dimensional faces of the mesh are collected in the setEhsuch thatEh=E_h^int∪E_h^ext, withE_h^int collecting interfaces andE_h^ext boundary faces. The faces of a generic elementK∈ Th are collected in the setEK. For anyK∈ Th(resp., anye∈ Eh),hK (resp.,he) denotes its diameter. For anyK∈ Th,T_K collects the elementsK^′ ∈ Th which share at least a vertex with K. Similarly,E_K collects the faces which share at least a vertex withK, and we setE^int_K :=E_K∩ E_h^int. For anye∈ Eh,nestands for the unit normal vector to e(the orientation is irrelevant, but fixed, for all e∈ E_h^int and points outward Ω for alle∈ E_h^ext) and, for any K∈ Th,nK stands for the outward unit normal vector toK.

We work with approximations that are possibly nonconforming, that is, not included in the space V. For this reason, we introduce the broken Sobolev spaceV(Th) :={v∈L^p(Ω), v|K ∈W^1,p(K) ∀K∈ Th}.

For any function v ∈V(Th),∇v denotes its broken gradient, that is, the distributional gradient evaluated elementwise. Moreover, [[v]] denotes the difference (evaluated along n_e) of the traces of v from the two adjacent mesh elements ife∈ E_h^int and the actual trace ofv ife∈ E_h^ext.

2.2 Approximate solution, approximate gradient, and error measure

Let a numerical scheme be used to discretize (1.2), and consider the k-th step,k≥1, of a nonlinear solver and the i-th step,i≥1, of a linear solver. We denoteu^k,i_h the corresponding discrete potential; we merely suppose thatu^k,i_h ∈V(Th). Separately fromu^k,i_h , we also consider a discrete gradientg^k,i_h ∈[L^p(Ω)]^d. This allows us to handle a wide class of discretization schemes in a unified setting. For conforming schemes,g_h^k,i is obtained by applying the usual gradient tou^k,i_h ; for various nonconforming schemes, the broken gradient can be used instead, but some schemes employ a more elaborate construction of g^k,i_h , taking into account, e.g., the jumps ofu^k,i_h . In all cases, wheneveru^k,i_h ∈V, there holdsg^k,i_h =∇u^k,i_h .

We measure the error as

Ju(u^k,i_h ,g^k,i_h ) =Ju,F(u^k,i_h ,g^k,i_h ) +Ju,NC(u^k,i_h ), (2.1) where

Ju,F(u^k,i_h ,g^k,i_h ) := sup

ϕ∈V;k∇ϕkp=1

σ(u,∇u)−σ(u^k,i_h ,g^k,i_h ),∇ϕ

, (2.2a)

Ju,NC(u^k,i_h ) :=

( X

K∈Th

X

e∈EK

α^s_eh^1−s_e k[[u−u^k,i_h ]]k^s_s,e )¹_q

. (2.2b)

The quantityJu,F(u^k,i_h ,g_h^k,i) measures the error in the fluxes (using a dual norm), see Chaillou and Suri [11, 12] and [18] for conforming discretizations. The quantity Ju,NC(u^k,i_h ) measures the nonconformity of the discrete potential (recall that a functionv∈V(Th) is inV if and only if [[v]] = 0 for alle∈ Eh, see, e.g., [17, Lemma 1.23]); a specific value for the weightsαeand the exponents≥1 is only needed in Section5.4. Owing to the well-posedness of (1.2), the characterization of conformity through jumps, and the above consistency of the discrete gradient inV, there holds Ju(u^k,i_h ,g_h^k,i) = 0 if and only ifu=u^k,i_h and∇u=g^k,i_h .

2.3 The algorithm

In Part I, we have derived anadaptive inexact Newton methodto solve problems of the form

A(U) =F, (2.3)

where A : R^N → R^N is a discrete nonlinear operator and F ∈ R^N a given vector, stemming from the discretization of problem (1.2) by a given numerical scheme. This algorithm is driven by stopping criteria based on a posteriori estimators distinguishing the three main error components, namely discretization, linearization, and algebraic errors. On a nonlinear solver step k, k ≥ 1, and linear solver step i, i ≥ 1, these estimators are respectively denoted byη_disc^k,i ,η_lin^k,i, andη^k,i_alg. A notion of algebraic remainder estimator

η^k,i_rem also appears. The evaluation of the estimators is discussed in Section2.4. Letγrem,γalg, andγlin be positive user-given weights, typically of order 0.1. The algorithm reads:

Algorithm 2.1 (Adaptive inexact Newton method). 1. Choose an initial vectorU⁰∈R^N. Setk:= 1.

2. From U^k−1, define a matrix A^k ∈ R^N,N and a vector F^k ∈ R^N. Consider the following system of linear algebraic equations:

A^kU^k=F^k. (2.4)

3. (a) DefineU^k,0:=U^k−1 and seti:= 1.

(b) Perform a step of a chosen iterative linear solver for the solution of the linear system (2.4), starting from the vectorU^k,i−1. This yields an approximation U^k,i toU^k which satisfies

A^kU^k,i=F^k−R^k,i, (2.5)

whereR^k,i ∈R^N is the algebraic residual vector on stepi.

(c) Perform ν >0 additional steps of the iterative linear solver yielding an approximationU^k,i+ν to U^k which satisfies

A^kU^k,i+ν =F^k−R^k,i+ν, (2.6)

whereR^k,i+ν ∈R^N is the algebraic residual vector on stepi+ν. The parameterνis progressively increased until

η_rem^k,i ≤γremmax

η^k,i_disc, η^k,i_lin, η_alg^k,i . (2.7) (d) Check the convergence criterion for the linear solver in the form

η^k,i_alg≤γalgmax

η_disc^k,i , η_lin^k,i . (2.8) If satisfied, setU^k:=U^k,i. If not, set i:=i+ν and go back to step 3b.

4. Check the convergence criterion for the nonlinear solver in the form

η^k,i_lin ≤γlinη_disc^k,i . (2.9)

If satisfied, finish. If not, setk:=k+ 1 and go back to step 2.

In addition to the estimatorsη^k,i_disc,η^k,i_lin,η_alg^k,i, andη_rem^k,i of Algorithm2.1, we also introduced in Part I the data oscillation estimator η_osc^k,i and the quadrature estimator η_quad^k,i . The evaluation of all the estimators is summarized in Section 2.4. In Part I, cf. Theorem 3.6, we derived the following guaranteed upper bound distinguishing the different error components:

Ju(u^k,i_h ,g^k,i_h )≤η_disc^k,i +η^k,i_lin +η_alg^k,i+η^k,i_rem+η_quad^k,i +η_osc^k,i, (2.10) see also Theorem 3.4 in Part I for a sharper bound without the distinction of error components. Moreover, under the criteria (2.7), (2.8), and (2.9) in Algorithm 2.1, global efficiency and robustness was derived in the form, see Theorem 5.4 of Part I,

η_disc^k,i +η^k,i_lin +η_alg^k,i+η_rem^k,i .Ju(u^k,i_h ,g_h^k,i) +η_quad^k,i +η_osc^k,i, (2.11) where A.B stands for the inequalityA ≤CB with a generic constantC independent of the mesh sizes hK andhe, the domain Ω, the nonlinear flux function σ, and the Lebesgue exponentp, but depending on the shape regularity of the mesh family{Th}h and on the polynomial degrees of the discretization and the various reconstructions.

Remark 2.2 (Local stopping criteria and local efficiency). Local, elementwise, stopping criteria are to be considered in (2.7), (2.8), and (2.9) in conjunction with adaptive mesh refinement. They yield local efficiency in a slightly different error measure, see Theorem 5.3in Part I.

2.4 Evaluation of the estimators

The evaluation of our estimators hinges on a few ingredients. First, we use three (vector-valued, piecewise polynomial) flux reconstructions d^k,i_h , l^k,i_h , and a^k,i_h , whered^k,i_h is meant to approximate the discrete flux

−σ(u^k,i_h ,g^k,i_h ), l^k,i_h represents the linearization error, and a^k,i_h the algebraic error. Moreover, we use a (piecewise polynomial) functionfh to approximate the datumf, a (piecewise polynomial) functionρ^k,i_h to represent the algebraic remainder, and a (vector-valued, piecewise polynomial) functionσ^k,i_h to approximate σ(u^k,i_h ,g^k,i_h ). With these ingredients, the estimators can be evaluated as follows: for all K∈ Th,

η_disc,K^k,i := 2^1/p kσ^k,i_h +d^k,i_h kq,K+η_NC,K^k,i

, (2.12a)

η_NC,K^k,i :=

( X

e∈EK

α^s_eh^1−s_e k[[u^k,i_h ]]k^s_s,e )¹_q

, (2.12b)

η_lin,K^k,i :=kl^k,i_h kq,K, η_alg,K^k,i :=ka^k,i_h kq,K, η_rem,K^k,i :=hΩkρ^k,i_h kq,K, (2.12c) η^k,i_osc,K:=CP,phKkf−fhkq,K, η_quad,K^k,i :=kσ(u^k,i_h ,g^k,i_h )−σ^k,i_h kq,K, (2.12d) where hΩ is the diameter of Ω andCP,p=π^−p2d^12−1p forp≥2 andCP,p=p^1p2^(p−1)p otherwise. The global versions of the estimators areη^k,i· :=n

P

K∈Th η^k,i_·,Kqo1/q

.

2.5 Construction of the ingredients

The construction of our ingredients (that is,d^k,i_h ,l^k,i_h ,a^k,i_h ,fh,ρ^k,i_h , andσ^k,i_h ) proceeds in three steps. In the first step, we construct the sum (d^k,i_h +l^k,i_h ), the functionfh, and the preliminary algebraic remainderr^k,i_h . The remainder r^k,i_h is a piecewise polynomial constructed from the residual vector R^k,i_h in (2.5), whereas the functionfh has to verify (fh,1)K = (f,1)K for allK∈ Th. We require the following quasi-equilibration property:

Assumption 2.3 (Quasi-equilibration for (d^k,i_h +l^k,i_h )). The function (d^k,i_h +l^k,i_h ) is in H^q(div,Ω) and satisfies

∇·(d^k,i_h +l^k,i_h ) =fh−r^k,i_h . (2.13) Our second step consists in considering the ν additional steps of the iterative linear solver (2.6). We then construct the fluxes (d^k,i+ν_h +l^k,i+ν_h ) and remainderr^k,i+ν_h as in Assumption2.3. We finally define the flux a^k,i_h and the algebraic remainderρ^k,i_h as

a^k,i_h := (d^k,i+ν_h +l^k,i+ν_h )−(d^k,i_h +l^k,i_h ), (2.14a)

ρ^k,i_h :=r^k,i+ν_h . (2.14b)

Finally, our third step consists in specifying the reconstruction d^k,i_h , and hence by subtraction the reconstruction l^k,i_h , and the approximationσ^k,i_h . We let

η_♯,^k,iT_K:=

( X

K^′∈T_K

h^q_K^′kfh+∇·σ^k,i_h k^q_q,K^′+ X

e∈E^int

K

hek[[σ^k,i_h ·ne]]k^q_q,e )¹_q

. (2.15)

Set the shorthand notationη^k,i_·,T_K :=n P

K^′∈T_K η^k,i_·,K^′qo¹_q

. The goal is to achieve:

Assumption 2.4 (Local approximation for d^k,i_h and σ^k,i_h and convergence for l^k,i_h ). For all K ∈ Th, d^k,i_h andσ^k,i_h satisfy the following local approximation property:

kσ^k,i_h +d^k,i_h kq,K .η^k,i_♯,T_K+η^k,i_NC,T_K+η_osc,^k,iT_K. (2.16) Moreover, kl^k,i_h kq,K →0 as the nonlinear solver converges.

Remark 2.5(Tighter approximation property). In most cases, we actually prove kσ^k,i_h +d^k,i_h kq,K .η_♯,^k,iT_K. η^k,i_osc,TK appears for conforming finite elements in the lowest-order settingm= 1 andl= 0, see Section4.4, while η^k,i_NC,T_K appears for interior penalty discontinuous Galerkin when using full prescription for the flux reconstruction, see Section 5.4.

In Section3–Section7, we show how to apply Algorithm2.1 and the a posteriori error estimate (2.10) to various discretizations methods, to the fixed point and Newton linearizations, and to arbitrary iterative linear solvers. This consists in identifying the functional forms of (2.3), (2.4), and (2.5), the approximate solutionu^k,i_h and the approximate gradientg^k,i_h , in constructing all the above ingredients, and in verifying Assumptions2.3and2.4.

2.6 Flux reconstructions

We reconstruct the fluxes d^k,i_h , l^k,i_h , and a^k,i_h in the Raviart–Thomas–N´ed´elec spaces. We consider three approaches: (i) full prescription of the degrees of freedom, similarly to Destuynder and M´etivet [14, 15];

(ii) solving local mixed finite element problems with prescribed boundary conditions, similarly to Luce and Wohlmuth [30] and to [20]; and (iii) solving local mixed finite element problems without prescribed boundary conditions, similarly to Braess and Sch¨oberl [5]. The first approach is explicit, whereas the two other approaches are implicit, necessitating the solution of local linear systems, typically on a patch of elements.

For an integerl ≥0, P^l(Th) denotes the broken polynomial space spanned by those functions vh such that, for all K∈ Th,vh|K ∈Pl(K). For φ∈L¹(Ω), Πlφ∈Pl(Th) is defined such that, for allvh ∈Pl(Th), (φ−Πlφ, vh) = 0; Π_l denotes the operator acting componentwise as Πl on vector-valued functions. For K ∈ Th and l ≥ 0, let RTN_l(K) := [Pl(K)]^d+xPl(K) be the Raviart–Thomas–N´ed´elec finite element space of order l. Functions v_h ∈ RTN_l(K) are such that, cf. Brezzi and Fortin [6], ∇·v_h ∈ Pl(K) and v_h·n_e ∈Pl(e) for alle ∈ EK. We then set RTN⁻¹_l (Th) :={v_h ∈[L^q(Ω)]^d;v_h|K ∈RTN_l(K) ∀K ∈ Th} and RTN_l(Th) := RTN⁻¹_l (Th)∩H^q(div,Ω). Functions in RTN_l(Th) have, in particular, a continuous normal component across interfaces. We use a similar notation for these spaces on various patches of elements. Finally, let I^RTN_l stand for the broken Raviart–Thomas–N´ed´elec interpolation operator; for a smooth enough function v, I^RTN_l v ∈ RTN⁻¹_l (Th) is such that, for all K ∈ Th, letting hw, vie stand for R

ew(s)v(s) ds,

h(I^RTN_l v−v)|K·ne, qhie= 0 ∀e∈ EK, ∀qh∈Pl(e), (2.17a) (I^RTN_l v−v,r_h)K = 0 ∀rh∈[Pl−1(K)]^d. (2.17b)

3 Nonconforming finite elements

We treat here the discretization of problem (1.2) by nonconforming finite elements. We first present a simple elementwise flux reconstruction by full prescription. Then, we outline a slightly tighter reconstruction using a dual mesh, already considered in Section 6 of Part I, which is equivalent to solving local mixed finite element problems with prescription. Finally, we outline an equivalent viewpoint based on local mixed problems without prescription.

3.1 Discretization

Setfh:= Π0f. The nonconforming Crouzeix–Raviart finite element spaceVh is spanned by piecewise affine polynomials on Th such that the interface jumps and boundary face values have zero mean value over the corresponding face. The discretization of problem (1.2) reads: finduh∈Vh such that

(σ(uh,∇uh),∇vh) = (fh, vh) ∀vh∈Vh. (3.1) The basis functions inVhare associated with the interfaces and are denoted{ψe}_e∈E^int

h . Testing (3.1) against these functions yields the nonlinear algebraic system (2.3).

3.2 Linearization

Letu⁰_h∈Vh, fixing the initial vector U⁰ in Algorithm2.1. The linearization of (3.1), fork≥1, reads: find u^k_h∈Vhsuch that

(σ^k−1(u^k_h,∇u^k_h),∇ψe) = (fh, ψe) ∀e∈ E_h^int, (3.2) which is the functional form of the algebraic system (2.4). Two common ways to define the flux function σ^k−1 are the fixed point linearization where

σ^k−1(v,ξ) :=A(u^k−1_h ,∇u^k−1_h )ξ, (3.3) and the Newton linearization where

σ^k−1(v,ξ) :=A(u^k−1_h ,∇u^k−1_h )ξ+ (v−u^k−1_h )∂vA(u^k−1_h ,∇u^k−1_h )∇u^k−1_h

+ (∂ξA(u^k−1_h ,∇u^k−1_h )·∇u^k−1_h )·(ξ− ∇u^k−1_h ). (3.4)

3.3 Algebraic solution

On the i-th step, i ≥ 1, of an iterative linear solver for the algebraic system (2.4), we obtain the alge- braic residual vector R^k,i in (2.5) with components associated with interfaces, R^k,i = {R^k,i_e }_e∈E^int

h . For convenience, we set R^k,i_e := 0 for alle∈ E_h^ext. The functional form of (2.5) is: find u^k,i_h ∈Vh such that

(σ^k−1(u^k,i_h ,∇u^k,i_h ),∇ψe) = (fh, ψe)−R^k,i_e ∀e∈ E_h^int. (3.5)

3.4 Flux reconstruction by full prescription

LetK∈ Th. We definef_h(x)|K := ^fh_d^|K(x−x_K), withx_K the barycenter ofK. For alle∈ EK, leta_K,e be the vertex ofK opposite to the face e. LetTestand for the patch of elements sharing the facee. We first prescribe (d^k,i_h +l^k,i_h ) elementwise inRTN⁻¹₀ (Th) (as shown in Lemma3.4below, (d^k,i_h +l^k,i_h ) turns out to be inRTN₀(Th)).

Definition 3.1 (Construction of (d^k,i_h +l^k,i_h )). Set, for all K∈ Th, (d^k,i_h +l^k,i_h )|K := −Π0σ^k−1(u^k,i_h ,∇u^k,i_h ) +fh

|K− X

e∈EK

|Te|⁻¹R^k,i_e

d (x−aK,e). (3.6) The construction of d^k,i_h mimics that of (d^k,i_h +l^k,i_h ) with σ(u^k,i_h ,∇u^k,i_h ) in place of σ^k−1(u^k,i_h ,∇u^k,i_h ).

Specifically, let

R¯e^k,i:= (fh, ψe)−(σ(u^k,i_h ,∇u^k,i_h ),∇ψe) ∀e∈ Eh^int, (3.7) and ¯R^k,i_e := 0 for alle∈ E_h^ext. We prescribed^k,i_h (and hence, alsol^k,i_h by subtraction):

Definition 3.2 (Construction ofd^k,i_h ). Set, for allK∈ Th, d^k,i_h |K := −Π0σ(u^k,i_h ,∇u^k,i_h ) +f_h

|K− X

e∈EK

|Te|⁻¹R¯^k,i_e

d (x−a_K,e). (3.8)

Definition 3.3 (Error measure, data oscillation, quadrature, and algebraic remainder). Use u^k,i_h and g^k,i_h := ∇u^k,i_h in the error measure (2.1) and set fh := Π0f, σ^k,i_h := Π₀σ(u^k,i_h ,∇u^k,i_h ), and r^k,i_h |K :=

P

e∈EK|Te|⁻¹R^k,i_e for all K∈ Th.

We now verify the assumptions of Section2.5:

Lemma 3.4 (Quasi-equilibration). Assumption2.3 holds.

Proof. The proof exploits the link between nonconforming finite elements and mixed finite elements, cf.

Marini [31] or Destuynder and M´etivet [14]. For all K ∈ Th and all e∈ EK, we introduce the geometric weightωe,K :=|K|/|Te|. Note that 0< ωe,K ≤1 andωe,K = 1 only on boundary faces. For any interface e ∈ E_h^int such that e = ∂K∩∂K^′, K, K^′ ∈ Th, observing that ωe,K +ωe,K^′ = 1, we define the weighted average of a piecewise polynomial functionv_hateas{{vh}}ω:=ωe,K^′(vh|K)|e+ωe,K(vh|K^′)|e. On boundary faces e∈ E_h^ext, we set{{vh}}ω:=v_h|e. We first show that, for all for all K∈ Thand alle∈ EK,

(d^k,i_h +l^k,i_h )|K·ne={{−Π0σ^k−1(u^k,i_h ,∇u^k,i_h ) +f_h}}ω·ne. (3.9) This is obvious fore∈ E_h^ext. Let nowe∈ E_h^int. Setwh:=−Π0σ^k−1(u^k,i_h ,∇u^k,i_h ) +fh. It is readily seen that (σ^k−1(u^k,i_h ,∇u^k,i_h ),∇ψe) =|e|[[Π0σ^k−1(u^k,i_h ,∇u^k,i_h )]]·neand (fh, ψe) =|e|[[fh]]·ne(recall that [[·]] denotes the jump across e in the direction of n_e). Hence, owing to (3.5), [[wh]]·ne =|e|⁻¹R_e^k,i. The result (3.9) then follows from

w_h|K·ne={{wh}}ω·ne+ωe,K[[wh]]·nK (3.10) and (3.6). Now, (3.9) shows that (d^k,i_h +l^k,i_h ) has continuous normal component across interfaces, so that (d^k,i_h +l^k,i_h ) ∈ RTN₀(Th). Finally, the property (2.13) follows by taking the divergence of (3.6) and considering the definition of r_h^k,i.

Lemma 3.5 (Local approximation and convergence). Assumption2.4holds.

Proof. The requirement on l^k,i_h is obvious from Definitions 3.1 and 3.2. Turning to d^k,i_h , we let vh :=

σ^k,i_h +d^k,i_h ∈RTN⁻¹₀ (Th) and use, for allK∈ Th, the estimatekvhkq,K .{P

e∈EKhekvh|K·nek^q_q,e}^1q shown in [18, Section A.4]. Lete∈ EK. Ife∈ E_h^ext, using ¯R^k,i_e := 0 in (3.8),|x−x_K| ≤ hK, a q-robust inverse inequality (see [18, Section A.1 and A.4]), the fact thatfh is constant onK, and∇·σ^k,i_h = 0 yields

hekvh|K·nek^q_q,e=hekfh|Kd⁻¹(x−x_K)·nek^q_q,e≤h^1+q_K kfh|Kk^q_q,e .h^q_Kkfhk^q_q,K =h^q_Kkfh+∇·σ^k,i_h k^q_q,K.

If e ∈ E_h^int, reasoning as in the proof of Lemma 3.4 yields d^k,i_h ·ne = {{−σ^k,i_h +f_h}}ω·ne (so that d^k,i_h ∈ RTN0(Th)). Using this relation, (3.10) to evaluatevh|K·ne, and the continuity of the normal component of d^k,i_h yieldsv_h|K·ne={{fh}}ω·ne+ωe,K[[σ^k,i_h ]]·nK. We conclude by proceeding as in the first part of the proof.

3.5 Flux reconstruction by local mixed problems with prescription

A slightly tighter flux reconstruction was considered in Section 6 of Part I. For all K∈ Th and alle∈ EK, let Ke be the sub-simplex of K formed by the face e and the barycenter x_K. Let De regroup the two (or one for boundary faces) sub-simplices which share e. Then, the tighter flux reconstruction consists in replacing in the last terms of (3.6) and (3.8) the vertexa_K,eby the barycenterx_Kand|Te|⁻¹by|De|⁻¹. The advantage is that, using local stopping criteria, elementwise efficiency (without neighbors) can be proven on each element of the dual mesh Dh ={De}e∈Eh. Moreover, for allDe∈ Dh with outward normal n_De, letting SDe collect the sub-simplices inDe,

(d^k,i_h +l^k,i_h )|De = arg inf

{v_h∈RTN^N

0(SDe),∇·v_h=f_h−r^k,i_h }

kΠ0σ^k−1(u^k,i_h ,∇u^k,i_h ) +v_hkDe,

whereRTN^N₀(SD_e) fixes to− Π0σ^k−1(u^k,i_h ,∇u^k,i_h )

·nD_e the normal component on those faces of∂Dethat are inside Ω, see [20], [21, Section 4.5], or [25, Section 7.3], andr^k,i_h |D_e =R^k,i_e |De|⁻¹. This construction stems from the face-centered finite volume method, where the equivalent of (3.5), yielding the same solution u^k,i_h , is,

− hΠ0σ^k−1(u^k,i_h ,∇u^k,i_h )·nDe,1i∂De = (fh,1)De−R_e^k,i ∀e∈ E_h^int. (3.11) A similar construction can be devised for d^k,i_h .

3.6 Flux reconstruction by local mixed problems without prescription

We finally present a construction inspired by the approach of Braess and Sch¨oberl [5] for conforming finite elements, see Section 4.4. For all e ∈ Eh, let RTN^N,0₀ (Te) denote the subspace of RTN0(Te) with zero normal flux through ∂Te fore∈ E_h^int and through that part of the boundary of Te which lies inside Ω for e∈ E_h^ext. LetP^∗0(Te) be spanned by piecewise constants onTe with zero mean onTe when e∈ E_h^int; when e∈ E_h^ext, the mean value condition is not imposed. Define (d^k,i_e +l^k,i_e )∈RTN^N,0₀ (Te) andqe∈P^∗0(Te) by the solution of the following mixed finite element problem onTe:

(d^k,i_e +l^k,i_e ,vh)Te−(qe,∇·vh)Te =−(ψeΠ0σ^k−1(u^k,i_h ,∇u^k,i_h ),vh)Te,

(∇·(d^k,i_e +l^k,i_e ), φh)Te= (fhψe−σ^k−1(u^k,i_h ,∇u^k,i_h )·∇ψe, φh)Te−(R^k,i_e , φh)Te|Te|⁻¹, for all (vh, φh)∈RTN^N,0₀ (Te)×P^∗0(Te). Then, setd^k,i_h +l^k,i_h :=P

e∈Eh(d^k,i_e +l^k,i_e ). In the above problems, we can takeφh∈P0(Te) since (3.5) yields, for alle∈ E_h^int, the Neumann compatibility condition

(fh, ψe)Te−(σ^k−1(u^k,i_h ,∇u^k,i_h ),∇ψe)Te−R^k,i_e = 0. (3.12) It can be shown that the constructions of (d^k,i_h +l^k,i_h ) from Definition3.1 and the present one coincide. A similar construction can be devised for d^k,i_h .

4 Conforming finite elements

We treat here the discretization of problem (1.2) by conforming finite elements. We focus on flux recon- struction through local mixed problems without prescription, following Braess and Sch¨oberl [5], cf. also Destuynder and M´etivet [15]. A slightly tighter construction without prescription is possible in the piece- wise affine case. Similarly to Section 3.5, this construction exploits the links between the piecewise affine finite element method and the vertex-centered finite volume method and allows for a fully local statement (without neighbors) of the local efficiency result on a vertex-centered dual mesh, cf. Luce and Wohlmuth [30]

and [18].

4.1 Discretization

Let Vh := Pm(Th)∩V, m ≥ 1, be the usual finite element space of continuous, piecewise m-th order polynomial functions. The corresponding discretization of problem (1.2) reads: find uh∈Vh such that

(σ(uh,∇uh),∇vh) = (fh, vh) ∀vh∈Vh. (4.1) Letψj ∈Vh,j∈ C:={1, . . . ,dim(Vh)}, denote the basis functions ofVh. Employing these functions in (4.1) gives rise to the nonlinear algebraic system (2.3).

4.2 Linearization

Letu⁰_h∈Vh, fixing the initial vector U⁰ in Algorithm2.1. The linearization of (4.1), fork≥1, reads: find u^k_h∈Vhsuch that

(σ^k−1(u^k_h,∇u^k_h),∇ψj) = (f, ψj) ∀j∈ C, (4.2) which is the functional form of the algebraic system (2.4). Two common ways to define the flux function σ^k−1 are the fixed point linearization (3.3) and the Newton linearization (3.4).

4.3 Algebraic solution

On the i-th step,i≥1, of an iterative linear solver for the algebraic system (2.4), we obtain the algebraic residual vectorR^k,i in (2.5), with components associated with the setC, R^k,i={R_j^k,i}j∈C. The functional form of (2.5) is: findu^k,i_h ∈Vh such that

(σ^k−1(u^k,i_h ,∇u^k,i_h ),∇ψj) = (f, ψj)−R^k,i_j ∀j∈ C. (4.3)

4.4 Flux reconstruction by local mixed problems without prescription

We construct (d^k,i_h +l^k,i_h )∈RTN_l(Th) with l :=m−1 or l :=m. The construction uses the partition of unity together with local mixed finite element problems posed on patches around mesh vertices, as outlined in Section 3.6. Let Vh denote the set of mesh vertices with subsets V_h^int for interior vertices and V_h^ext for boundary ones. Let ψ^a∈P1(Th)∩C⁰(Ω) stand for the classical hat basis function associated with vertex a ∈ Vh. To distribute the algebraic residual onto vertices, we set, for all a ∈ V_h^int, R^k,ia := P

j∈CβjR_j^k,i where the coefficientsβj are such thatψ^a=P

j∈Cβjψj, while, fora∈ V_h^ext, we setRa^k,i:= 0. Furthermore, for all a∈ Vh, let T^abe the patch of elements of Th that sharea, and letRTN^N,0_l (T^a) be the subspace of RTN_l(T^a) with zero normal flux through∂T^afora∈ V_h^intand through that part of∂T^awhich lies inside Ω fora∈ V_h^ext. LetP^∗l(T^a) be spanned by piecewisel-th order polynomials onT^awith zero mean onT^awhen a∈ V_h^int; when a∈ V_h^ext, the mean value condition is not imposed.

Definition 4.1 (Construction of (d^k,i_h +l^k,i_h )). For all vertices a∈ Vh, define(d^k,i_a +l^k,i_a )∈RTN^N,0_l (T^a) andq^a∈P^∗l(T^a)by

(d^k,ia +l^k,ia ,vh)T^a−(q^a,∇·vh)T^a=−(I^RTN_l (ψ^aΠlσ^k−1(u^k,i_h ,∇u^k,i_h )),vh)T^a, (4.4a) (∇·(d^k,ia +l^k,ia ), φh)T^a= (f ψ^a−σ^k−1(u^k,i_h ,∇u^k,i_h )·∇ψ^a, φh)T^a−(R^k,ia , φh)T^a|T^a|⁻¹, (4.4b) for all (vh, φh)∈RTN^N,0_l (T^a)×P^∗l(T^a). Then, setd^k,i_h +l^k,i_h :=P

a∈Vh(d^k,ia +l^k,ia ).

In the above problems, we can takeφh ∈Pl(T^a) since multiplying (4.3) by the coefficientsβj, summing over allj∈ C, and using the definition ofR^k,ia , yields, for alla∈ V_h^int, the Neumann compatibility condition (σ^k−1(u^k,i_h ,∇u^k,i_h ),∇ψ^a)T^a = (f, ψ^a)T^a−R^k,ia . (4.5) Moreover, it can be shown that the construction of Definition4.1is equivalent to the one proposed by Braess and Sch¨oberl [5]. We proceed similarly ford^k,i_h . Set

R¯^k,ia := (f, ψ^a)T^a−(σ(u^k,i_h ,∇u^k,i_h ),∇ψ^a)T^a ∀a∈ V_h^int, (4.6) and for any a∈ V_h^ext, set ¯Ra^k,i:= 0.

Definition 4.2 (Construction of d^k,i_h ). Defined^k,ia ∈RTN^N,0_l (T^a) and q¯^a∈ P^∗l(T^a) by solving the mixed finite element problems (4.4) with σ(u^k,i_h ,∇u^k,i_h ) in place of σ^k−1(u^k,i_h ,∇u^k,i_h ) and R¯^k,ia in place of R^k,ia . Then, set d^k,i_h :=P

a∈Vhd^k,ia .

Definition 4.3 (Error measure, data oscillation, quadrature, and algebraic remainder). Use u^k,i_h and g^k,i_h := ∇u^k,i_h in the error measure (2.1) and set fh := Πlf, σ^k,i_h := Π_lσ(u^k,i_h ,∇u^k,i_h ), and r^k,i_h |K :=

P

a∈VK|Ta|⁻¹R^k,ia for allK∈ Th, whereVK collects the vertices of the element K.

We now verify the assumptions of Section2.5:

Lemma 4.4 (Quasi-equilibration). Assumption2.3 holds.

Proof. Let K ∈ Th and let vh ∈P^l(K) (and zero elsewhere) be fixed. For anya ∈ VK, by (4.5), we can takevh as test function φh in (4.4b). Since P

a∈VKψ^a|K = 1 andP

a∈VK∇ψ^a|K= 0 (ψ^aform a partition of unity onK), we infer

(∇·(d^k,i_h +l^k,i_h ), vh)K = X

a∈VK

(∇·(d^k,ia +l^k,ia ), vh)K = (f, vh)K−X

a∈VK

(R^k,ia , vh)K|T^a|⁻¹,

whence the assertion of the lemma follows from the definition ofr^k,i_h .

Lemma 4.5 (Local approximation and convergence). Assumption2.4holds.