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Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov–Galerkin, and Monotone Mixed Methods

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Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov–Galerkin, and Monotone Mixed Methods Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov{Galerkin, and Monotone Mixed Methods I. Muga† · K.G. van der Zee‡ 5th June, 2018 Abstract This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms, and its inex- act version using discrete dual norms. It is shown that this development, in the case of strictly-convex reflexive Banach spaces with strictly-convex dual, gives rise to a class of nonlinear Petrov{Galerkin methods and, equiv- alently, abstract mixed methods with monotone nonlinearity. Crucial in the formulation of these methods is the (nonlinear) bijective duality map. Under the Fortin condition, we prove discrete stability of the abstract inexact method, and subsequently carry out a complete error analysis. As part of our analysis, we prove new bounds for best-approximation projec- tors, which involve constants depending on the geometry of the underlying Banach space. The theory generalizes and extends the classical Petrov{ Galerkin method as well as existing residual-minimization approaches, such as the discontinuous Petrov{Galerkin method. Keywords Operators in Banach spaces · Residual minimization · Petrov{Galerkin discretization · Error analysis · Quasi-optimality · Duality mapping · Best approximation · Geometric constants Mathematics Subject Classification 41A65 · 65J05 · 46B20 · 65N12 · 65N15 arXiv:1511.04400v3 [math.NA] 29 Jul 2018 †Pontificia Univ. Cat´olicade Valpara´ıso,Instituto de Mathem´aticas,[email protected] ‡University of Nottingham, School of Mathematical Sciences, [email protected] 1 Contents 1 Introduction3 1.1 Petrov{Galerkin discretization and residual minimization . .4 1.2 Residual minimization in discrete dual norms . .4 1.3 Main results . .5 1.4 Discussion: Unifying aspects . .9 1.5 Outline of paper . .9 2 Preliminaries: Duality mappings and best approximation9 2.1 The duality mapping . .9 ∗ 2.1.1 Strict convexity of Y ......................... 10 2.1.2 Strict convexity of Y .......................... 11 2.1.3 Reflexivity of Y ............................. 11 2.1.4 Reflexive smooth setting . 12 2.1.5 Subdifferential property . 12 2.2 Best approximation in Banach spaces . 13 2.2.1 Existence, uniqueness and characterization . 13 2.2.2 Banach{Mazur constant and nonlinear projector estimate . 14 2.2.3 A priori bound I . 15 2.2.4 Asymmetric-orthogonality constant . 16 2.2.5 A priori bound II . 19 2.3 Proof of Lemma 2.9.............................. 19 2.4 Proof of Lemma 2.17.............................. 21 3 Residual minimization, nonlinear Petrov{Galerkin and monotone mixed formulation 22 3.1 Equivalent best-approximation problem . 22 3.2 Analysis of residual minimization . 23 3.3 Characterization of residual minimization . 24 4 Analysis of the inexact method 27 4.1 Equivalent discrete settings . 27 4.2 Well-posedness of the inexact method . 29 4.3 Error analysis of the inexact method . 32 4.4 Direct a priori error analysis of the inexact method . 34 4.5 Proof of Proposition 4.8............................ 36 5 Connection to other theories 37 5.1 Petrov{Galerkin methods . 38 5.2 Residual minimization . 39 5.3 Inexact method in Hilbert spaces . 40 Acknowledgements 40 2 Discretization in Banach Spaces 3 1 Introduction In the setting of Banach spaces, we consider the abstract problem Find u 2 U : ∗ Bu = f in V ; (1.1) where U and V are infinite-dimensional Banach spaces and the data f is a given element in the dual space V∗. The operator B : U ! V∗ is a continuous, bounded- below linear operator, that is, there is a continuity constant MB > 0 and bounded- below constant γB > 0 such that γ kwk ≤ kBwk ≤ M kwk ; 8w 2 :1 (1.2) B U V∗ B U U Problem (1.1) is equivalent to the variational statement Bu; v = f; v 8v 2 ; V∗;V V∗;V V commonly encountered in the weak formulation of partial differential equations (PDEs), i.e., when Bu; v := b(u; v) and b : × ! is a bilinear form. V∗;V U V R Note that the above Banach-space setting allows the consideration of PDEs in non-standard (non-Hilbert) settings suitable for dealing with rough data (e.g., measure-valued sources) and irregular solutions (e.g., in W 1;p, p 6= 2, or in BV ). A central problem in numerical analysis is to devise a discretization method that, for a given family fUngn2N of discrete (finite-dimensional) subspaces Un ⊂ U, is guaranteed to provide a near-best approximation un 2 Un to the solution u 2 U of (1.1). This means that, for some constant C ≥ 1 independent of n, the approximation un satisfies the error bound ku − u k ≤ C inf ku − w k : (1.3) n U n U wn2Un A discretization method for which this is true, is said to be quasi-optimal. It is the purpose of this paper to propose and analyze a new quasi-optimal discretization method for the problem in (1.1) that generalizes and improves upon existing methods. 1 The smallest possible constant MB coincides with kBk := sup kBwk ∗ =kwk , while the V U w2Unf0g −1 −1 largest possible constant γB coincides with 1=kB k, where B : Im(B) ! U . 4 I. Muga and K.G. van der Zee 1.1 Petrov{Galerkin discretization and residual minimiza- tion A standard method for (1.1) is the Petrov{Galerkin discretization: ( Find un 2 Un : Bun; vn = f; vn 8vn 2 n ; (1.4) V∗;V V∗;V V with Vn ⊂ V a discrete subspace of the same dimension as Un. It is well-known however, that the fundamental difficulty of (1.4) is stability: One must come up with a test space Vn that is precisely compatible with Un in the sense that they have the same dimension and the discrete inf{sup condition is satisfied; see, e.g., [22, Chapter 2] and [33, Chapter 10]. Incompatible pairs (Un; Vn) may lead to spurious numerical artifacts and non-convergent approximations. An alternative method, which is not common, is residual minimization: ( Find un 2 Un : un = arg min f − Bwn ∗ ; (1.5) wn2Un V where the dual norm is given by hr; vi ∗ krk = sup V ;V ; for any r 2 ∗ : (1.6) V∗ kvk V v2Vnf0g V The residual-minimization method is appealing for its stability and quasi- optimality without requiring additional conditions. This was proven by Guer- mond [26], who studied residual minimization abstractly in Banach spaces, and focussed on the case where the residual is in an Lp space, for 1 ≤ p < 1. If V is a Hilbert space, residual minimization corresponds to the familiar least-squares minimization method [5]; otherwise it requires the minimization of a convex (non- quadratic) functional. 1.2 Residual minimization in discrete dual norms Although residual minimization is a quasi-optimal method, an essential complica- tion is that the dual norm (1.6) may be non-computable in practice, because it requires a supremum over V that may be intractable. This is the case, for example, 1;p d for the Sobolev space V = W0 (Ω), with Ω ⊂ R a bounded d-dimensional domain, ∗ 1;p ∗ −1;q for which the dual is the negative Sobolev space V = [W0 (Ω)] =: W (Ω) (see, e.g., [1]), where p−1 + q−1 = 1. Situations with non-computable dual norms are very common in weak formulations of PDEs and, therefore, such complications can not be neglected. Discretization in Banach Spaces 5 A natural replacement that makes such dual norms computationally-tractable is obtained by restricting the supremum to discrete subspaces Vm ⊂ V. This idea leads to the following inexact residual minimization: ( Find un 2 Un : un = arg min f − Bwn ( )∗ ; (1.7) wn2Un Vm where the discrete dual norm is now given by hr; vmiV∗;V ∗ krk ∗ = sup ; for any r 2 V : (1.8) (Vm) kv k vm2Vmnf0g m V Note that a notation with a separate parametrization (·)m is used to highlight the fact that Vm need not necessarily be related to Un. 1.3 Main results The main objective of this work is to present a comprehensive abstract analysis of the inexact residual-minimization method (1.7) in the setting of Banach spaces. As part of our analysis, we also obtain new abstract results for the (exact) residual- minimization method (1.5) when specifically applied in non-Hilbert spaces. We use the remainder of the introduction to announce the main results in this work and discuss their significance. Most of our results are valid in the case that V is a reflexive Banach space such that V and V∗ are strictly convex 2, which we shall refer to as the reflexive smooth setting. The reflexive smooth setting includes Hilbert spaces, but also important non-Hilbert spaces, since Lp(Ω) (as well as p-Sobolev spaces) for p 2 (1; 1) are reflexive and strictly convex, however not so for p = 1 and p = 1 (see [14, Chapter II] and [8, Section 4.3]). We assume this special setting throughout the remainder of Section1. Indispensable in our analysis is the duality mapping, which is a well-studied operator in nonlinear functional analysis that can be thought of as the extension to Banach spaces of the well-known Riesz map (which is a Hilbert-space construct). ∗ In the reflexive smooth setting, the duality mapping JV : V ! V is a bijective monotone operator that is nonlinear in the non-Hilbert case. To give a specific 1;p example, if V = W0 (Ω) then JV is a (normalized) p-Laplace-type operator. We refer to Section 2.1 for details and other relevant properties. 2 A normed space Y is said to be strictly convex if, for all y1; y2 2 Y such that y1 6= y2 and ky1k = ky2k = 1, it holds that kθy1 + (1 − θ)y2k < 1 for all θ 2 (0; 1), see e.g., [17,8, 13].
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