Rate optimal adaptive FEM with inexact solver for strongly monotone operators

Transcription

1 ASC Report No. 19/2016 Rate optimal adaptive FEM with inexact solver for strongly monotone operators G. Gantner, A. Haberl, D. Praetorius, B. Stiftner Institute for Analysis and Scientific Computing Vienna University of Technology TU Wien ISBN

2 Most recent ASC Reports 19/2016 H. Woracek Directing functionals and de Branges space completions in almost Pontryagin spaces 18/2016 X. Chen, E.S. Daus, and A. Jüngel Global existence analysis of cross-diffusion population systems for multiple species 17/2016 M. Halla and L. Nannen Two scale Hardy space infinite elements for scalar waveguide problems 16/2016 D. Stürzer, A. Arnold, and A. Kuggi Closed-loop stability analysis of a gantry crane with heavy chain 15/2016 L. Nannen, M. Wess Spurious modes of the complex scaled Helmholtz equation 14/2016 A. Bespalov, A. Haberl, and D. Praetorius Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems 13/2016 J.M. Melenk and A. Rieder Runge-Kutta convolution quadrature and FEM-BEM coupling for the time dependent linear Schrödinger equation 12/2016 A. Arnold and C. Negulescu Stationary Schrödinger equation in the semi-classical limit: Numerical coupling of oscillatory and evanescent regions 11/2016 P.M. Lima, M.L. Morgado, M. Schöbinger und E.B. Weinmüller A novel computational approach to singular free boundary problems in ordinary differential equations 10/2016 M. Hanke, R. März, C. Tischendorf, E. Weinmüller, S. Wurm Least-Squares Collocation for Higher Index Differential-Algebraic Equtions Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstraße Wien, Austria admin@asc.tuwien.ac.at WWW: FAX: ISBN c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors.

3 Rate optimal adaptive FEM with inexact solver for strongly monotone operators Dirk Praetorius (joint work with Gregor Gantner, Alexander Haberl, and Bernhard Stiftner) Meanwhile, the mathematical understanding of adaptive FEM has reached a mature state; see [9, 14, 3, 15, 6, 11] for some milestones for linear elliptic PDEs, [17, 8, 2, 13] for non-linear problems, and [4] for some general framework. Optimal adaptive FEM with inexact solvers has already been addressed in [15, 1, 4] for linear PDEs and in [5] for eigenvalue problems. However, for problems involving nonlinear operators, optimal adaptive FEM with inexact solvers has not been analyzed yet. Our work [12] aims to close the gap between convergence analysis (e.g. [4]) and empirical evidence (e.g. [10]) by analyzing an algorithm from [7]. Model problem. We follow [4] and present our results from [12] in an abstract framework, while precise examples for our setting are given, e.g., in [2, 13]. Let H be a separable Hilbert space over K {R, C} with norm and scalar product (, ) H. With the duality pairing, between H and its dual H, let A : H H be a nonlinear operator which satisfies the following assumptions: (O1) A is strongly monotone: There exists α > 0 such that α u v 2 Re Au Av, u v for all u, v H. (O2) A is Lipschitz continuous: There exists L > 0 such that Au Av := Au Av, w sup L u v for all u, v H. w H\{0} w (O3) A has a potential: There exists a Gâteaux differentiable function P : H K with Gâteaux derivative dp = A, i.e., P (u + rv) P (u) Au, v = lim for all u, v H. r 0 r Let F H. According to the main theorem on strongly monotone operators [18], (O1) (O2) imply the existence and uniqueness of u H such that (1) Au, v = F, v for all v H. To sketch the proof, let I : H H denote the Riesz mapping defined by Iu, v = (u, v) H for all u, v H. Then, Φ : H H, Φ(u) := u α L 2 I 1 (Au F ) satisfies (2) Φ(u) Φ(v) q u v for all u, v H, where q := (1 α 2 /L 2 ) 1/2 < 1. Hence, the Banach fixpoint theorem proves the existence and uniqueness of u H with Φ(u ) = u which is equivalent to (1). In particular, the Picard iteration u n := Φ(u n 1 ) with arbitrary initial guess u 0 H converges to u, and it holds (3) u u n q 1 q un u n 1 qn 1 q u1 u 0 for all n 1. With (O3), (1) (resp. (4) below) is equivalent to energy minimization, and v u 2 is equivalent to the energy difference. This guarantees the quasi-orthogonality [4]. 1

4 2 Oberwolfach Report 44/2016 FEM with iterative solver based on Picard iteration. Let X H be a discrete subspace. As in the continuous case, there is a unique u X such that (4) Au, v = F, v for all v X. According to (O1) (O2), it holds the Céa-type quasi-optimality u u L α u v for all v X. To solve the nonlinear system (4), we use the Picard iteration (applied in X ): Given u n 1 X, we compute u n = Φ (u n 1 ) as follows: Solve the linear system (w, v ) H = Au n 1 F, v for all v X. Define u n := u n 1 α L w 2. Applying (3) on the discrete level, we infer from [7] that (5) u u n u u + q 1 q un u n 1 L α min u v + qn v X 1 q u1 u 0. A posteriori error estimator. We suppose that all considered discrete spaces X H are associated with a conforming triangulation T of a bounded Lipschitz domain Ω R d, d 2. For all T T and all v X, we suppose an a posteriori computable refinement indicator η (T, v ) 0. We then define (6) η (v ) := η (T, v ) and η (U, v ) 2 := η (T, v ) 2 for U T. T U We suppose that there exist constants C ax > 0 and 0 < ϱ ax < 1 such that for all T and all refinements T of T, the following properties (A1) (A3) from [4] hold: (A1) Stability on non-refined element domains: η (T T, v ) η (T T, v ) C ax v v for all v X, v X. (A2) Reduction on refined element domains: η (T \T, v ) ϱ ax η (T \T, v ) + C ax v v for all v X, v X. (A3) Discrete reliability: u u C ax η (T \T, u ). Note that (A1) (A2) are required for all discrete functions (and follow from inverse estimates), while (A3) is only required for the discrete solutions u resp. u of (4). Adaptive algorithm. With adaptivity parameters 0 < θ 1, λ > 0, and C mark 1, an initial conforming triangulation T 0, and an initial guess u 0 0 X 0, our adaptive algorithm iterates the following steps (i) (iii) for all l = 0, 1, 2,... (i) Repeat (a) (b) for all n = 1, 2, 3,..., until u n l un 1 l λ η l (u n l ). (a) Compute discrete Picard iterate u n l X l. (b) Compute refinement indicators η l (T, u n l ) for all T T l (ii) Define u l := u n l and determine a set M l T l of minimal cardinality, up to the multiplicative factor C mark, such that θ η l (u l ) η l (M l, u l ). (iii) Employ newest vertex bisection [16] to generate the coarsest conforming refinement T l+1 of T l such that M l T l \T l+1 (i.e., all marked elements have been refined) and define u 0 l+1 := u l X l X l+1. In step (iii), we suppose that mesh-refinement leads to nested discrete spaces.

5 Adaptive Algorithms 3 Lucky break-down of adaptive algorithm. First, if the repeat loop in step (i) does not terminate, it holds u X l. Moreover, there exists C > 0 with (7) u u n l + η l (u n l ) C q n n 0. Second, if the repeat loop in step (i) terminates with M l = in step (ii), then u = u k as well as M k = for all k l. Overall, we may thus suppose that the repeat loop in step (i) terminates and that #T l < #T l+1 for all l 0. Bounded number of Picard iterations in step (i). There exists C > 0 such that nested iteration u 0 l := u l 1 X l 1 X l guarantees [ ( { u l = u n l with n C 1 + log max 1, η l 1(u l 1 ) })] (8) for all l 1. η l (u l ) Linear convergence. For 0 < θ 1 and all sufficiently small λ > 0, there exist constants 0 < ϱ < 1 and C > 0 such that (9) η l+n (u l+n ) Cϱ n η l (u l ) for all l, n 0. In particular, there exists C > 0 such that (10) u u l C η l (u l ) C Cϱ l η 0 (u 0 ) l 0. Optimal algebraic convergence rates. For sufficiently small 0 < θ 1, sufficiently small λ > 0, and all s > 0, there exists C > 0 such that (11) η l (u l ) C ( #T l #T 0 + 1) s for all l 0, provided that the rate s is possible with respect to certain nonlinear approximation classes [4, 6]. Optimal computational complexity. Currently, the proof of optimal computational complexity is open, but it is observed in numerical experiments. References [1] Arioli, Georgoulis, Loghin. SIAM J. Sci. Comput., 35(3):A1537 A1559, [2] Belenki, Diening, Kreuzer. IMA J. Numer. Anal., 32(2): , [3] Binev, Dahmen, DeVore. Numer. Math., 97(2): , [4] Carstensen, Feischl, Page, Praetorius. Comput. Math. Appl., 67(6): , [5] Carstensen, Gedicke. SIAM J. Numer. Anal., 50(3): , [6] Cascon, Kreuzer, Nochetto, Siebert. SIAM J. Numer. Anal., 46(5): , [7] Congreve, Wihler. J. Comput. Appl. Math., 311: , [8] Diening, Kreuzer. SIAM J. Numer. Anal., 46(2): , [9] Dörfler. SIAM J. Numer. Anal., 33(3): , [10] Ern, Vohralík. SIAM J. Sci. Comput., 35(4):A1761 A1791, [11] Feischl, Führer, Praetorius. SIAM J. Numer. Anal., 52(2): , [12] Gantner, Haberl, Praetorius, Stiftner. in preparation, [13] Garau, Morin, Zuppa. Numer. Math. Theory Methods Appl., 5(2): , [14] Morin, Nochetto, Siebert. SIAM J. Numer. Anal., 38(2): , [15] Stevenson. Found. Comput. Math., 7(2): , [16] Stevenson. Math. Comp., 77(261): , [17] Veeser. Numer. Math., 92(4): , [18] Zeidler. Part II/B, Springer, 1990.