Space-time Finite Element Methods for Parabolic Evolution Problems

Transcription

1 Space-time Finite Element Methods for Parabolic Evolution Problems with Variable Coefficients Ulrich Langer, Martin Neumüller, Andreas Schafelner Johannes Kepler University, Linz Doctoral Program Computational Mathematics Numerical Analysis and Symbolic Computation 7 August 2018 A. Schafelner JKU Linz 1 / 39

2 Outline Introduction Space-time Variational Formulations Space-time Finite Element Methods Numerical Experiments Conclusions & Outlook A. Schafelner JKU Linz 2 / 39

3 Introduction Space-time Variational Formulations Space-time Finite Element Methods Numerical Experiments Conclusions & Outlook A. Schafelner JKU Linz 3 / 39

4 Introduction : A parabolic model problem Let Q = Q T := Ω (0, T ) be our space-time cylinder and Σ := Ω (0, T ), Σ 0 := Ω {0} and Σ T := Ω {T }. Then: Given f, g, ν and u 0, find u such that u t (x, t) div x(ν(x, t) x u(x, t)) =f(x, t), (x, t) Q, u(x, t) =g(x, t) = 0, (x, t) Σ, u(x, 0) =u 0 (x) = 0, x Ω, where ν is a given uniformly positive and bounded coefficient. Examples: diffusion, heat-conduction and 2D eddy current. A. Schafelner JKU Linz 3 / 39

5 Introduction : References Time-parallel methods: Gander (2015): Historical overview on 50 years time-parallel methods Space-time methods for parabolic evolution problems Steinbach, Yang (2018): Overview on space-time methods. Steinbach (2015): Theory for the limit case of our method A. Schafelner JKU Linz 4 / 39

6 Introduction Space-time Variational Formulations Space-time Finite Element Methods Numerical Experiments Conclusions & Outlook A. Schafelner JKU Linz 5 / 39

7 Space-time Variational Formulations : Space-time Variational Formulations I (1) Line Variational Formulation and Vertical Method of Lines discretize first in space, then in time (2) Line Variational Formulation and Horizontal Method of Lines discretize first in time, then in space (3) Space-time Variational Formulation discretize all at once on unstructured decompositions of the space-time cylinder considered in this talk Why? A. Schafelner JKU Linz 5 / 39

8 Space-time Variational Formulations : Space-time Variational Formulations II several ways to obtain wellposedness of the continuous problem Space-time v.f. in Sobolev spaces on the space-time cylinder, see, e.g., Ladyšhenskaya! in H kx,kt (Q) + b.c. Space-time v.f. in Bochner spaces of abstract functions, see, e.g., Lions, Friedman! in H kt (0, T ; H kx (Ω)) + b.c. A. Schafelner JKU Linz 6 / 39

9 Space-time Variational Formulations : A (very) weak space-time variational formulation Find u H 1,0 0 (Q) s.t. a(u, v) = l(v) v H 1,1 0,0 (Q), with a(u, v) := u(x, t) t v(x, t) + ν(x, t) x u(x, t) x v(x, t) d(x, t), Q l(v) := f(x, t) v(x, t) d(x, t) + u 0 (x) v(x, 0) dx. Q Ω A. Schafelner JKU Linz 7 / 39

10 Space-time Variational Formulations : Regularity results from Ladyžhenskaya et al If u 0 L 2 (Ω), f L 2,1 (Q T ) := {v : T 0 v(, t) L 2 (Ω) dt < }, and 0 < ν ν(x, t) ν, then there exist a unique generalized solution u H 1,0 0 (Q T ) that is a generalized solution in V 1,0 2,0 (Q T ). For ν 1, f L 2 (Q T ) and u 0 H0 1 (Ω), Ladyžhenskaya proved that the generalized solution u even belongs to space H,1 0 (Q T ) = {v H 1 0 (Q T ) : x v L 2 (Q T )}, and continuously depends on t in the norm of the space H 1 0 (Ω). A. Schafelner JKU Linz 8 / 39

11 Space-time Variational Formulations : Maximal parabolic regularity For the class of problems t u div x (ν(x, t) x u) = f }{{} =Lu with u 0 = 0, we have maximal parabolic regularity if t u X + Lu X C f X, where X = L p (0, T ; L q (Ω)) = L p,q (Q), 1 < p, q <. For p = q = 2, we denote the space of such functions by H L,1 (Q) = {v L 2 (Q) : t u L 2 (Q) Lu L 2 (Q)}. A. Schafelner JKU Linz 9 / 39

12 Introduction Space-time Variational Formulations Space-time Finite Element Methods Numerical Experiments Conclusions & Outlook A. Schafelner JKU Linz 10 / 39

13 Space-time Finite Element Methods : Space-time FEM with time-upwind stabilization The main idea: for each element K, define individual upwind test function v h,t (x, t) := v h (x, t) + θ K h K t v h (x, t), for all (x, t) K, with θ K positive parameter, and h K := diam(k), for each element K, multiply the PDE by v h,t (x, t), sum up over all elements K, proceed as usual, IbyP, etc. A. Schafelner JKU Linz 10 / 39

14 Space-time Finite Element Methods : Derivation of the FE scheme Let u H L,1 (Q) be the solution of our IBVP, i.e. t u div x (ν x u) = f in L 2 (K). Multiplication of the PDE with v h,t, integration over a single element K, and summing up over all elements yields t u(v h + θ K h K t v h ) d(x, t) K T h K + div x (ν x u)(v h + θ K h K t v h ) d(x, t) K = f(v h + θ K h K t v h ) d(x, t) K T h K A. Schafelner JKU Linz 11 / 39

15 Space-time Finite Element Methods : Derivation of the FE scheme (cont.) The time derivative and the r.h.s. remain unchanged, IbyP wrt x on the div x -term gives us div x (ν x u)(v h + θ K h K t v h ) d(x, t) K = (ν x u) x v h + θ K h K (ν x u) x ( t v h ) d(x, t) K (n x (ν x u))v h + θ K h K (n x (ν x u)) t v h ds (x,t) K A. Schafelner JKU Linz 11 / 39

16 Space-time Finite Element Methods : Derivation of the FE scheme (cont.) We keep the volume integrals and take a closer look at the boundary terms. Summing again up gives (n x (ν x u)) v h ds (x,t) + θ K h K (n x (ν x u)) t v h ds (x,t) K T h K If u is sufficiently smooth, we have at a common edge/face/volume K (n x (ν x u)) K = ( n x (ν x u) ) K A. Schafelner JKU Linz 11 / 39

17 Space-time Finite Element Methods : Derivation of the FE scheme (cont.) Hence, the first integral vanishes at all inner edges/faces/volumes. Since v h Σ = 0 and n x 0 on Σ 0 and Σ T, the first integral vanishes on the entire boundary Q = Σ Σ 0 Σ T, i.e., (n x (ν x u)) v h ds (x,t) = 0 K T h K The second boundary term however does not vanish, as in general θ K θ K and h K h K for K K. A. Schafelner JKU Linz 11 / 39

18 Space-time Finite Element Methods : The space-time FE scheme Find u h V 0h := {v C(Q) : v(x K ( )) P p (K), v Σ Σ0 = 0} : a h (u h, v h ) = l h (v h ), v h V 0h, (1) where a h (u h, v h ) := t u h v h + θ K h K t u h t v h d(x, t) K T h K + (ν x u h ) x v h + θ K h K (ν x u h ) x ( t v h ) d(x, t) K θ K h K (n x (ν x u h )) t v h ds (x,t), K l h (v h ) := f (v h + θ K h K t v h ) d(x, t). K T K h A. Schafelner JKU Linz 12 / 39

19 Space-time Finite Element Methods : Ellipticity = Stability Definition 3.1 (Mesh dependent norm). v h 2 h := K T h [ ν 1/2 x v h 2 L 2(K) + θ Kh K t v h 2 L 2(K)] v h 2 L 2(Σ T ). Lemma 3.2 (Coercivity on the FE space). There exits a constant µ a such that a h (v h, v h ) µ a v h 2 h, v h V 0h, with µ a = min K Th { 1 ci,divx νk θ K 4h K } 1 2 for θ K h K c 2 I,divx ν K, i.e., µ a = 1 2 for θ K = h K c 2 I,divx ν K. A. Schafelner JKU Linz 13 / 39

20 Space-time Finite Element Methods : The constant c I,divx the constant c I,divx comes from the inverse inequality div x (νw h ) L2 (K) c I,divx h 1 K νw h L2 (K), w h W h K, with W h K := {w h : w h = x v h, v h V 0h K } and ν W (T 1 h ), c I,divx is independent of h K, but depends on polynomial degree p and the dimension d, c I,divx is computable! to be precise, we can compute a sharp lower bound for c I,divx h 1 K A. Schafelner JKU Linz 14 / 39

21 Space-time Finite Element Methods : The constant c I,divx ci,divx 10 1 Values of c I,divx p d = 1 d = 2, min d = 2, max d = 3, min d = 3, max p 2 on the coarsest triangulation of the d + 1 hypercube. A. Schafelner JKU Linz 14 / 39

22 Space-time Finite Element Methods : Boundedness Definition 3.3. v 2 h, = v 2 h + [ (θk h K ) 1 v 2 L 2 (K) + θ Kh K v 2 ] H 2 (K) K T h Lemma 3.4. The bilinear form a h (.,. ) is uniformly bounded on V 0h, V 0h, i.e., a h (u, v h ) µ b u h, v h h, where V 0h, := V 0h + H 2 (T h ) H 1,0 0 (Q), and µ b = { max K Th 2(1 + θk h 1 K c2 ν 2 K T r ν ), 2c 2 K T r ν2 K, 2 + c2 I,1, 1 + (c I,νθ K ) 2} 1/2 that is bounded provided that θ K = O(h K ). A. Schafelner JKU Linz 15 / 39

23 Space-time Finite Element Methods : A Cea-like estimate Lemma 3.5. Let u V 0 H 2 (T h ) be the exact solution, and u h V 0h the solution of the finite element scheme (1). Then there holds the Cea-like estimate u u h h ( 1 + µ ) b µ a inf v h V 0h u v h h, Remark. We can again weaken the assumptions on u, in particular, it is enough to require u H L,1 (Q). A. Schafelner JKU Linz 16 / 39

24 Space-time Finite Element Methods : An a prior error estimate I Theorem 3.6. Let s and k be positive integers with s [2, p + 1] and k > (d + 1)/2. Furthermore, let u V 0 H k (Q) H s (T h ) be the exact solution, and u h V 0h the solution of the finite element scheme (1). Then there holds the a priori error estimate ( 1/2 u u h h c h 2(s 1) K u 2 H (K)) s. (2) K T h A. Schafelner JKU Linz 17 / 39

25 Space-time Finite Element Methods : An a prior error estimate II Remark. u H k (Q) with k > (d + 1)/2 is a rather strong assumption needed because of Lagrange Interpolant Π h can be weakened Duan et al. have shown (v I h v) L2 (Q) C M i=1 for v H s (T (Q)) with s = (s 1,..., s M ) and h s i 1 i v H s i(qi ), H s (T (Q)) := {v L 2 (Q) : v Qi H si (Q i ), for all i = 1,..., M} A. Schafelner JKU Linz 18 / 39

26 Introduction Space-time Variational Formulations Space-time Finite Element Methods Numerical Experiments Conclusions & Outlook A. Schafelner JKU Linz 19 / 39

27 Numerical Experiments : Numerical Experiments Key information Space-time FEM was implemented with MFEM, Linear systems were solved by means of hypre, Meshes generated by NETGEN (2D, 3D) and Karabelas & Neumüller (4D), Tests were performed on RADON1, c I,divx was computed numerically, Convergence rates in L 2 -norm and. h -norm. A. Schafelner JKU Linz 19 / 39

28 Numerical Experiments : Examples Example 4.1 (Smooth solution, uniform coefficient). Consider space-time cylinder Q = (0, 1) d+1, ν 1 and choose the exact solution d u(x, t) = sin(x i π) sin(tπ). i=1 Example 4.1: Solution for d = 2 A. Schafelner JKU Linz 20 / 39

29 Numerical Experiments : Examples u uh h d = 2, p = 1 d = 2, p = 2 d = 2, p = 3 d = 3, p = 1 d = 3, p = 2 d = 3, p = 3 d = 2, O(h) d = 2, O(h 2 ) d = 2, O(h 3 ) d = 3, O(h) d = 3, O(h 2 ) d = 3, O(h 3 ) dofs Example 4.1. A. Schafelner JKU Linz 21 / 39

30 Numerical Experiments : Examples Example 4.2 (Smooth solution, discont. coefficient). Consider space-time cylinder Q = (0, 1) d+1, space-time dependent coefficient ν(x, t) = and choose the smooth solution u(x, t) = { , for 2x 1 t < 1 2, , for 2x 1 t > 1 2, ( ( sin 9π 2x 1 t 1 ) 2 ) (x 1 x 2 2 1) d ( ( sin 40π 2x 1 t 1 2 sin(4πt) sin(πx i ) ) i=2 2 ) (x 1 x 2 1)(t t 2 ), for 2x 1 t 1 2,, otherwise, d sin(πx i ) i=2 A. Schafelner JKU Linz 22 / 39

31 Numerical Experiments : Examples (a) (b) (a) Initial mesh for d = 2; (b) Inital mesh for d = 3, sliced in t direction A. Schafelner JKU Linz 23 / 39

32 Numerical Experiments : Examples u uh h d = 2, p = 1 d = 2, p = 2 d = 2, p = 3 d = 3, p = 1 d = 3, p = 2 d = 2, O(h) d = 2, O(h 2 ) d = 2, O(h 3 ) d = 3, O(h) d = 3, O(h 2 ) dofs Example 4.2. A. Schafelner JKU Linz 24 / 39

33 Numerical Experiments : Examples Example 4.3 (Adaptivity). Choose space-time cylinder Q = (0, 1) 3, ν 1, and exact solution u(x, t) = (x 2 1 x 1 )(x 2 2 x 2 )(t 2 t)e 100((x1 t)2 +(x 2 t) 2 ) A. Schafelner JKU Linz 25 / 39

34 Numerical Experiments : Examples An error indicator We use the residual-based error indicator by Steinbach & Yang (2017) where η K := ( h 2 K R h (u h ) 2 L 2 (K) + h K J h (u h ) 2 L 2 ( K)) 1/2, R h (u h ) := f + div x (ν x u h ) t u h in K, J h (u h ) := [ν x u h ] e on e K, with maximum marking strategy, i.e., MARK K T h if η K σ max K T h η K A. Schafelner JKU Linz 26 / 39

35 Numerical Experiments : Examples (a) (b) (c) Plot and mesh at (a) t = 0.25, (b) t = 0.5, (c) t = 0.75 A. Schafelner JKU Linz 27 / 39

36 Numerical Experiments : Examples (a) (b) (c) Evolution of the mesh at (a) t = 0.25, (b) t = 0.5, (c) t = 0.75 A. Schafelner JKU Linz 28 / 39

37 Numerical Experiments : Examples u uh h σ = 0, p = 1 σ = 0, p = 2 σ = 0, p = 3 σ = 0.5, p = 1 σ = 0.5, p = 2 σ = 0.5, p = dofs Example 4.3 Error rates for d = 2. A. Schafelner JKU Linz 29 / 39

38 Numerical Experiments : Examples Example 4.4 (Adaptivity II). Choose space-time cylinder Q = (0, 1) 3, ν 1, and exact solution ( ) 1 u(x, t) = sin 1 10π + x x2 2 + t2 A. Schafelner JKU Linz 30 / 39

39 Numerical Experiments : Examples u uh h σ = 0, p = 1 σ = 0, p = 2 σ = 0, p = 3 σ = 0.5, p = 1 σ = 0.5, p = 2 σ = 0.5, p = dofs Example 4.4 Error rates for d = 2. A. Schafelner JKU Linz 31 / 39

40 Numerical Experiments : Examples (a) (b) (a) Mesh at the origin (0, 0, 0) after 13 adaptive refinements; (b) Evolution of the mesh in the origin over 13 adaptive refinements. A. Schafelner JKU Linz 32 / 39

41 Numerical Experiments : Examples Efficiency index 100 Example 4.3, p = 1 Example 4.3, p = 2 Example 4.3, p = 3 Example 4.4, p = 1 Example 4.4, p = 2 Example 4.4, p = 3 10 Ieff dofs Efficiency Indices. I eff = ( K T h η K) 1/2 A. Schafelner JKU Linz / u u h h 33 / 39

42 Numerical Experiments : Scaling Strong scaling I 10 4 BoomerAMG + GMRES, p = 1 BoomerAMG + GMRES, p = 2 BoomerAMG + GMRES, p = 3 perfect strong scaling 10 3 time in s cores Example 4.1 with d = 3 and dofs. A. Schafelner JKU Linz 34 / 39

43 Numerical Experiments : Scaling Strong scaling II perfect strong scaling BoomerAMG + GMRES, p = 1 BoomerAMG + GMRES, p = 2 BoomerAMG + GMRES, p = 3 time in s cores Example 4.2 with d = 3 and dofs. A. Schafelner JKU Linz 35 / 39

44 Introduction Space-time Variational Formulations Space-time Finite Element Methods Numerical Experiments Conclusions & Outlook A. Schafelner JKU Linz 36 / 39

45 Conclusions & Outlook : Conclusions stability of the FE scheme provided that θ K = O(h K ), we compute the optimal θ K, we proved an a priori estimate for. h, the numerical experiments are in accordance with the theory, fully parallel space-time FEM. A. Schafelner JKU Linz 36 / 39

46 Conclusions & Outlook : Outlook application to more complex domains, e.g., cross-section of an electrical motor, a posteriori error estimators for AFEM (adaptivity), improve the solving of the huge linear system (preconditioning), combine AFEM with Nested Iterations, main future goal: non-linear parabolic problems and eddy-current problems (electrical engineering). A. Schafelner JKU Linz 37 / 39

47 Conclusions & Outlook : A. Schafelner JKU Linz 38 / 39

48 Conclusions & Outlook : A. Schafelner JKU Linz 38 / 39

49 Conclusions & Outlook : [1] Karabelas, E., and Neumüller, M. Generating admissible space-time meshes for moving domains in d+1-dimensions. Tech. Rep. 07, Institute of Computational Mathematics, Johannes Kepler University Linz, [2] Ladyžhenskaya, O. A. The boundary value problems of mathematical physics, vol. 49 of Applied Mathematical Sciences. Springer-Verlag, New York, Translated from the Russian by Jack Lohwater [Arthur J. Lohwater]. [3] Langer, U., Moore, S., and Neumüller, M. Space-time isogeometric analysis of parabolic evolution problems. Comput. Methods Appl. Mech. Engrg. 306 (2016), [4] Langer, U., Neumüller, M., and S., A. Space-time finite element methods for parabolic evolution problems with variable coefficients. In Revision. [5] Steinbach, O. Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15, 4 (2015), [6] Steinbach, O., and Yang, H. Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3d and 4d. Numerical Linear Algebra with Applications 25, 3 (2018), 2018;25:e2143. Thank you! A. Schafelner JKU Linz 39 / 39