Past, present and space-time

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Past, present and space-time Arnold Reusken Chair for Numerical Mathematics RWTH Aachen Utrecht, 12.11.2015 Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 1 / 20

Outline Past. Past present. Space-time. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 2 / 20

Past I: 1983-1984 Diploma thesis Stabiliteit en instabiliteit van een Hopscotchmethode Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 3 / 20

Past II: 1984-1988 (PhD thesis) Convergence Analysis of Nonlinear Multigrid Methods Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 4 / 20

Past present A: 1978-1989 UU, B: 1989-1997 TUE C: 1997-present RWTH Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 5 / 20

Space-time Finite Element Method for PDEs on Evolving Surfaces Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 6 / 20

Motivation: simulation of two-phase incompressible flows system: n-butanol/water Model: Navier-Stokes equations + coupling conditions Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 7 / 20

Rising droplet with surfactant transport solution gravity-driven butanol-droplet in water Velocity field determined from NS-equations. + surfactant eqn. Ṡ D Γ Γ S + ( Γ u)s = 0 Elliptic PDE on evolving surface Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 8 / 20

Surfactant PDE Γ(0) smooth surface in R 3, Γ(t) =. Γ(t), t [0, T ], advected by smooth w = w(x, t) R 3. Model for diffusive mass transport on Γ(t): Diffusion equation u + (div Γ w)u Γ u = 0 on Γ(t), t (0, T ] with u = u t + w u. Initial condition u(x, 0) = u 0 (x) for x Γ(0). Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 9 / 20

Weak formulations Space-time manifold Γ = Γ(t) {t}, Γ R 4 t (0,T ] Suitable (Sobolev) spaces W (trial), H (test) on Γ. a(u, v) = ( Γ u, Γ v) L 2 (Γ ), u, v H. Well-posed weak formulation determine u W such that u, v + a(u, v) = (f, v) L 2 (Γ ) for all v H. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 10 / 20

Weak formulations Space-time manifold Γ = Γ(t) {t}, Γ R 4 t (0,T ] Suitable (Sobolev) spaces W (trial), H (test) on Γ. a(u, v) = ( Γ u, Γ v) L 2 (Γ ), u, v H. Well-posed weak formulation determine u W such that u, v + a(u, v) = (f, v) L 2 (Γ ) for all v H. Theorem [Olshanskii,R. SINUM 2014] Weak formulation is well-posed. Analysis based on continuity and inf-sup property Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 10 / 20

A time-discontinuous Eulerian weak formulation Time slabs: t j = j t, I n := (t n 1, t n ], Γ n := t In Γ(t). Broken space W b := N n=1 W n. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 11 / 20

A time-discontinuous Eulerian weak formulation Time slabs: t j = j t, I n := (t n 1, t n ], Γ n := t In Γ(t). Broken space W b := N n=1 W n. N a(u, v) = a n (u, v), a n (u, v) = n=1 tn t n 1 Γ(t) Γ u Γ v ds dt Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 11 / 20

A time-discontinuous Eulerian weak formulation Time slabs: t j = j t, I n := (t n 1, t n ], Γ n := t In Γ(t). Broken space W b := N n=1 W n. N a(u, v) = a n (u, v), a n (u, v) = n=1 tn t n 1 Γ(t) Γ u Γ v ds dt N d(u, v) = d n (u, v), d n (u, v) = [u] n 1 v+ n 1 ds n=1 Γ(t n 1 ) Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 11 / 20

A time-discontinuous Eulerian weak formulation Time slabs: t j = j t, I n := (t n 1, t n ], Γ n := t In Γ(t). Broken space W b := N n=1 W n. N a(u, v) = a n (u, v), a n (u, v) = n=1 tn t n 1 Γ(t) Γ u Γ v ds dt N d(u, v) = d n (u, v), d n (u, v) = [u] n 1 v+ n 1 ds n=1 Γ(t n 1 ) N u, v b = u n, v n n n=1 Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 11 / 20

A time-discontinuous Eulerian weak formulation Time slabs: t j = j t, I n := (t n 1, t n ], Γ n := t In Γ(t). Broken space W b := N n=1 W n. N a(u, v) = a n (u, v), a n (u, v) = n=1 tn t n 1 Γ(t) Γ u Γ v ds dt N d(u, v) = d n (u, v), d n (u, v) = [u] n 1 v+ n 1 ds n=1 Γ(t n 1 ) N u, v b = u n, v n n n=1 Time-discontinuous weak formulation (allows time-stepping) Find u W b such that u, v b + a(u, v) + d(u, v) = F (v) for all v W b. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 11 / 20

Galerkin FEM based on time-discontinuous formulation Key ideas: W b replaced by FE space Wh Γ on Γ. For FE space we use trace of standard outer space-time FE space. Γ n is approximated (zero level of discrete level set function). Γ(t n ) t n Γ(t n 1 ) t n 1 Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 12 / 20

Trace FE spaces Space-time slab: Q n = Ω (t n 1, t n ] R d+1. T n : triangulation of Ω. V n : standard FE space on T n (piecewise linears). Γ(t n ) t n Γ(t n 1 ) t n 1 Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 13 / 20

Trace FE spaces Space-time slab: Q n = Ω (t n 1, t n ] R d+1. T n : triangulation of Ω. V n : standard FE space on T n (piecewise linears). Γ(t n ) t n Γ(t n 1 ) t n 1 W n,h := { w : Q n R w(x, t) = φ 0 (x) + tφ 1 (x), φ 0, φ 1 V n } Wn,h Γ := { v : Γ n R v = w Γ n, w W n,h, t }, 1 n N. Wh Γ := N n=1wn,h Γ Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 13 / 20

Galerkin FEM Galerkin trace-fem Find u h = u h, t W Γ h such that u h, v h b + a(u h, v h ) + d(u h, v h ) = F (v h ) for all v h W Γ h Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 14 / 20

Galerkin FEM Galerkin trace-fem Find u h = u h, t W Γ h such that u h, v h b + a(u h, v h ) + d(u h, v h ) = F (v h ) for all v h W Γ h Time-stepping procedure For n = 1,..., N, do: Find u h = u h, t Wn,h Γ such that u h, v h n +a n (u h, v h )+ u h + v h + ds = t n 1 uh n 1 t n 1 v + h ds+f (v h) v h W Γ n,h Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 14 / 20

Error analysis [Olshanskii,AR, SINUM 2014] Discrete stability N u 2 h := u 2 H + max n=1,...,n un 2 t n + [u] n 1 2 t n 1. inf u W b n=1 u, v b + a(u, v) + d(u, v) sup c s v W b v h u h Global stability result. No conditions on t. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 15 / 20

Error analysis [Olshanskii,AR, SINUM 2014] Discrete stability N u 2 h := u 2 H + max n=1,...,n un 2 t n + [u] n 1 2 t n 1. inf u W b n=1 u, v b + a(u, v) + d(u, v) sup c s v W b v h u h Global stability result. No conditions on t. Discretization error bounds. Assume t h u u h h ch( u H 2 (Γ ) + u u h H 1 (Γ ) ch 2 ( u H 2 (Γ ) + sup u H 2 (Γ(t))). t [0,T ] sup u H 2 (Γ(t))). t [0,T ] Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 15 / 20

Experiment: diffusion on a moving+deforming ellipsoid Γ(t) is zero level of φ(x, y, z, t) = ( x ) 2 + y 2 + z 2 1, t [0, 4]. 1 + 0.25 sin(t) cos(t) Velocity field w(x, y, z, t) = 0.25 1+0.25 sin(t) (x, 0, 0)T. Solution prescribed: u(x, y, z, t) = e t xy. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 16 / 20

Structure of the algorithm Coarse regular tetrahedral triangulation of Ω = [ 1.5, 1.5] 3. FE space V h : piecewise linears. Per time slab Local spatial refinement close to Γ n. Outer space-time FE space; linear in t. Approximation of space-time surface Γ n. FE space on Γ n : trace space. Apply Galerkin discretization with this FE space. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 17 / 20

Results: discrete L t L 2 x-errors max t [0,4] e h L2 (Γ(t)) 1 1 0.1 0.1 0.01 0.01 0.001 1 2 3 4 5 6 0.0001 0.01 0.1 1 -log 2 h x 0 0.001 1 2 3 4 5 6 0.0001 0.01 0.1 1 -log 2 h t h x = h constant h t = t constant. Observations: very stable method; second order convergence. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 18 / 20

Evolving surface with topological singularity h = 1/16, t = 1/128 Domain Ω = ( 3, 3) ( 2, 2) 2, t [0, 1]. Prescribed level set function φ, determines Γ(t). Space-time interpolation yields Γ n. Surfactant transport equation. h = 1/16, t = 1/4 u 0 (x) = 3 x 1 for x 1 0, zero otherwise. Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 19 / 20

Summary and outlook The space-time journey started with the Hopscotch stability (1983-1984) under the supervision of Gerard Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 20 / 20

Summary and outlook The space-time journey started with the Hopscotch stability (1983-1984) under the supervision of Gerard Bedankt voor de zeer motiverende begeleiding... Ik wens je alle goeds voor de tijd die komen gaat Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 20 / 20

Summary and outlook The space-time journey started with the Hopscotch stability (1983-1984) under the supervision of Gerard Bedankt voor de zeer motiverende begeleiding... Ik wens je alle goeds voor de tijd die komen gaat Past present future Reusken (RWTH Aachen) Past, present and space-time Utrecht, 12.11.2015 20 / 20

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