Coupled Elliptic Solvers for Embedded Mesh and Interface Problems

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1 Coupled Elliptic Solvers for Embedded Mesh and Interface Problems Natalie Beams, Andreas Klöckner, Luke Olson University of Illinois at Urbana-Champaign 5 January 2017

Complicated geometry, time-dependent problems... 1.

changes with time? Source: Ramamurti et al., J Exp Biol.

2 Complicated geometry, time-dependent problems Conforming/refined mesh Lose advantages of structured mesh What if domain changes with time? Source: Ramamurti et al., J Exp Biol. 205, (2002) N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

3 Complicated geometry, time-dependent problems Conforming/refined mesh Lose advantages of structured mesh What if domain changes with time? 2. Embedded domain methods How best to enforce the true boundary conditions? Can we do this in a high-order way? Ω N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

4 Embedded domain types Γ Γ Ω i Interior problem on Ω i embedded in a fictitious domain Ω e Problem in Ω e is treated as a domain with an exclusion N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

5 Embedded domains & finite elements Finite cell methods (e.g., Parvizian et al., 2007) Fictitious domain methods (e.g., Glowinski et al, 1994) Immersed finite element methods Ω Ω Ω Finite element-integral equation method (Rüberg & Cirak, 2010) + Ω FE Layer potential IE N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

6 Embedded mesh problems: convergence High-order implementation of immersed boundary methods, immersed interface methods, etc. can be tricky Our FE-IE implementation is high order when components are high order N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

7 Embedded mesh problems: convergence Consider test problem: u(x) = 1 x Ω u(x) = 0 x Ω (1) Domain Ω is circle centered at (0, 0) with radius r = 0.5; Embedded in = [ 0.6, 0.6] [0.6, 0.6] Ω N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

8 Embedded mesh problems: convergence Ω Splitting is: [FE] u 1 (x) = 1 x u 1 = 0 x [IE] u 2 (x) = 0 x Ω u 2 = u 1 x Ω (2) N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

9 Embedded mesh problems: convergence Convergence of our implementation for the interior FE-IE problem: FE basis order qbx h fe, h ie error error 0 order , , , , , , , , , N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

10 Embedded mesh problems: convergence Now consider the complementary problem (domain with exclusion): Ω u(x) = f(x) x \Ω u(x) = g(x) x Ω u(x) = ĝ(x) x (3) With = [ 1, 1] [1, 1] N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

11 FE-IE for exclusion(s): new splitting Interior FE problem & purely exterior IE problem, coupled through boundary conditions coupled system Split problems are now: [FE] u 1 (x) = f(x) x u 1 = ĝ(x) u 2 x [IE] u 2 (x) = 0 x R 2 \Ω u 2 = g(x) u 1 x Ω (4) Ω Ω Original problem domain \Ω FE domain = [ 1, 1] [1, 1] IE domain R 2 \Ω Achieve same order of convergence N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

12 Coupled subproblems for interface problems Consider the interface problem β u(x) = f(x) in Ω i Ω e, with u i (x) = cu e (x) + a(x) on Γ, and u i (x) n = κ ue (x) n + b(x) on Γ (5) with two domains Ω i and Ω e separated by an interface Γ: Γ Ω i Ω e N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

13 Coupled subproblems for interface problems Ω i Interior problem on Ω i embedded in a fictitious domain Γ + Γ Ω e Problem in Ω e is treated as a domain with an exclusion = coupled interior & exterior solutions N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

14 Coupled subproblems for interface problems Flexible representation through combinations of single & double layer potentials Can handle non-homogeneous jump conditions in derivative... N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

15 Coupled subproblems for interface problems Flexible representation through combinations of single & double layer potentials Can handle non-homogeneous jump conditions in derivative... and value N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

16 Interface problems: sample coupled system IE self op. FE eval. IE eval. FE eval. FE matrix FE eval. IE self op. FE eval. IE off- IE off- FE matrix curve eval. curve eval. σ i U i σ e U e = jump cond. interior r.h.s. jump cond. exterior r.h.s N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

17 Summary Combine best aspects of FE and IE solvers Flexible representation for interior domains, domains with exclusions, and many interface problems Computational mechanics behind FE and IE solvers remain largely unchanged High-order convergence, even near the embedded boundary N. Beams, A. Klöckner, L. Olson Coupled Elliptic Solvers/Interface Problems HKUST, 5 Jan

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