A posteriori error estimates for space-time domain decomposition - PowerPoint PPT Presentation

A posteriori error estimates for space-time domain decomposition method for two-phase flow problem Sarah Ali Hassan, Elyes Ahmed, Caroline Japhet, Michel Kern, Martin Vohralík INRIA Paris & ENPC (project-team SERENA), University Paris 13 (LAGA), UPMC Work supported by ANDRA, ANR DEDALES and ERC GATIPOR PINT, 7th Workshop on Parallel-in-Time methods, Roscoff Marine Station, May 02–05, 2018 1 / 25

OUTLINE Motivations and problem setting Robin domain decomposition for a two-phase flow problem 1 2 Estimates and stopping criteria in a two-phase flow problem 3 Numerical experiments 2 / 25

Motivations and problem setting OUTLINE Motivations and problem setting Robin domain decomposition for a two-phase flow problem 1 2 Estimates and stopping criteria in a two-phase flow problem 3 Numerical experiments 3 / 25

Motivations and problem setting Geological disposal of nuclear waste Deep underground repository (High-level radioactive waste) Challenges: Different materials → strong heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. 4 / 25

Motivations and problem setting Geological disposal of nuclear waste Deep underground repository (High-level radioactive waste) Challenges: Different materials → strong heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. Use space-time DD methods 4 / 25

Motivations and problem setting Geological disposal of nuclear waste Deep underground repository (High-level radioactive waste) Challenges: � Estimate the error at each iteration of Different materials → strong the DD method heterogeneity, different time scales. Large differences in spatial scales. Long-term computations. Use space-time DD methods 4 / 25

Motivations and problem setting Geological disposal of nuclear waste Deep underground repository (High-level radioactive waste) Challenges: � Estimate the error at each iteration of Different materials → strong the DD method heterogeneity, different time scales. � Develop stopping criteria to stop the Large differences in spatial scales. DD iterations as soon as the Long-term computations. discretization error has been reached Use space-time DD methods 4 / 25

Robin domain decomposition for a two-phase flow problem OUTLINE Motivations and problem setting Robin domain decomposition for a two-phase flow problem 1 2 Estimates and stopping criteria in a two-phase flow problem 3 Numerical experiments 5 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Discretize in time and apply the DD algorithm at each time step: 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Discretize in time and apply the DD algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Discretize in time and apply the DD algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Discretize in time and apply the DD algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Discretize in time and apply the DD algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Discretize in time and apply the DD algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Discretize in time and apply the DD algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Solve time-dependent problems in the Discretize in time and apply the DD subdomains, in parallel, algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Solve time-dependent problems in the Discretize in time and apply the DD subdomains, in parallel, algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Solve time-dependent problems in the Discretize in time and apply the DD subdomains, in parallel, algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Solve time-dependent problems in the Discretize in time and apply the DD subdomains, in parallel, algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Solve time-dependent problems in the Discretize in time and apply the DD subdomains, in parallel, algorithm at each time step: Solve stationary problems in the subdomains, in parallel, Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Solve time-dependent problems in the Discretize in time and apply the DD subdomains, in parallel, algorithm at each time step: Exchange information through the Solve stationary problems in the space-time interface · · · Following subdomains, in parallel, [Halpern-Nataf-Gander (03), Martin (05)] Exchange information through the interface Same time step on the whole domain. 6 / 25

Robin domain decomposition for a two-phase flow problem Two phase flow equation and DD in time Domain decomposition in space Space-time domain decomposition Solve time-dependent problems in the Discretize in time and apply the DD subdomains, in parallel, algorithm at each time step: Exchange information through the Solve stationary problems in the space-time interface · · · Following subdomains, in parallel, [Halpern-Nataf-Gander (03), Martin (05)] Exchange information through the interface Different time steps can be used in each subdomain according to its Same time step on the whole domain. physical properties. · · · Following [Halpern-C.J.-Szeftel (12), Hoang-C.J.-Jaffré-Kern-Roberts (13)] 6 / 25

Robin domain decomposition for a two-phase flow problem Multidomain problem: Physical form Two–phase immiscible flow with discontinuous capillary pressure curves · · · Following [Enchery-Eymard-Michel 06] Nonlinear (degenerate) diffusion equation in each subdomain For f ∈ L 2 (Ω × ( 0 , T )) and a final time T > 0, find u i : Ω i × [ 0 , T ] → [ 0 , 1 ] , i = 1 , 2, such that: ∂ t u i − ∆ ϕ i ( u i ) = f , in Ω i × ( 0 , T ) , u i ( · , 0 ) = u 0 , in Ω i , on Γ D u i = g i , i × ( 0 , T ) . Kirchhoff transform ϕ i � u i λ i ( a ) π ′ ϕ i ( u i ) = i ( a ) d a 0 Capillary pressure Global mobility of the gas π i ( u i ) : [ 0 , 1 ] → R λ i ( u i ) : [ 0 , 1 ] → R Ω ⊂ R d , d = 2 , 3 u 0 initial gas saturation u scalar unknown gas saturation g boundary gas saturation 1 − u is the water saturation 7 / 25

Robin domain decomposition for a two-phase flow problem Multidomain problem: Physical form with the nonlinear interface conditions ( physical transmission conditions ) ∇ ϕ 1 ( u 1 ) · n 1 = −∇ ϕ 2 ( u 2 ) · n 2 , on Γ × ( 0 , T ) , π 1 ( u 1 ) = π 2 ( u 2 ) , on Γ × ( 0 , T ) , Γ × ( 0 , T ) 7 π 2 (1) 6 π 1 (1) 5 4 3 π 2 ( u ) 2 π 1 ( u ) π 2 (0) 1 Ω 1 Ω 2 π 1 (0) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u ∗ u 1 ∗ u 2 u 1 2 8 / 25