A Hybridized DG / Mixed Method For Nonlinear Convection-Diffusion - PowerPoint PPT Presentation

A Hybridized DG / Mixed Method For Nonlinear Convection-Diffusion Problems Aravind Balan , Michael Woopen, Jochen Sch¨ utz and Georg May AICES Graduate School, RWTH Aachen University, Germany WCCM 2012, S˜ ao Paulo, Brazil July 9, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 1 / 24

Outline Introduction 1 BDM Mixed Method for Diffusion 2 Hybridized BDM Mixed Method for Diffusion 3 4 Hybridized DG-BDM (HDG-BDM) for Advection-Diffusion 5 Hybridized DG (HDG) for Advection-Diffusion 6 Numerical Results Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 2 / 24

Background HDG-BDM method for Advection-Diffusion equations. ∇ · ( f ( u ) − f v ( u, ∇ u )) = 0 1 H. Egger and J. Sch¨ oberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010 2 J. Sch¨ utz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24

Background HDG-BDM method for Advection-Diffusion equations. ∇ · ( f ( u ) − f v ( u, ∇ u )) = 0 Discontinuous Galerkin for Advection; known to work well ∇ · f ( u ) = 0 1 H. Egger and J. Sch¨ oberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010 2 J. Sch¨ utz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24

Background HDG-BDM method for Advection-Diffusion equations. ∇ · ( f ( u ) − f v ( u, ∇ u )) = 0 Discontinuous Galerkin for Advection; known to work well ∇ · f ( u ) = 0 BDM Mixed method for Diffusion; known to work well ∇ · f v ( u, σ ) = 0 σ = ∇ u 1 H. Egger and J. Sch¨ oberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010 2 J. Sch¨ utz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24

Background HDG-BDM method for Advection-Diffusion equations. ∇ · ( f ( u ) − f v ( u, ∇ u )) = 0 Discontinuous Galerkin for Advection; known to work well ∇ · f ( u ) = 0 BDM Mixed method for Diffusion; known to work well ∇ · f v ( u, σ ) = 0 σ = ∇ u Hybridization to reduce the global coupled degrees of freedom λ ≈ u | Γ 1 H. Egger and J. Sch¨ oberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010 2 J. Sch¨ utz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24

Background HDG-BDM method for Advection-Diffusion equations. ∇ · ( f ( u ) − f v ( u, ∇ u )) = 0 Discontinuous Galerkin for Advection; known to work well ∇ · f ( u ) = 0 BDM Mixed method for Diffusion; known to work well ∇ · f v ( u, σ ) = 0 σ = ∇ u Hybridization to reduce the global coupled degrees of freedom λ ≈ u | Γ oberl 1 Linear case : Proposed by H. Egger and J. Sch¨ 1 H. Egger and J. Sch¨ oberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010 2 J. Sch¨ utz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24

Background HDG-BDM method for Advection-Diffusion equations. ∇ · ( f ( u ) − f v ( u, ∇ u )) = 0 Discontinuous Galerkin for Advection; known to work well ∇ · f ( u ) = 0 BDM Mixed method for Diffusion; known to work well ∇ · f v ( u, σ ) = 0 σ = ∇ u Hybridization to reduce the global coupled degrees of freedom λ ≈ u | Γ oberl 1 Linear case : Proposed by H. Egger and J. Sch¨ Non-Linear case : Proposed by J. Sch¨ utz and G. May (Promising results for N-S equations 2 ) 1 H. Egger and J. Sch¨ oberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010 2 J. Sch¨ utz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24

Features of the HDG-BDM scheme Reduces to DG for pure advection, to BDM mixed for pure diffusion 3 B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004 4 N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24

Features of the HDG-BDM scheme Reduces to DG for pure advection, to BDM mixed for pure diffusion No additional parameter in the intermediate range 3 B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004 4 N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24

Features of the HDG-BDM scheme Reduces to DG for pure advection, to BDM mixed for pure diffusion No additional parameter in the intermediate range Using local solvers 3 to make it a system for λ 3 B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004 4 N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24

Features of the HDG-BDM scheme Reduces to DG for pure advection, to BDM mixed for pure diffusion No additional parameter in the intermediate range Using local solvers 3 to make it a system for λ Solution can be post-processed to get better convergence 3 B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004 4 N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24

Features of the HDG-BDM scheme Reduces to DG for pure advection, to BDM mixed for pure diffusion No additional parameter in the intermediate range Using local solvers 3 to make it a system for λ Solution can be post-processed to get better convergence It can be easily modified to the well known Hybridized Discontinuous Galerkin (HDG) scheme 4 3 B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004 4 N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24

Features of the HDG-BDM scheme Reduces to DG for pure advection, to BDM mixed for pure diffusion No additional parameter in the intermediate range Using local solvers 3 to make it a system for λ Solution can be post-processed to get better convergence It can be easily modified to the well known Hybridized Discontinuous Galerkin (HDG) scheme 4 It can be even mixed with the HDG scheme due to hybridization. 3 B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004 4 N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24

BDM Mixed Method for Diffusion Consider Laplace equation −∇ · ∇ u = S in Ω u = g in ∂ Ω Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24

BDM Mixed Method for Diffusion Consider Laplace equation −∇ · ∇ u = S in Ω u = g in ∂ Ω Introducing new variable, σ = ∇ u Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24

BDM Mixed Method for Diffusion Consider Laplace equation −∇ · ∇ u = S in Ω u = g in ∂ Ω Introducing new variable, σ = ∇ u σ = ∇ u in Ω −∇ · σ = S in Ω = in ∂ Ω u g Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24

BDM Mixed Method for Diffusion Consider Laplace equation −∇ · ∇ u = S in Ω u = g in ∂ Ω Introducing new variable, σ = ∇ u σ = ∇ u in Ω −∇ · σ = S in Ω = in ∂ Ω u g The solution spaces : u h ∈ V h , σ h ∈ ˜ H h Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24

BDM Mixed Method for Diffusion Consider Laplace equation −∇ · ∇ u = S in Ω u = g in ∂ Ω Introducing new variable, σ = ∇ u σ = ∇ u in Ω −∇ · σ = S in Ω = in ∂ Ω u g The solution spaces : u h ∈ V h , σ h ∈ ˜ H h { ϕ ∈ L 2 (Ω) : ϕ | Ω k ∈ P m − 1 (Ω k ) } := V h Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24

BDM Mixed Method for Diffusion Consider Laplace equation −∇ · ∇ u = S in Ω u = g in ∂ Ω Introducing new variable, σ = ∇ u σ = ∇ u in Ω −∇ · σ = S in Ω = in ∂ Ω u g The solution spaces : u h ∈ V h , σ h ∈ ˜ H h { ϕ ∈ L 2 (Ω) : ϕ | Ω k ∈ P m − 1 (Ω k ) } := V h ˜ τ | Ω k ∈ P m (Ω k ) × P m (Ω k ) } H h := { τ ∈ H ( div, Ω) : Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24

BDM Mixed Method for Diffusion Consider Laplace equation −∇ · ∇ u = S in Ω u = g in ∂ Ω Introducing new variable, σ = ∇ u σ = ∇ u in Ω −∇ · σ = S in Ω = in ∂ Ω u g The solution spaces : u h ∈ V h , σ h ∈ ˜ H h { ϕ ∈ L 2 (Ω) : ϕ | Ω k ∈ P m − 1 (Ω k ) } := V h ˜ τ | Ω k ∈ P m (Ω k ) × P m (Ω k ) } H h := { τ ∈ H ( div, Ω) : BDM Mixed method � � � ∀ τ ∈ ˜ σ h · τ + ( ∇ · τ ) u h − ( τ · n ) g = 0 H h Ω Ω ∂ Ω � � − ∇ · σ h ϕ = Sϕ ∀ ϕ ∈ V h Ω Ω Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24