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Space-Time Discontinuous Petrov-Galerkin Finite Elements for Transient Fluid Mechanics Truman Ellis a , Leszek Demkowicz b , Jesse Chan c , Nate Roberts a and Robert Moser b a Sandia Laboratories b ICES, University of Texas at Austin c Dept. of

  1. Locally Conservative DPG Local Conservation for Convection-Diffusion The local conservation law in convection diffusion is � � ˆ t = g , ∂ K K which is equivalent to having v K := { v , τ } = { 1 K , 0 } in the test space. In general, this is not satisfied by the optimal test functions. Following Moro et al 2 (also Chang and Nelson 3 ), we can enforce this condition with Lagrange multipliers: � � � L ( u h , λ ) = 1 � R − 1 � 2 V ( Bu h − l ) V − λ K � Bu h − l , v K � , � �� � 2 K � ˆ t , 1 K � ∂ K −� g , 1 K � K where λ = { λ 1 , · · · , λ N } . 2 D. Moro, N.C. Nguyen, and J. Peraire. ‘‘A Hybridized Discontinuous Petrov-Galerkin Scheme for Scalar Conservation Laws’’. In: Int. J. Num. Meth. Eng. (2011). 3 C.L. Chang and J.J. Nelson. ‘‘Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation’’. In: SIAM J. Num. Anal. 34 (1997), pp. 480–489. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 17 / 62

  2. Locally Conservative DPG Locally Conservative Saddle Point System Finding the critical points of L ( u , λ ) , we get the following equations. Locally Conservative Saddle Point System ∂ L ( u h , λ ) = b ( u h , R − 1 V B δ u h ) − l ( R − 1 V B δ u h ) ∂ u h � − λ K b ( δ u h , v K ) = 0 ∀ δ u h ∈ U h K ∂ L ( u h , λ ) = − b ( u h , v K ) + l ( v K ) = 0 ∀ K ∂λ K A few consequences: Minimization problem turns into a constrained minimization problem. Optimal test function are in the orthogonal complement of constants. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 18 / 62

  3. Locally Conservative DPG Optimal Test Functions For each u = { u , σ , ˆ u , ˆ t } ∈ U h , find v u = { v u , τ u } ∈ V such that ( v u , w ) V = b ( u , w ) ∀ w ∈ V where V becomes V p +∆ p in order to make this computationally tractable. We recently developed this modification to the robust test norm 4 which behaves better in the presence of singularities. Convection-Diffusion Test Norm � � � � 2 � � 1 1 � � � ( v , τ ) � 2 + �∇ · τ − β · ∇ v � 2 V , Ω h = √ ǫ, � � min � τ � � | K | + � β · ∇ v � 2 + ǫ �∇ v � 2 + � v � 2 � �� � No longer necessary 4 J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 19 / 62

  4. Locally Conservative DPG Optimal Test Functions For each u = { u , σ , ˆ u , ˆ t } ∈ U h , find v u = { v u , τ u } ∈ V such that ( v u , w ) V = b ( u , w ) ∀ w ∈ V where V becomes V p +∆ p in order to make this computationally tractable. We recently developed this modification to the robust test norm 4 which behaves better in the presence of singularities. Convection-Diffusion Test Norm � � � � 2 � � 1 1 � � � ( v , τ ) � 2 + �∇ · τ − β · ∇ v � 2 V , Ω h = √ ǫ, � � min � τ � � | K | � � 2 � + � β · ∇ v � 2 + ǫ �∇ v � 2 + 1 K v | K | � �� � Scaling term 4 J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 19 / 62

  5. Locally Conservative DPG Stability and Robustness Analysis 5 We follow Brezzi’s theory for an abstract mixed problem:   u ∈ U , p ∈ Q  a ( u , w ) + c ( p , w ) = l ( w ) ∀ w ∈ U   c ( q , u ) = g ( q ) ∀ q ∈ Q where a , c , l , g denote the appropriate bilinear and linear forms. a ( u , w ) = b ( u , R − 1 V B w ) = ( R − 1 V B u , R − 1 V B w ) V c ( p , w ) = � K λ K � ˆ t , 1 K � ∂ K Locally conservative DPG satisfies inf-sup and inf-sup in kernel conditions. Robustness is proved by switching to energy norm in Brezzi analysis. 5 T.E. Ellis, L.F. Demkowicz, and J.L. Chan. ‘‘Locally Conservative Discontinuous Petrov-Galerkin Finite Elements For Fluid Problems’’. In: Comp. Math. Appl. 68.11 (2014), pp. 1530 –1549. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 20 / 62

  6. Numerical Experiments Stokes Flow Around a Cylinder Horizontal Velocity 1 Refinement 1 Refinement 6 Refinements 6 Refinements Nonconservative Conservative T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 21 / 62

  7. Numerical Experiments Stokes Flow Around a Cylinder Percent Mass Loss at x = [ − 1 , − 0 . 95 , 0 , 0 . 95 , 3 ] 1e 8 100 2.5 1844 DOFs 1844 DOFs 4526 DOFs 3980 DOFs 2.0 17552 DOFs 12380 DOFs 80 66056 DOFs 1.5 33272 DOFs 92084 DOFs 75326 DOFs 1.0 195674 DOFs 183326 DOFs percent mass loss percent mass loss 60 339545 DOFs 358739 DOFs 0.5 0.0 40 0.5 1.0 20 1.5 0 2.0 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x location x location Nonconservative Conservative T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 21 / 62

  8. Numerical Experiments Stokes Flow Over a Backward Facing Step Horizontal Velocity Initial Mesh Initial Mesh 8 Refinements 8 Refinements Nonconservative Conservative T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 22 / 62

  9. Numerical Experiments Stokes Flow Over a Backward Facing Step Percent Mass Loss at x = [ 0 , 0 . 5 , · · · , 9 . 5 , 10 ] 1e 12 40 2.0 8110 DOFs 8110 DOFs 12084 DOFs 10107 DOFs 35 14760 DOFs 12104 DOFs 16757 DOFs 14101 DOFs 30 1.5 18754 DOFs 16098 DOFs 20751 DOFs 18095 DOFs percent mass loss percent mass loss 25 22748 DOFs 20092 DOFs 24745 DOFs 22089 DOFs 20 1.0 26742 DOFs 24086 DOFs 15 10 0.5 5 0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 x location x location Nonconservative Conservative T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 22 / 62

  10. Table of Contents Motivation: Automating Scientific Computing 1 DPG: A Framework for Computational Mechanics 2 Locally Conservative DPG 3 Space-Time Convection-Diffusion 4 Space-Time Incompressible Navier-Stokes 5 Space-Time Compressible Navier-Stokes 6 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 23 / 62

  11. Space-Time DPG Extending DPG to Transient Problems Time stepping techniques are not ideally suited to highly adaptive grids Space-time FEM proposed as a solution ✦ Unified treatment of space and time ✦ Local space-time adaptivity (local time stepping) ✦ Parallel-in-time integration (space-time multigrid) ✪ Spatially stable FEM methods may not be stable in space-time ✪ Need to support higher dimensional problems DPG provides necessary stability and adaptivity Courtesy of XBraid by LLNL T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 24 / 62

  12. Space-Time DPG for Convection-Diffusion Space-Time Divergence Form Equation is parabolic in space-time. ∂ u ∂ t + β · ∇ u − ǫ ∆ u = f This is just a composition of a constitutive law and conservation of mass. σ − ǫ ∇ u = 0 ∂ u ∂ t + ∇ · ( β u − σ ) = f We can rewrite this in terms of a space-time divergence. 1 ǫ σ − ∇ u = 0 � � β u − σ ∇ xt · = f u T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 25 / 62

  13. Space-Time DPG for Convection-Diffusion Ultra-Weak Formulation with Discontinuous Test Functions Multiply by test function and integrate by parts over space-time element K. � � 1 ǫ σ , τ + ( u , ∇ · τ ) K − � ˆ u , τ · n x � ∂ K = 0 K �� � � β u − σ + � ˆ − , ∇ xt v t , v � ∂ K = f u K where Support of Trace Variables ˆ u := tr ( u ) ˆ t := tr ( β u − σ ) · n x ˆ ˆ ˆ t t t ˆ ˆ ˆ ˆ u t u t + tr ( u ) · n t ˆ t ˆ ˆ ˆ ˆ u t u t Trace ˆ u defined on spatial ˆ ˆ u ˆ t u ˆ t boundaries t ˆ ˆ ˆ t t t Flux ˆ t defined on all x boundaries T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 26 / 62

  14. Space-Time Convection-Diffusion L 2 Equivalent Norms Analytical Solution Bilinear form with group variables: e − lt ( e λ 1 ( x − 1 ) − e λ 2 ( x − 1 ) ) u ) , v ) = ( u , A ∗ b (( u , ˆ h v ) L 2 (Ω h ) + � � u , [ [ v ] ] � Γ h � ǫ x � 1 + 1 − e For conforming v ∗ satisfying A ∗ v ∗ = u L 2 (Ω h ) = b ( u , v ∗ ) = b ( u , v ∗ ) � v ∗ � V � u � 2 � v ∗ � V | b ( u , v ∗ ) | � v ∗ � = � u � E � v ∗ � V ≤ sup � v ∗ � v ∗ � = 0 Necessary robustness condition: � v ∗ � V � � u � L 2 (Ω h ) ⇒ � u � L 2 (Ω h ) � � u � E T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 27 / 62

  15. Space-Time Convection-Diffusion L 2 Equivalent Norms A norm should be: bounded by � u � L 2 (Ω h ) , have good conditioning, not produce boundary layers in the optimal test function. 10 -1 10 0 10 -1 10 -2 10 -2 10 -3 10 -3 Error Error 10 -4 10 -4 10 -5 10 -5 ǫ =10 − 2 L 2 Error ǫ =10 − 6 L 2 Error ǫ =10 − 2 L 2 Error ǫ =10 − 6 L 2 Error 10 -6 ǫ =10 − 2 V ∗ Error ǫ =10 − 6 V ∗ Error ǫ =10 − 2 V ∗ Error ǫ =10 − 6 V ∗ Error 10 -6 ǫ =10 − 4 L 2 Error ǫ =10 − 8 L 2 Error ǫ =10 − 4 L 2 Error ǫ =10 − 8 L 2 Error ǫ =10 − 4 V ∗ Error ǫ =10 − 8 V ∗ Error ǫ =10 − 4 V ∗ Error ǫ =10 − 8 V ∗ Error 10 -7 10 -7 10 2 10 3 10 4 10 5 10 6 10 7 10 2 10 3 10 4 10 5 10 6 10 7 � DOFs � � DOFs � 2 2 � ( v , τ ) � 2 = � � � ( v , τ ) � 2 = � � � ∇ · τ − ˜ � ∇ · τ − ˜ β · ∇ xt v β · ∇ xt v � � � � � � 2 � � h 2 , 1 1 1 + � v � 2 + � τ � 2 � τ � 2 � � + min + ǫ τ + ∇ v � � ǫ + ǫ �∇ v � 2 + � β · ∇ v � 2 + � v � 2 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 28 / 62

  16. Steady Convection-Diffusion Ideal Optimal Shape Functions Graph Norm Coupled Robust Norm v v 0.06 0.0015 0.04 0.0010 0.02 0.0005 0.00 0.0000 0.02 0.0005 0.04 0.0010 0.06 0.08 0.0015 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 τ τ 0.08 0.10 0.07 0.08 0.06 0.06 0.05 0.04 0.04 0.03 0.02 0.02 0.00 0.01 0.02 0.00 0.01 0.04 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 29 / 62

  17. Steady Convection-Diffusion Approximated ( p = 3 ) Optimal Shape Functions Graph Norm Coupled Robust Norm v v 0.06 0.0015 0.04 0.0010 0.02 0.0005 0.00 0.0000 0.02 0.0005 0.04 0.0010 0.06 0.08 0.0015 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 τ τ 0.08 0.10 0.07 0.08 0.06 0.06 0.05 0.04 0.04 0.03 0.02 0.02 0.00 0.01 0.02 0.00 0.01 0.04 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 30 / 62

  18. Steady Convection-Diffusion Two Robust Norms for Steady Convection-Diffusion The following norms are robust for steady convection-diffusion. The robust norm was derived in 6 : � ǫ � � ( v , τ ) � 2 = � β · ∇ v � 2 + ǫ �∇ v � 2 + min � v � 2 h 2 , 1 � � h 2 , 1 1 + �∇ · τ � 2 + min � τ � 2 . ǫ The case for the coupled robust norm was made in 7 : � ǫ � � ( v , τ ) � 2 = � β · ∇ v � 2 + ǫ �∇ v � 2 + min � v � 2 h 2 , 1 � � h 2 , 1 1 + �∇ · τ − β · ∇ v � 2 + min � τ � 2 . ǫ 6 J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795. 7 J.L. Chan. ‘‘A DPG Method for Convection-Diffusion Problems’’. PhD thesis. University of Texas at Austin, 2013. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 31 / 62

  19. Space-Time Convection-Diffusion Two Robust Norms for Transient Convection-Diffusion � � � � ∇ v β Let ˜ β := and ∇ xt v := . ∂ v 1 ∂ t The following norms are robust for space-time convection-diffusion. Robust Norm: � � � ǫ � � � 2 � ( v , τ ) � 2 = + ǫ �∇ v � 2 + min � ˜ � v � 2 β · ∇ xt v h 2 , 1 � � � h 2 , 1 1 + �∇ · τ � 2 + min � τ � 2 . ǫ Coupled Robust Norm � � � ǫ � � � 2 � ( v , τ ) � 2 = + ǫ �∇ v � 2 + min � ˜ � v � 2 β · ∇ xt v � h 2 , 1 � � � � 2 h 2 , 1 1 � � � τ � 2 . � ∇ · τ − ˜ + β · ∇ xt v � + min ǫ T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 32 / 62

  20. Robust Norms for Transient Convection-Diffusion Adjoint Operator Consider the problem with homogeneous boundary conditions 1 ǫ σ − ∇ u = 0 ˜ β · ∇ xt u − ∇ · σ = f β n u − ǫ∂ u ∂ n = 0 on Γ − u = 0 on Γ + u = u 0 on Γ 0 . The adjoint operator A ∗ is given by � � 1 A ∗ ( v , τ ) = ǫ τ + ∇ v , − ˜ β · ∇ xt v + ∇ · τ . T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 33 / 62

  21. Robust Norms for Transient Convection-Diffusion Controlling Different Field Variables We decompose the continuous adjoint problem A ∗ ( τ , v ) = ( σ , u ) into Continuous part with forcing u 1 τ 1 · n x = 0 on Γ − ǫ τ 1 + ∇ v 1 = 0 v 1 = 0 on Γ + − ˜ β · ∇ xt v 1 + ∇ · τ 1 = u v 1 = 0 on Γ T Continuous part with forcing σ 1 τ 2 · n x = 0 on Γ − ǫ τ 2 + ∇ v 2 = σ v 2 = 0 on Γ + − ˜ β · ∇ xt v 2 + ∇ · τ 2 = 0 v 2 = 0 on Γ T T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 34 / 62

  22. Robust Norms for Transient Convection-Diffusion Proved Bounds at Our Disposal Proofs of these lemmas can be found in 8 . Lemma (1) If ∇ · β = 0 , we can bound � v � 2 + ǫ �∇ v � 2 ≤ � u � 2 + ǫ � σ � 2 where v = v 1 + v 2 . Lemma (2) � � � ∇ β − 1 � 2 ∇ · β I L ∞ ≤ C β , we can bound If � � � � � ˜ β · ∇ xt v 1 � � � u � . 8 T. Ellis, J. Chan, and L. Demkowicz. ‘‘Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations,x Eds. G.R. Barrenechea et al.’’ In: vol. 114. Lecture Notes in Computational Science and Engineering. in print, see also ICES Report 2015/21. Springer, 2016. Chap. Robust DPG Methods for Transient Convection-Diffusion. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 35 / 62

  23. Robust Norms for Transient Convection-Diffusion Control of u Bound on � ( v 1 , τ 1 ) � � � � � � ˜ Lemma (2) ⇒ β · ∇ xt v 1 � � � u � � � � � � ˜ Lemma (2) ⇒ �∇ · τ 1 � ≤ � u � + β · ∇ xt v 1 � � 2 � u � � � � � � ∇ · τ 1 − ˜ Lemma (2) ⇒ β · ∇ xt v 1 � = � u � � v 1 � 2 + ǫ �∇ v 1 � 2 ≤ � u � 2 Lemma (1) ⇒ 1 Lemma (1) ⇒ ǫ � τ 1 � = ǫ �∇ v 1 � ≤ � u � We can guarantee robust control � ( u , 0 ) � L 2 (Ω h ) � � ( u , σ ) � E . T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 36 / 62

  24. Robust Norms for Transient Convection-Diffusion Control of σ Bound on � ( v 2 , τ 2 ) � � � � � � ∇ · τ 2 − ˜ Definition ⇒ β · ∇ xt v 2 � = 0 ≤ � σ � � v 2 � 2 + ǫ �∇ v 2 � 2 ≤ ǫ � σ � 2 Lemma (1) ⇒ 1 Lemma (1) ⇒ ǫ � τ 2 � = � σ � + ǫ �∇ v 2 � = ( 1 + ǫ ) � σ � � � � � � ˜ β · ∇ xt v 2 � or �∇ · τ 2 � . We have not been able to prove bounds on We can not guarantee robust control � ( 0 , σ ) � L 2 (Ω h ) ✓ � � ( u , σ ) � E . ✓ T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 37 / 62

  25. Robust Norms for Transient Convection-Diffusion Transient Analytical Solution Transient impulse decays to Eriksson-Johnson steady state solution. u = exp ( − lt ) [ exp ( λ 1 x ) − exp ( λ 2 x )] + cos ( π y ) exp ( s 1 x ) − exp ( r 1 x ) exp ( − s 1 ) − exp ( − r 1 ) t = 0 . 5 t = 0 . 0 t = 1 . 0 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 38 / 62

  26. Robust Norms for Transient Convection-Diffusion Robust Convergence to Analytical Solution 10 0 10 0 L 2 p =1 Robust L 2 p =1 CoupledRobust V ∗ p =1 Robust V ∗ p =1 CoupledRobust L 2 p =2 Robust L 2 p =2 CoupledRobust 10 -1 V ∗ p =2 Robust V ∗ p =2 CoupledRobust L 2 p =4 Robust L 2 p =4 CoupledRobust V ∗ p =4 Robust V ∗ p =4 CoupledRobust 10 -2 Error Error 10 -1 10 -3 L 2 p =1 Robust L 2 p =1 CoupledRobust V ∗ p =1 Robust V ∗ p =1 CoupledRobust L 2 p =2 Robust L 2 p =2 CoupledRobust 10 -4 V ∗ p =2 Robust V ∗ p =2 CoupledRobust L 2 p =4 Robust L 2 p =4 CoupledRobust V ∗ p =4 Robust V ∗ p =4 CoupledRobust 10 -5 10 -2 10 4 10 5 10 7 10 4 10 5 10 7 10 2 10 3 10 6 10 2 10 3 10 6 DOFs DOFs ǫ = 10 − 2 ǫ = 10 − 4 10 0 10 0 10 -1 10 -1 Error Error L 2 p =1 Robust L 2 p =1 CoupledRobust L 2 p =1 Robust L 2 p =1 CoupledRobust 10 -2 10 -2 V ∗ p =1 Robust V ∗ p =1 CoupledRobust V ∗ p =1 Robust V ∗ p =1 CoupledRobust L 2 p =2 Robust L 2 p =2 CoupledRobust L 2 p =2 Robust L 2 p =2 CoupledRobust V ∗ p =2 Robust V ∗ p =2 CoupledRobust V ∗ p =2 Robust V ∗ p =2 CoupledRobust L 2 p =4 Robust L 2 p =4 CoupledRobust L 2 p =4 Robust L 2 p =4 CoupledRobust V ∗ p =4 Robust V ∗ p =4 CoupledRobust V ∗ p =4 Robust V ∗ p =4 CoupledRobust 10 -3 10 -3 10 2 10 3 10 4 10 5 10 6 10 7 10 2 10 3 10 4 10 5 10 6 10 7 DOFs DOFs ǫ = 10 − 6 ǫ = 10 − 8 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 39 / 62

  27. Table of Contents Motivation: Automating Scientific Computing 1 DPG: A Framework for Computational Mechanics 2 Locally Conservative DPG 3 Space-Time Convection-Diffusion 4 Space-Time Incompressible Navier-Stokes 5 Space-Time Compressible Navier-Stokes 6 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 40 / 62

  28. Space-Time Incompressible Navier-Stokes Nonlinear Form Space-time divergence form: 1 ν D − ∇ u = 0 � � u ⊗ u − D + p I ∇ xt · = f u ∇ · u = 0 Multiply by S ∈ H ( div , Q ) , v ∈ H 1 xt ( Q ) , q ∈ H 1 ( Q ) , and integrate by parts: � � 1 + ( u , ∇ · S ) − � ˆ u , S · n x � = 0 ν D , S �� � � u ⊗ u − D + p I + � ˆ − , ∇ xt v t , v � = ( f , v ) u − ( u , ∇ q ) + � ˆ u · n , q � = 0 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 41 / 62

  29. Space-Time Incompressible Navier-Stokes Robust Norms Recall the adjoint and robust norm for convection-diffusion: ( σ , 1 ǫ τ + ∇ v ) + ( u , ∇ · τ − β · ∇ v − ∂ v ∂ t ) � � � ǫ � � � 2 � β · ∇ v + ∂ v � ( v , τ ) � 2 � � + ǫ �∇ v � 2 � v � 2 V , K := K + min h 2 , 1 � K ∂ t K � � 1 ǫ , 1 + �∇ · τ � 2 � τ � 2 K + min K h 2 For incompressible Navier-Stokes the adjoint comes from: � � � � �� u · ( ∇ v ) T + ∂ v ∆ D , 1 ν S + ∇ v + ∆ u , ∇ · S − ∇ q − u · ∇ v + ˜ ˜ ∂ t + ( p , −∇ · v ) T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 42 / 62

  30. Space-Time Incompressible Navier-Stokes Norms for Navier-Stokes come from analogy Convection-Diffusion Navier-Stokes ǫ → ν τ → S ∇ v → ∇ v ∇ · τ → ∇ · S − ∇ q u · ( ∇ v ) T + ∂ v β · ∇ v + ∂ v u · ∇ v + ˜ ˜ → ∂ t ∂ t � � 2 � � u · ( ∇ v ) T + ∂ v � � � ( v , D , q ) � 2 + ν �∇ v � 2 V , K := � ˜ u · ∇ v + ˜ � K ∂ t K � ν � Robust norm: � v � 2 K + �∇ · S − ∇ q � 2 + min h 2 , 1 K � � 1 ν , 1 � S � 2 K + �∇ · v � 2 K + � q � 2 + min K h 2 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 43 / 62

  31. Space-Time Incompressible Navier-Stokes Taylor-Green Vortex � � Coupled Robust Norm 10 0 sin x cos y u = e − 2 Re t − cos x sin y 10 -1 10 -2 Error 10 -3 10 -4 L 2 Re =10 1 p =1 V ∗ Re =10 1 p =1 L 2 Re =10 1 p =2 V ∗ Re =10 1 p =2 L 2 Re =10 1 p =4 V ∗ Re =10 1 p =4 10 -5 L 2 Re =10 3 p =1 V ∗ Re =10 3 p =1 L 2 Re =10 3 p =2 V ∗ Re =10 3 p =2 L 2 Re =10 3 p =4 V ∗ Re =10 3 p =4 L 2 Re =10 5 p =1 V ∗ Re =10 5 p =1 L 2 Re =10 5 p =2 V ∗ Re =10 5 p =2 L 2 Re =10 5 p =4 V ∗ Re =10 5 p =4 10 -6 10 3 10 4 10 5 10 6 10 7 Degrees of Freedom T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 44 / 62

  32. Space-Time Incompressible Navier-Stokes Flow Over a Cylinder, Initial Mesh T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 45 / 62

  33. Space-Time Incompressible Navier-Stokes Flow Over a Cylinder, 4 Refinements T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 45 / 62

  34. Space-Time Incompressible Navier-Stokes Solve Times and Strong Scaling Transient Flow Over a Cylinder 1 Node 4 Nodes 32 Nodes Ref Elems DOFs Time Time Scaling vs 1 Time Scaling vs 4 0 80 31,304 1,772 453 3.91 451 1.01 1 605 225,908 8,190 3,574 2.29 717 4.98 2 3,013 1,081,598 32,008 12,076 2.65 2,648 4.56 3 9,726 3,429,384 28,744 6,319 4.54 4 11,742 4,144,674 8,510 Computations on Lonestar, 1 node = 24 processors 32,008 seconds = 8.8 hours 28,744 seconds = 8.0 hours 8,510 seconds = 2.4 hours T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 46 / 62

  35. Table of Contents Motivation: Automating Scientific Computing 1 DPG: A Framework for Computational Mechanics 2 Locally Conservative DPG 3 Space-Time Convection-Diffusion 4 Space-Time Incompressible Navier-Stokes 5 Space-Time Compressible Navier-Stokes 6 T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 47 / 62

  36. Space-Time Compressible Navier-Stokes First Order System with Primitive Variables Assuming Stokes hypothesis, ideal gas law, and constant viscosity: 1 µ D − ∇ u = 0 Pr C p µ q + ∇ T = 0 � � ρ u ∇ xt · = f c ρ � � � � D + D T − 2 ρ u ⊗ u + ρ RT I − 3 tr ( D ) I ∇ xt · = f m ρ u � � � � � � D + D T − 2 C v T + 1 2 u · u + ρ RT u + q − u · 3 tr ( D ) I ρ u ∇ xt · � � = f e C v T + 1 ρ 2 u · u T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 48 / 62

  37. Space-Time Compressible Navier-Stokes Compact Notation Conserved quantities Euler fluxes C c := ρ F c := ρ u C m := ρ u F m := ρ u ⊗ u + ρ RT I � � C e := ρ ( C v T + 1 C v T + 1 2 u · u ) F e := ρ u 2 u · u + ρ RT u Viscous fluxes Viscous variables K c := 0 M D := D K m := D + D T − 2 M q := Pr 3 tr ( D ) I q C p � � D + D T − 2 G D := 2 u K e := − q + u · 3 tr ( D ) I G q := − T Use change of variables to get conservation or entropy variables. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 49 / 62

  38. Space-Time Compressible Navier-Stokes Conservation Variables (Popular for Time-Stepping) Change of variables: ρ = ρ m = ρ u � � C v T + 1 E = ρ 2 u · u Euler fluxes: F c c = m � � m = m ⊗ m E − m · m F c + ( γ − 1 ) I ρ 2 ρ ρ − ( γ − 1 ) m · m e = γ E m F c 2 ρ 2 m T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 50 / 62

  39. Space-Time Compressible Navier-Stokes Entropy Variables (Symmetrize the Bubnov-Galerkin Stiffness Matrix) Change of variables: � � �� ( γ − 1 )( E − 1 2 ρ m · m ) 1 − E + ( E − 2 ρ m · m ) γ + 1 − ln ρ γ V c = 1 E − 2 ρ m · m m V m = 1 E − 2 ρ m · m − ρ V e = 1 E − 2 ρ m · m Euler fluxes: � � � γ − 1 � 1 1 − γ + V c − 2 V e V m · V m γ − 1 F e c = V m exp ( − V e ) γ γ − 1 � � � � γ − 1 � � 1 1 − γ + V c − 2 V e V m · V m − V m ⊗ V m γ − 1 F e m = + ( γ − 1 ) I exp ( − V e ) γ γ − 1 V e � � � γ − 1 � � � 1 1 − γ + V c − 2 V e V m · V m γ − 1 V m 1 F e e = V m · V m − γ exp ( − V e ) γ γ − 1 V e 2 V e T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 51 / 62

  40. Space-Time Compressible Navier-Stokes Define Group Variables Group terms Group variables C := { C c , C m , C e } W := { ρ , u , T } � � ˆ u , − ˆ F := { F c , F m , F e } 2 ˆ W := T K := { K c , K m , K e } Σ := { D , q } M := { M D , M q } ˆ t := { ˆ t e , ˆ t m , , ˆ t e } G := { G D , G q } Ψ := { S , τ } f := { f c , f m , f e } V := { v c , v m , , v e } Navier-Stokes variational formulation is � � � ˆ � 1 µ M , Ψ + ( G , ∇ · Ψ) − W , Ψ · n x = 0 �� � � F − K − , ∇ xt V + � ˆ t , V � = ( f , V ) C T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 52 / 62

  41. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 1 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  42. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 2 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  43. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 3 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  44. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 4 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  45. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 5 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  46. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 6 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  47. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 7 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  48. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 8 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  49. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 9 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  50. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 10 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  51. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 11 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  52. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 12 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  53. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 13 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  54. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 14 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  55. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Mesh 15 Primitive Variables Conservation Variables Entropy Variables T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 53 / 62

  56. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 0 Refinements Velocity 0 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 0 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  57. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 1 Refinements Velocity 1 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 1 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  58. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 2 Refinements Velocity 2 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 2 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  59. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 3 Refinements Velocity 3 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 3 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  60. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 4 Refinements Velocity 4 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 4 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  61. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 5 Refinements Velocity 5 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 5 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  62. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 6 Refinements Velocity 6 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 6 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  63. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 7 Refinements Velocity 7 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 7 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  64. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 8 Refinements Velocity 8 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 8 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  65. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 9 Refinements Velocity 9 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 9 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  66. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 10 Refinements Velocity 10 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 10 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  67. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 11 Refinements Velocity 11 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 11 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  68. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 12 Refinements Velocity 12 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 12 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  69. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 13 Refinements Velocity 13 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 13 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  70. Space-Time Compressible Navier-Stokes Sod Shock Tube with µ = 10 − 5 Density 14 Refinements Velocity 14 Refinements 1.2 Exact Exact Primal 1.0 1.0 Conservation Primal Entropy 0.8 Conservation 0.6 velocity Entropy 0.8 0.4 0.2 0.0 density 0.6 0.2 0.0 0.2 0.4 0.6 0.8 1.0 x Pressure 14 Refinements Exact 1.0 Primal 0.4 Conservation Entropy 0.8 pressure 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 54 / 62

  71. Space-Time Compressible Navier-Stokes Entropy Scaled Test Norms Let W , U , and V be the set of primitive, conservation, and entropy variables. The entropy function H = − ρ log ( p ρ − γ ) provides a natural residual for the Navier-Stokes equations. A 0 = H , UU = V , U is known as the symmetrizer and ( U , A 0 U ) provides a natural metric for the linearized Euler equations. In primitive variables: ( U , A 0 U ) = ( U , W W , V , U U , W W ) = ( W , U T , W V , U U , W W ) = ( W , A 0 ( W ) W ) where   γ − 1 0 0 ρ   A 0 ( W ) = U T ρ , W V , U U , W =   0 0 C v T ρ 0 0 T 2 ( W , A 0 W ) has units of density. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 55 / 62

  72. Space-Time Compressible Navier-Stokes Entropy Scaled Test Norms Bilinear form with group variables: �� � � � ˆ � W , ˆ = ( W , A ∗ , v h v ) L 2 (Ω h ) + W , [ [ v ] ] b W Γ h For conforming v ∗ satisfying A ∗ v ∗ = A 0 W � � � � 2 = b ( W , v ∗ ) = b ( W , v ∗ ) 1 � � � v ∗ � V � A 0 W 2 � � v ∗ � V | b ( W , v ∗ ) | � v ∗ � = � W � E � v ∗ � V . ≤ sup � v ∗ � v ∗ � = 0 Necessary robustness condition: � � � � 1 � v ∗ � V � � � � A 0 W 2 � L 2 (Ω h ) � � � � 1 � � ⇒ 2 � � W � E � A 0 W � L 2 (Ω h ) T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 56 / 62

  73. Space-Time Compressible Navier-Stokes Entropy Scaled Test Norms We load our adjoint equations with A 0 W : 1 µ M ∗ (Ψ) + K ∗ ( ∇ V ) = 0 � � F ∗ ( ∇ xt V ) + G ∗ ( ∇ Ψ) = A 0 W − C ∗ This leads to the entropy scaled robust norm: � � � � � � 2 � � 2 − 1 − 1 0 ( F ∗ + C ∗ ) � ( V , Ψ) � 2 � � � K ∗ � V , K := 2 + µ 2 � A � A � � 0 K K � � � � � � µ 2 2 � � � � − 1 − 1 � � � � G ∗ + min h 2 , 1 2 + 2 � A V � A � � 0 0 K K � � � � � � 2 µ, 1 1 − 1 � M ∗ � + min � A 2 � 0 h 2 K T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 57 / 62

  74. Space-Time Compressible Navier-Stokes Entropy Scaled Test Norms We load our adjoint equations with A 0 W : 1 µ M ∗ (Ψ) + K ∗ ( ∇ V ) = 0 � � F ∗ ( ∇ xt V ) + G ∗ ( ∇ Ψ) = A 0 W − C ∗ This leads to the entropy scaled robust norm: � � � � � � 2 � � 2 − 1 − 1 0 ( F ∗ + C ∗ ) � ( V , Ψ) � 2 � � � K ∗ � V , K := 2 + µ 2 � A � A � � 0 K K � � � � � � µ 2 2 � � � � − 1 − 1 � � � � G ∗ + min h 2 , 1 2 + 2 � A V � A � � 0 0 K K � � � � � � 2 µ, 1 1 − 1 � M ∗ � + min � A 2 � 0 h 2 K Numerical results were disappointing. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 57 / 62

  75. Space-Time Compressible Navier-Stokes Noh Implosion with Primitive Variables, Robust Norm, µ = 10 − 3 After 5 refinements 5.0 Exact Initial 4 Elements 4.5 5 Refinements 4.0 10 Refinements 3.5 3.0 density After 10 refinements 2.5 2.0 1.5 1.0 0.5 1.0 0.8 0.6 0.4 0.2 0.0 x T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 58 / 62

  76. Space-Time Compressible Navier-Stokes Piston with µ = 10 − 2 √ ˆ t c = 2 ( − ρ u + ρ ) ˆ u = 1 √ 2 ( − ρ u 2 − ρ RT + ρ u ) ˆ t m = ˆ t c = 0 √ 2 ( − ρ u ( C v T + 1 2 u 2 ) − u ρ RT + ρ ( C v T + 1 ˆ t m − ˆ t e = 0 2 u 2 )) ˆ t e = Density Velocity T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 59 / 62

  77. Space-Time Compressible Navier-Stokes Piston with µ = 10 − 2 Mesh after 8 adaptive refinements T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 60 / 62

  78. Future Directions Improve performance: line smoothing for multigrid Shock capturing: DPG makes no promises when it comes to Gibbs phenomenon Non-Hilbert DPG: L 1 is known to limit oscillations Anisotropic refinements: necessary for time slabs More extensive 2D results: shedding vortex problems, 2D Incompressible Flow Over a Cylinder shock problems 3D results: will not be cheap T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 61 / 62

  79. Thank You! Recommended References ◮ J.L. Chan. ‘‘A DPG Method for Convection-Diffusion Problems’’. PhD thesis. University of Texas at Austin, 2013. ◮ D. Moro, N.C. Nguyen, and J. Peraire. ‘‘A Hybridized Discontinuous Petrov-Galerkin Scheme for Scalar Conservation Laws’’. In: Int. J. Num. Meth. Eng. (2011). ◮ C.L. Chang and J.J. Nelson. ‘‘Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation’’. In: SIAM J. Num. Anal. 34 (1997), pp. 480–489. ◮ J. Chan et al. ‘‘A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms’’. In: Comp. Math. Appl. 67.4 (2014), pp. 771 –795. ◮ T.E. Ellis, L.F. Demkowicz, and J.L. Chan. ‘‘Locally Conservative Discontinuous Petrov-Galerkin Finite Elements For Fluid Problems’’. In: Comp. Math. Appl. 68.11 (2014), pp. 1530 –1549. ◮ T. Ellis, J. Chan, and L. Demkowicz. ‘‘Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations,x Eds. G.R. Barrenechea et al.’’ In: vol. 114. Lecture Notes in Computational Science and Engineering. in print, see also ICES Report 2015/21. Springer, 2016. Chap. Robust DPG Methods for Transient Convection-Diffusion. ◮ L.F. Demkowicz and J. Gopalakrishnan. ‘‘Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations (eds. X. Feng, O. Karakashian, Y. Xing)’’. In: vol. 157. IMA Volumes in Mathematics and its Applications, 2014. Chap. An Overview of the DPG Method, pp. 149–180. ◮ N.V. Roberts. ‘‘Camellia: A Software Framework for Discontinuous Petrov-Galerkin Methods’’. In: Comp. Math. Appl. 68.11 (2014), pp. 1581 –1604. ◮ L.F. Demkowicz and N. Heuer. ‘‘Robust DPG Method for Convection-Dominated Diffusion Problems’’. In: SIAM J. Numer. Anal. 51.5 (2013), pp. 1514–2537. ◮ N. Roberts, T. Bui-Thanh, and L. Demkowicz. ‘‘The DPG method for the Stokes problem’’. In: Comp. Math. Appl. 67.4 (2014), pp. 966 –995. T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 62 / 62

  80. Scaling Issues Multigrid and Convection-Diffusion Convection-diffusion, 0 10 ǫ = 10 − 2 , 64 × 64 mesh -1 10 Conjugate gradient Geometric multigrid -2 10 residual preconditioner -3 Multiplicative V-cycle 10 Overlapping additive -4 10 Schwarz smoother Hierarchy of p − coarsening -5 10 0 500 1000 1500 2000 iteration followed by h − coarsening T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 1 / 14

  81. Scaling Issues Incompressible Flow Over a Cylinder, Initial Mesh T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 2 / 14

  82. Scaling Issues Incompressible Flow Over a Cylinder, 4 Refinements T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 2 / 14

  83. Scaling Issues Solve Times and Strong Scaling Transient Flow Over a Cylinder 1 Node 4 Nodes 32 Nodes Ref Elems DOFs Time Time Scaling vs 1 Time Scaling vs 4 0 80 31,304 1,772 453 3.91 451 1.01 1 605 225,908 8,190 3,574 2.29 717 4.98 2 3,013 1,081,598 32,008 12,076 2.65 2,648 4.56 3 9,726 3,429,384 28,744 6,319 4.54 4 11,742 4,144,674 8,510 Computations on Lonestar, 1 node = 24 processors 32,008 seconds = 8.8 hours 28,744 seconds = 8.0 hours 8,510 seconds = 2.4 hours T. Ellis, L. D., J. Chan, N, Roberts, R. Moser Space-Time DPG for Fluid Mechanics Nov. 7,, 2016 3 / 14

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