New DPG techniques for designing numerical schemes

Transcription

1 New DPG techniques for designing numerical schemes Jay Gopalakrishnan University of Florida Collaborator: Leszek Demkowicz October 2009 Massachusetts Institute of Technology, Boston Thanks: NSF Jay Gopalakrishnan 1/31

2 New DPG techniques for designing numerical schemes Jay Gopalakrishnan University of Florida Collaborator: Leszek Demkowicz October 2009 Massachusetts Institute of Technology, Boston Thanks: NSF Jay Gopalakrishnan 1/31

3 The philosophy The new: Discontinuous Petrov-Galerkin (DPG) methods Remarkable stability (through natural test space design) The old: DG methods (upwind stabilization, or stability by penalty parameters) SUPG methods (stability through artificial streamline diffusion) Jay Gopalakrishnan 2/31

4 Outline 1 How does the new compare to the old? Sample comparisons between DPG and DG results. 2 Elements of design of schemes. The example of simple 1D transport equation. 3 DPG method for the transport equation. Extension of the 1D idea to 2D. The spectral DPG method. The composite DPG method on a mesh. 4 Extensions. The DPG-X method. Optimal test functions. hp-results. A method for all seasons? Jay Gopalakrishnan 3/31

5 Comparison: 1D, 1 element case 0 The spectral DG solutions u(x) Exact solution 2.5 p=1 p=3 p= x Experiment: Solve 1D transport equation using DG and DPG on one element. Exact solution has a sharp layer at x = 1. Jay Gopalakrishnan 4/31

6 Comparison: 1D, 1 element case 0 The spectral DPG solutions u(x) Exact solution 2.5 p=1 p=3 p= x Experiment: Solve 1D transport equation using DG and DPG on one element. Exact solution has a sharp layer at x = 1. DPG solutions oscillate an order of magnitude less. Jay Gopalakrishnan 4/31

7 Comparison: Crosswind diffusion Pure transport should not diffuse materials crosswind. But most numerical methods do. Experiment: Use DG and DPG for simulating vertically upward transport of linearly varying density from the bottom of the unit square. Jay Gopalakrishnan 5/31

8 Comparison: Crosswind diffusion DPG doesn t. DG has crosswind diffusion. Experiment: Use DG and DPG for simulating vertically upward transport of linearly varying density from the bottom of the unit square. Jay Gopalakrishnan 5/31

9 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed Jay Gopalakrishnan 6/31

10 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate Jay Gopalakrishnan 6/31

11 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG Jay Gopalakrishnan 6/31

12 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate Jay Gopalakrishnan 6/31

13 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes Jay Gopalakrishnan 6/31

14 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes [Peterson 1991]: On general meshes O(h p+1/2 ) is the best possible Jay Gopalakrishnan 6/31

15 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes [Peterson 1991]: On general meshes O(h p+1/2 ) is the best possible [Bey & Oden 1996]: First generalization to hp (adding reaction) Jay Gopalakrishnan 6/31

16 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes [Peterson 1991]: On general meshes O(h p+1/2 ) is the best possible [Bey & Oden 1996]: First generalization to hp (adding reaction) [Falk 1998]: A nice review Jay Gopalakrishnan 6/31

17 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes [Peterson 1991]: On general meshes O(h p+1/2 ) is the best possible [Bey & Oden 1996]: First generalization to hp (adding reaction) [Falk 1998]: A nice review [Houston, Schwab & Süli 2000]: Improved hp analysis Jay Gopalakrishnan 6/31

18 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes [Peterson 1991]: On general meshes O(h p+1/2 ) is the best possible [Bey & Oden 1996]: First generalization to hp (adding reaction) [Falk 1998]: A nice review [Houston, Schwab & Süli 2000]: Improved hp analysis [Cockburn, Dong & Guzmán 2008]: O(h p+1 )-rate on other special meshes Jay Gopalakrishnan 6/31

19 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes [Peterson 1991]: On general meshes O(h p+1/2 ) is the best possible [Bey & Oden 1996]: First generalization to hp (adding reaction) [Falk 1998]: A nice review [Houston, Schwab & Süli 2000]: Improved hp analysis [Cockburn, Dong & Guzmán 2008]: O(h p+1 )-rate on other special meshes [Nguyen, Peraire & Cockburn 2009]: HDG scheme for convection-diffusion Jay Gopalakrishnan 6/31

20 Comparison: Convergence rates Brief history of finite element methods for stationary transport: [Reed & Hill 1973]: First DG method proposed [Lasaint & Raviart 1974]: First analysis, proved O(h p )-rate [Hughes & Brooks 1979]: Invented SUPG [Johnson & Pitkäranta 1986]: Proved O(h p+1/2 )-rate [Richter 1988]: Showed O(h p+1 )-rate for special meshes [Peterson 1991]: On general meshes O(h p+1/2 ) is the best possible [Bey & Oden 1996]: First generalization to hp (adding reaction) [Falk 1998]: A nice review [Houston, Schwab & Süli 2000]: Improved hp analysis [Cockburn, Dong & Guzmán 2008]: O(h p+1 )-rate on other special meshes [Nguyen, Peraire & Cockburn 2009]: HDG scheme for convection-diffusion DPG: 1 st method with provably optimal h and p rates. Jay Gopalakrishnan 6/31

21 Comparison: Convergence rates L 2 error in u DG DPG 1 DPG A DPG X rate O(h 1.5 ) rate O(h 2 ) /h 2 (approx # degrees of freedom) Experiment: Apply DG and three different DPG methods (with p = 1) to Peterson s transport example. Jay Gopalakrishnan 7/31

22 Next 1 How does the new compare to the old? Sample comparisons between DPG and DG results. 2 Elements of design of schemes. The example of simple 1D transport equation. 3 DPG method for the transport equation. Extension of the 1D idea to 2D. The spectral DPG method. The composite DPG method on a mesh. 4 Extensions. The DPG-X method. Optimal test functions. A method for all seasons? Jay Gopalakrishnan 8/31

23 Petrov-Galerkin schemes Petrov-Galerkin schemes are distinguished by different trial and test spaces. [ P.D.E.+ The problem: boundary conditions. Find u in a trial space satisfying Variational form: b(u, v) = l(v) for all v in a test space. Find u n in a discrete trial space X n satisfying Discretization: b(u n, v n ) = l(v n ) for all v n in a discrete test space V n. Petrov-Galerkin schemes have X n V n. Jay Gopalakrishnan 9/31

24 Designing a simple PG scheme Example: A simple continuous Petrov-Galerkin (CPG) scheme [ u = f in (0, 1), 1D transport eq. u(0) = u 0 (inflow b.c.) Find u in H 1, satisfying u(0) = u 0, & 1 1 Variational form: u v = fv, for all v in L }{{}}{{} b(u,v) l(v) Spectral method: [ Find up P p, satisfying u p (0) = u 0, & b(u p, v) = l(v), v P p 1. u p : trial fn. v : test fn. (Notation: P p = set of polynomials of degree at most p.) Jay Gopalakrishnan 10/31

25 Babuška s theorem Let u X and u n X n trial space be exact and approximate solutions, b(u u n, v n ) = 0 v n V n test space, and b(, ) be bounded in X V n. Theorem (A simple version of Babuška s theorem) If then C 1 w n X sup v n V n b(w n, v n ) v n Vn w n X n, u u n X C 2 inf u w n X. w n X n Guiding principle: While we must choose trial spaces with good approximation properties, we may design test spaces solely to obtain good stability properties. Jay Gopalakrishnan 11/31

26 Choice of spaces Example: The 1D spectral CPG scheme (contd.) Spectral method: [ Find up P p, satisfying u p (0) = u 0, & b(u p, v) = l(v), v P p 1. u p : trial fn. v : test fn. Q: Why the choice of spaces P p and P p 1? A: 1 Since b(u, v) = u b(u, v) v, the fraction is maximized by v = u, 0 v L 2 which we call the the optimal test function for the given u. If u is in P p, then v = u is in P p 1. Babuška s theorem = stability, for these choice of spaces. Jay Gopalakrishnan 12/31

27 What is DPG? DPG schemes (Discontinuous Petrov-Galerkin schemes) uses nonequal DG spaces (no interlement continuity) for trial and test spaces. The name DPG was used previously for methods with DG test spaces augmented with bubbles etc: [Bottasso, Micheletti & Sacco 2002]: DPG for elliptic problems [Bottasso, Micheletti & Sacco 2005]: Multiscale DPG [Causin, Sacco & Bottasso, 2005]: DPG for advection diffusion. [Causin & Sacco 2005]: Hybridized DPG for Laplace s equation The DPG methods of this talk differs from the above works in our approach to the test space design. Jay Gopalakrishnan 13/31

28 Q: Why did we set the trial space to P p+1? Jay Gopalakrishnan 14/31 A simple DPG method Example: DPG for 1D transport equation [ u = f in (0, 1), 1D transport eq. u(0) = u 0 (inflow b.c.) Find u L 2, and a number û 1, satisfying L variational form: uv + û 1 v(1) = fv + u 0 v(0), v H }{{}}{{} b( (u,û 1 ), v) l(v) Spectral method: [ Find up P p, and a number û 1 satisfying b( (u p, û 1 ), v) = l(v), v P p+1. This leads to a stable discontinuous Petrov-Galerkin (DPG) scheme.

29 A simple DPG method Q: Why is the trial space P p+1? b((u p, û 1 ), v) = A: Because inf-sup condition is then satisfied. In more detail: 1 0 u p v + û 1 v(1) Jay Gopalakrishnan 14/31

30 A simple DPG method Q: Why is the trial space P p+1? b((u p, û 1 ), v) = A: Because inf-sup condition is then satisfied. In more detail: 1 Choose a test space norm, say v 2 V = v 2 L 2 + v(1) 2. 0 u p v + û 1 v(1) Jay Gopalakrishnan 14/31

31 A simple DPG method Q: Why is the trial space P p+1? b((u p, û 1 ), v) = A: Because inf-sup condition is then satisfied. In more detail: 1 Choose a test space norm, say v 2 V = v 2 L 2 + v(1) 2. 0 u p v + û 1 v(1) Then, sup v H 1 b( (u p, û 1 ), v) v V is attained by ṽ = û 1 + ṽ integrate transport direction This maximizing ṽ is the optimal test function. û 1 1 x u p (s) ds. Jay Gopalakrishnan 14/31

32 A simple DPG method Q: Why is the trial space P p+1? b((u p, û 1 ), v) = A: Because inf-sup condition is then satisfied. In more detail: 1 Choose a test space norm, say v 2 V = v 2 L 2 + v(1) 2. 0 u p v + û 1 v(1) Then, sup v H 1 b( (u p, û 1 ), v) v V is attained by ṽ = û 1 + ṽ integrate transport direction This maximizing ṽ is the optimal test function. û 1 1 If u p P p, then ṽ is in P p+1. Hence our trial space choice. x u p (s) ds. Jay Gopalakrishnan 14/31

33 What have we gained? Even in the simplest 1D 1-element case, we see that DPG makes a difference. Recall the initial results: 0 The spectral DG solutions 0 The spectral DPG solutions u(x) 1.5 u(x) Exact solution 2.5 p=1 p=3 p= x DG Exact solution 2.5 p=1 p=3 p= x DPG DPG exhibits enhanced stability. Jay Gopalakrishnan 15/31

34 Next 1 How does the new compare to the old? Sample comparisons between DPG and DG results. 2 Elements of design of schemes. The example of simple 1D transport equation. 3 DPG method for the transport equation. Extension of the 1D idea to 2D. The spectral DPG method. The composite DPG method on a mesh. 4 Extensions. The DPG-X method. Optimal test functions. A method for all seasons? Jay Gopalakrishnan 16/31

35 The 2D, one element case The 2D transport equation on one element : [ β u = f on, β in u = g on in (inflow boundary). = u β v + β nuv + β nuv = f v out in Jay Gopalakrishnan 17/31

36 The 2D, one element case The 2D transport equation on one element : [ β u = f on, β in u = g on in (inflow boundary). = u β v + φ v + β ngv = f v out in Jay Gopalakrishnan 17/31

37 The 2D, one element case The 2D transport equation on one element : [ β u = f on, β in u = g on in (inflow boundary). = u β v + φ v + β ngv = f v out in } {{ } b( (u,φ), v) Jay Gopalakrishnan 17/31

38 The 2D, one element case The 2D transport equation on one element : [ β u = f on, β in u = g on in (inflow boundary). = u β v + φ v + β ngv = f v out in Variational formulation } {{ } b( (u,φ), v) Find solution u L 2 () and outflux φ L 2 ( out ) satisfying b( (u, φ), v) = l(v), for all v L 2 () with β v L 2 (). Jay Gopalakrishnan 17/31

39 How to construct 2D test space? 1 1D case: v = û 1 + u(s) ds 2D case: x v û 1 integral flow direction β in The optimal test function v Jay Gopalakrishnan 18/31

40 How to construct 2D test space? 1 1D case: v = û 1 + u(s) ds 2D case: x v extension û 1 integral flow direction β in v = extension + integral = E out (û 1 ) + higher degree Jay Gopalakrishnan 18/31

41 How to construct 2D test space? 1 1D case: v = û 1 + u(s) ds 2D case: x v extension û φ 1 integral flow direction β in v = extension + integral = E out (û 1 ) + higher degree v = E out (φ) + higher degree in β-direction extend E out extends from outflow boundary, constantly along streamlines. Jay Gopalakrishnan 18/31

42 How to construct 2D test space? 1 1D case: v = û 1 + u(s) ds 2D case: x φ v extension û φ 1 in φ integral flow direction β β in v = extension + integral = E out (û 1 ) + higher degree v = E out (φ) + higher degree in β-direction extend discontinuity line E out extends from outflow boundary, constantly along streamlines. Even if φ is polynomial on each edge, E out (φ) need not be! E out (φ) can be discontinuous inside. Jay Gopalakrishnan 18/31

43 A new test space The new finite element that forms the test space is composed of: = interval/triangle/tetrahedron, V p () = E out (M p+1 ( out )) η 1 P p () (geometry), (space), Σ = the following set of moments: (degrees of freedom), 2 Z ( β v)q for all q P p(), 6 4 Z vµ for all µ P p+1(f ) for all faces of. F Possible to implement with standard finite element technology. Note: M p+1 ( out ) = set of functions that are polynomials of degree p + 1 on each edge of out. η 1 = streamline coordinate. Jay Gopalakrishnan 19/31 η 2 β η 1

44 The 2D spectral method Trial space = X p () = Test space = V p (), The spectral method on one element Find (u p, φ p+1 ) X p () satisfying u pβ v + φ p+1 v = for all v V p (). Theorem P p () M p+1 ( out ), solution u approximated in P p (), outflux φ approximated in M p+1 ( out ). introduced in the previous slide. out fv β n g, v, in The solution of the method (both u p and φ p+1 ) coincides with the (L 2 ) best possible approximations of the exact solution in the trial space. Jay Gopalakrishnan 20/31

45 The composite DPG method On a mesh of triangles, construct the composite method as follows: On each triangle, set test and trial space to X p () and V p () (no interelement continuity). Elements are coupled through single-valued outflux φ h in M h = {µ : µ E P p+1 (E) for all mesh edges E not on in Ω}, The DPG-1 method ( ) u hβ vh + φ h v h φ h v h = f v h β n gv h. out in \ in Ω Ω in Ω β We can solve the system by marching from the inflow boundary. Jay Gopalakrishnan 21/31

46 The composite DPG method On a mesh of triangles, construct the composite method as follows: On each triangle, set test and trial space to X p () and V p () (no interelement continuity). Elements are coupled through single-valued outflux φ h in M h = {µ : µ E P p+1 (E) for all mesh edges E not on in Ω}, The DPG-1 method ( ) u hβ vh + φ h v h φ h v h = f v h β n gv h. out in \ in Ω Ω in Ω β We can solve the system by marching from the inflow boundary. Jay Gopalakrishnan 21/31

47 The composite DPG method On a mesh of triangles, construct the composite method as follows: On each triangle, set test and trial space to X p () and V p () (no interelement continuity). Elements are coupled through single-valued outflux φ h in M h = {µ : µ E P p+1 (E) for all mesh edges E not on in Ω}, The DPG-1 method ( ) u hβ vh + φ h v h φ h v h = f v h β n gv h. out in \ in Ω Ω in Ω β We can solve the system by marching from the inflow boundary. Jay Gopalakrishnan 21/31

48 The composite DPG method On a mesh of triangles, construct the composite method as follows: On each triangle, set test and trial space to X p () and V p () (no interelement continuity). Elements are coupled through single-valued outflux φ h in M h = {µ : µ E P p+1 (E) for all mesh edges E not on in Ω}, The DPG-1 method ( ) u hβ vh + φ h v h φ h v h = f v h β n gv h. out in \ in Ω Ω in Ω β We can solve the system by marching from the inflow boundary. Jay Gopalakrishnan 21/31

49 Discretization errors of DPG-1 Theorem (Optimal error estimates) There is a constant C independent of h and p such that for all 0 s p + 1. u u h L 2 (Ω) C hs p s u H s+1 (Ω) This is the first known error estimate (for any FEM) for the transport equation that is optimal in h and p on general meshes. Yet, our techniques of proof need improvement: We did not obtain estimates with the usual regularity assumption on u. We could prove only suboptimal estimates for φ h (although all our numerical experiments indicate that φ h converges optimally). Jay Gopalakrishnan 22/31

50 Recall Peterson s example L 2 error in u DG DPG 1 DPG A DPG X rate O(h 1.5 ) rate O(h 2 ) /h 2 (approx # degrees of freedom) Experiment: Apply DG and three different DPG methods (with p = 1) to Peterson s transport example. Jay Gopalakrishnan 23/31

51 Example with a discontinuous solution We consider an example of [Houston, Schwab & Süli 2000]. (They used it to show that DG methods work better than SUPG in the presence of shock-like discontinuities when mesh is aligned with shocks.) Mesh Exact solution Experiment: Compare DPG and DG applied to this example. Jay Gopalakrishnan 24/31

52 Example with a discontinuous solution Results: 10 0 L 2 error in u DPG outperforms DG. Solid lines indicate h- refinement. Dotted lines indicate p- refinement DPG DG Degrees of Freedom hp optimal convergence rates are observed. Jay Gopalakrishnan 24/31

53 Next 1 How does the new compare to the old? Sample comparisons between DPG and DG results. 2 Elements of design of schemes. The example of simple 1D transport equation. 3 DPG method for the transport equation. Extension of the 1D idea to 2D. The spectral DPG method. The composite DPG method on a mesh. 4 Extensions. The DPG-X method. Optimal test functions. A method for all seasons? Jay Gopalakrishnan 25/31

54 The optimal test functions in 2D On each triangle, settestandtrialspacetox p () andv p () (no interelement continuity). Elements We are constructed coupled through the testsingle-valued functions of the outflux DPG-1 φ h in method heuristically (by simply generalizing the form of the optimal expression in 1D). M h = {µ : µ E P p+1 (E) for all mesh edges E not on in Ω}, DPG-1 method 1 u hβ 0.5 vh + φ h v h φ h v h = out in \ in Ω 0 =1 =1 0.5 We can solve the system by marching 1 from the inflow boundary. β 1.5 An outflux trial function on an edge. function E out(φ) in The corresponding test DPG-1. Ω fv h β n gv h. in Ω akrishnan 21/28 Jay Gopalakrishnan 26/31

55 The optimal test functions in 2D On each triangle, settestandtrialspacetox p () andv p () (no interelement continuity). Elements We are constructed coupled through the testsingle-valued functions of the outflux DPG-1 φ h in method heuristically (by simply generalizing the form of the optimal expression in 1D). M h = {µ : µ E P p+1 (E) for all mesh edges E not on in Ω}, But, they turn out to be not the optimal test functions in 2D... DPG-1 method u hβ vh + φ h v h φ h v h = fv h β n gv h. out in \ in Ω Ω in Ω =1 =1 = We can solve the system by marching from the inflow boundary. β An outflux trial function on an edge. function E out(φ) in DPG-1. The corresponding test The actual optimal test function. akrishnan 21/28 Jay Gopalakrishnan 26/31

56 Calculating the optimal test function Recall the variational formulation for the transport equation: X Z Z Z «Z uβ v + φv φv = out in \ in Ω {z } b( (u, φ), v) To maximize b( (u, φ), v) v V, first set V -norm by v 2 V = ( Ω Z fv β n gv. in Ω β v 2 + v ), 2 out Jay Gopalakrishnan 27/31

57 Calculating the optimal test function Recall the variational formulation for the transport equation: X Z Z Z «Z uβ v + φv φv = out in \ in Ω {z } b( (u, φ), v) b( (u, φ), v) v V, Ω Z fv β n gv. in Ω To maximize first set V -norm by v 2 V = ( and then solve a local problem for the optimal test function v: β v 2 + v ), 2 out Find v : (v, δ v ) V = b( (u, φ), δ v ), δ v. Jay Gopalakrishnan 27/31

58 Calculating the optimal test function Recall the variational formulation for the transport equation: X Z Z Z «Z uβ v + φv φv = out in \ in Ω {z } b( (u, φ), v) b( (u, φ), v) v V, Ω Z fv β n gv. in Ω To maximize first set V -norm by v 2 V = ( and then solve a local problem for the optimal test function v: β v 2 + v ), 2 out Find v : (v, δ v ) V = b( (u, φ), δ v ), δ v. The hand-calculated solution with u = 0, and φ =indicator function of an edge, was =1 =1 shown on the previous slide: Jay Gopalakrishnan 27/

59 DPG-X The use of the exactly optimal test functions leads to a new method, which we call the DPG-X method. Its performance is comparable to DPG-1 method. L 2 error in u DG DPG 1 DPG A DPG X rate O(h 1.5 ) rate O(h 2 ) /h 2 (approx # degrees of freedom) DG & DPG on Peterson s mesh Jay Gopalakrishnan 28/31

60 DPG-X The use of the exactly optimal test functions leads to a new method, which we call the DPG-X method. Its performance is comparable to DPG-1 method. L 2 error in u DG DPG 1 DPG A DPG X rate O(h 1.5 ) rate O(h 2 ) While DPG-1 can be solved by marching from the inflow, DPG-X requires the solution of a symmetric positive definite system! /h 2 (approx # degrees of freedom) DG & DPG on Peterson s mesh Jay Gopalakrishnan 28/31

61 The abstract idea For any bilinear form b(u, v) in the DPG setting, the optimal test functions can be locally computed: v i = Tu i : (Tu i, δ v ) V = b(u i, δ v ), δ v. This idea is not restricted to the transport equation. Methods now immediately generalize to variable β, convection-diffusion, and all other problems which can be formulated in DPG form! We only need to approximate the optimal test function problem. Stiffness matrix is symmetric (even for the pure transport problem). B ij = b(u j, v i ) = (Tu j, v i ) V = (Tu j, Tu i ) V = (Tu i, Tu j ) V = b(v i, u j ) = B ji. Jay Gopalakrishnan 29/31

62 Stability The method is of least squares type. The novelty is in the potential for local computation of optimal test functions. With the optimal test space, inf-sup condition is obvious in the norm b(u, v) u E = sup. v V v V Error estimates follow immediately in E. It can be a theoretically difficult problem to obtain error estimates in other norms. However, hp-adaptivity can proceed by estimators in the E -norm. All our numerical experiments show extraordinary stability with h and p variations. Jay Gopalakrishnan 30/31

63 Conclusions We presented a DPG method for transport equation. The DPG method outperforms DG in computations. We proved optimal theoretical convergence estimates. The concept of optimal test functions leads to a new paradigm in designing numerical schemes. Methods are waiting to be discovered. Full manuscripts: 1 L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part I: The transport equation, Submitted, (2009). 2 L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions, Submitted, (2009). Preprints available online. Jay Gopalakrishnan 31/31