Discontinuous Petrov Galerkin (DPG) Method

Transcription

1 Discontinuous Petrov Galerkin (DPG) Method Leszek Demkowicz ICES, The University of Texas at Austin ETH Summer School on Wave Propagation Zurich, August 22-26, 2016

2 Relevant Collaboration: ICES: S. Nagaraj, S. Petrides Portland State: J. Gopalakrishnan, N. Olivares, P. Sepulveda Humboldt U: C. Carstensen Vienna TH: J. Schoeberl KIT: Ch. Wieners ETH Zurich, August 2016 DPG Method 2 / 180

3 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 3 / 180

4 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 4 / 180

5 Three interpretations ETH Zurich, August 2016 DPG Method 5 / 180

6 Abstract variational problem U, V - Hilbert spaces, b(u, v) - bilinear (sesquilinear) continuous form on U V, b(u, v) b u U v V, }{{} =:M l(v) - linear (antilinear) continuous functional on V, l(v) l V v V The abstract variational problem: { u U b(u, v) = l(v) v V Bu = l B : U V < Bu, v >= b(u, v) v V ETH Zurich, August 2016 DPG Method 6 / 180

7 Banach - Babuška - Nečas Theorem If b satisfies the inf-sup condition ( B is bounded below), inf sup u U =1 v V =1 b(u, v) =: γ > 0 sup v V and l satisfies the compatibility condition: where l(v) = 0 v V 0 b(u, v) v V V 0 := N (B ) = {v V : b(u, v) = 0 u U} γ u U then the variational problem has a unique solution u that satisfies the stability estimate: u 1 γ l V. Proof: Direct reinterpretation of Banach Closed Range Theorem. see e.g. Oden, D, Functional Analysis, Chapman & Hall, 2nd ed., 2010, p.518 ETH Zurich, August 2016 DPG Method 7 / 180

8 Petrov-Galerkin Method and (improved) Babuška Theorem U h U, V h V, dim U h = dim V h - finite-dimensional trial and test (sub)spaces { uh U h b(u h, v h ) = l(v h ), v h V h Theorem (Babuška ). The discrete inf-sup condition b(u h, v h ) sup γ h u h U v h V h v h V implies existence, uniqueness and discrete stability u h U γ 1 h l V h I. Babuska, Error-bounds for Finite Element Method., Numer. Math, 16, 1970/1971. ETH Zurich, August 2016 DPG Method 8 / 180

9 Petrov-Galerkin Method and (improved) Babuška Theorem U h U, V h V, dim U h = dim V h - finite-dimensional trial and test (sub)spaces { uh U h b(u h, v h ) = l(v h ), v h V h Theorem (Babuška ). The discrete inf-sup condition b(u h, v h ) sup γ h u h U v h V h v h V implies existence, uniqueness and discrete stability and convergence u h U γ 1 h l V h u u h U M γ h inf w h U h u w h U (Uniform) discrete stability and approximability imply convergence. I. Babuska, Error-bounds for Finite Element Method., Numer. Math, 16, 1970/1971. ETH Zurich, August 2016 DPG Method 8 / 180

10 Lemma (Del Pasqua, Ljance, Kato, Szyld) Let U, (, ) be a Hilbert space and P : U U a linear projection, i.e. P 2 = P. Then I P = P Proof: Let X = R(P ) and Y = N (P ). It is well known that U = X Y. Pick an arbitrary unit vector u U. Let u = x + y, x X, y Y be the unique decomposition of u. By the properties of a scalar product, 1 = u 2 = (x + y, x + y) = x 2 + y 2 + 2Re(x, y). D.B. Szyld. The many proofs of an identity on the norm of oblique projections. Numer. Algor., 42: , ETH Zurich, August 2016 DPG Method 9 / 180

11 Lemma (Del Pasqua, Ljance, Kato), cont. Consider now a symmetric image w of u, w = x + ȳ, x = y x x, Vector w has a unit length as well. Indeed, ȳ = x y y. w 2 = ( x+ȳ, x+ȳ) = x 2 + ȳ 2 +2Re( x, ȳ) = y 2 + x 2 +2Re We have now, P u = x = ȳ = (I P )w I P w = I P Taking supremum over u = 1 finishes the proof. ETH Zurich, August 2016 DPG Method 10 / 180 ( ) y x (x, y) = 1. x y

12 Proof of Babuška Theorem Observe that the Petrov Galerkin discretization executes a linear projection P h : U U h, P h u = u h, b(p h u u, v h ) = 0 v h V h. The stability estimate implies an estimate on the norm of the projection, We have then: P h u U = u h 1 l V γ h = 1 sup b(u, v h ) M u U. h γ h v h =1 γ h u u h U = (I P h )u U (definition of P h ) = (I P h )(u w h ) U (P h w h = w h w h V h ) I P h u w h = P h u w h (Lemma) M γ h u w h ( P h M γ h ), and we conclude the proof by taking infimum over w h U h. ETH Zurich, August 2016 DPG Method 11 / 180

13 Babuška Theorem, cont. If γ h admit a positive lower bound, i.e. a uniform discrete inf sup condition holds, then, u u h }{{} approximation error inf h γ h =: γ 0 > 0, M γ 0 }{{} inf u w h U w h U h }{{} stability constant best approximation error i.e. the actual and the best approximation errors must converge at the same rate. The result has coined the famous phrase: (Uniform) discrete stability and approximability imply convergence.. ETH Zurich, August 2016 DPG Method 12 / 180

14 Optimal test functions The main trouble: continuous inf-sup condtion = / discrete inf-sup condition b(u h, v) b(u h, v h ) sup γ u h U = / sup γ u h U v V v V v h V h v h V ETH Zurich, August 2016 DPG Method 13 / 180

15 Optimal test functions The main trouble: unless continuous inf-sup condtion = / discrete inf-sup condition b(u h, v) b(u h, v h ) sup γ u h U = / sup γ u h U v V v V v h V h v h V L.D., J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, , ETH Zurich, August 2016 DPG Method 13 / 180

16 Optimal test functions The main trouble: continuous inf-sup condtion = / discrete inf-sup condition b(u h, v) b(u h, v h ) sup γ u h U = / sup γ u h U v V v V v h V h v h V unless we employ special test functions that realize the supremum: v h = arg max v V b(u h, v) v L.D., J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, , ETH Zurich, August 2016 DPG Method 13 / 180

17 Optimal test functions The main trouble: unless continuous inf-sup condtion = / discrete inf-sup condition b(u h, v) b(u h, v h ) sup γ u h U = / sup γ u h U v V v V v h V h v h V we employ special test functions that realize the supremum: Recall that the Riesz operator R V v h = arg max v V b(u h, v) v : V V is an isometry. Then: sup v b(u h,v) v = Bu h V = R 1 V Bu h }{{} V = =v h = Bu h,v h v h V = b(u h,v h ) v h V (R 1 V Bu h,v h ) V v h V Variational definition of v h : { vh V (v, δv) V = b(u h, δv) δv V. The operator T := R 1 B : U h V will be called the trial to test operator. V L.D., J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, , ETH Zurich, August 2016 DPG Method 13 / 180

18 DPG is a Minimum Residual Method With the optimal test functions in place, γ h γ, and the Galerkin method is automatically stable. Trade now the original norm in U for an energy norm : Two points: u E := R 1 b(u, v) V Bu V = Bu V = sup v V v V With respect to the new, energy norm, both continuity constant M and inf-sup constant γ are unity. The use of optimal test functions (their construction is independent of the choice of trial norm) implies that γ h γ = 1. Thus, by the Babuška Theorem, u u h E M γ h }{{} =1 inf u w h E. w h U h In other words, FE solution u h is the best approximation of the exact solution u in the energy norm. We have arrived through a back door at a Minimum Residual Method. Residual norm really... ETH Zurich, August 2016 DPG Method 14 / 180

19 Moral of the story The minimum residual method, with the residual measured in the dual test norm, is the most stable Petrov-Galerkin method you can have. ETH Zurich, August 2016 DPG Method 15 / 180

20 DPG is a minimum residual method { u U b(u, v) = l(v) v V Bu = l B : U V Bu, v = b(u, v) J.H. Bramble, R.D. Lazarov, J.E. Pasciak, A Least-squares Approach Based on a Discrete Minus One Inner Product for First Order Systems Math. Comp, 66, , L.D., J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, , ETH Zurich, August 2016 DPG Method 16 / 180

21 DPG is a minimum residual method { u U b(u, v) = l(v) v V Bu = l B : U V Bu, v = b(u, v) Minimum residual method: U h U, 1 2 Bu h l 2 V min u h U h J.H. Bramble, R.D. Lazarov, J.E. Pasciak, A Least-squares Approach Based on a Discrete Minus One Inner Product for First Order Systems Math. Comp, 66, , L.D., J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, , ETH Zurich, August 2016 DPG Method 16 / 180

22 DPG is a minimum residual method { u U b(u, v) = l(v) v V Bu = l B : U V Bu, v = b(u, v) Minimum residual method: U h U, Riesz operator: 1 2 Bu h l 2 V min u h U h is an isometry, R V v V = v V. R V : V V, R V v, δv = (v, δv) V J.H. Bramble, R.D. Lazarov, J.E. Pasciak, A Least-squares Approach Based on a Discrete Minus One Inner Product for First Order Systems Math. Comp, 66, , L.D., J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, , ETH Zurich, August 2016 DPG Method 16 / 180

23 DPG is a minimum residual method { u U b(u, v) = l(v) v V Bu = l B : U V Bu, v = b(u, v) Minimum residual method: U h U, Riesz operator: 1 2 Bu h l 2 V min u h U h R V : V V, R V v, δv = (v, δv) V is an isometry, R V v V = v V. Minimum residual method reformulated: 1 2 Bu h l 2 V = 1 2 R 1 V (Bu h l) 2 V min u h U h J.H. Bramble, R.D. Lazarov, J.E. Pasciak, A Least-squares Approach Based on a Discrete Minus One Inner Product for First Order Systems Math. Comp, 66, , L.D., J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, , ETH Zurich, August 2016 DPG Method 16 / 180

24 DPG is a minimum residual method Taking Gâteaux derivative, (R 1 V (Bu h l), R 1 V Bδu h) V = 0 δu h U h ETH Zurich, August 2016 DPG Method 17 / 180

25 DPG is a minimum residual method Taking Gâteaux derivative, (R 1 V (Bu h l), R 1 V Bδu h) V = 0 δu h U h or Bu h l, R 1 V Bδu h = 0 δu h U h ETH Zurich, August 2016 DPG Method 17 / 180

26 DPG is a minimum residual method Taking Gâteaux derivative, (R 1 V (Bu h l), R 1 V Bδu h) V = 0 δu h U h or Bu h l, R 1 V Bδu h = 0 δu h U h }{{} v h ETH Zurich, August 2016 DPG Method 17 / 180

27 DPG is a minimum residual method Taking Gâteaux derivative, (R 1 V (Bu h l), R 1 V Bδu h) V = 0 δu h U h or Bu h l, v h = 0 v h = R 1 V Bδu h ETH Zurich, August 2016 DPG Method 17 / 180

28 DPG is a minimum residual method Taking Gâteaux derivative, (R 1 V (Bu h l), R 1 V Bδu h) V = 0 δu h U h or Bu h, v h = l, v h v h = R 1 V Bδu h ETH Zurich, August 2016 DPG Method 17 / 180

29 DPG is a minimum residual method Taking Gâteaux derivative, (R 1 V (Bu h l), R 1 V Bδu h) V = 0 δu h U h or where b(u h, v h ) = l(v h ) { vh V (v h, δv) V = b(δu h, δv) δv V ETH Zurich, August 2016 DPG Method 17 / 180

30 DPG is a mixed method An alternate route, ( R 1 V (Bu h l), R 1 V }{{} Bδu h) V = 0 δu h U h =:ψ(error representation function) W. Dahmen, Ch. Huang, Ch. Schwab, and G. Welper. Adaptive Petrov Galerkin methods for first order transport equations, SIAM J. Num. Anal. 50(5): , 2012 ETH Zurich, August 2016 DPG Method 18 / 180

31 DPG is a mixed method An alternate route, ( R 1 V (Bu h l), R 1 V }{{} Bδu h) V = 0 δu h U h =:ψ(error representation function) or { ψ = R 1 V (Bu h l) (ψ, R 1 V Bδu h) V = 0 δu h U h W. Dahmen, Ch. Huang, Ch. Schwab, and G. Welper. Adaptive Petrov Galerkin methods for first order transport equations, SIAM J. Num. Anal. 50(5): , 2012 ETH Zurich, August 2016 DPG Method 18 / 180

32 DPG is a mixed method An alternate route, ( R 1 V (Bu h l), R 1 V }{{} Bδu h) V = 0 δu h U h =:ψ(error representation function) or { (ψ, δv)v b(u h, δv) = l(δv) δv V b(δu h, ψ) = 0 δu h U h W. Dahmen, Ch. Huang, Ch. Schwab, and G. Welper. Adaptive Petrov Galerkin methods for first order transport equations, SIAM J. Num. Anal. 50(5): , 2012 ETH Zurich, August 2016 DPG Method 18 / 180

33 DPG method, a summary so far Stiffness matrix is always hermitian and positive-definite (it is a generalization of the least squares method...). J.T.Oden, L.D., T.Strouboulis and Ph. Devloo, Adaptive Methods for Problems in Solid and Fluid Mechanics, in Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley & Sons, London 1986 ETH Zurich, August 2016 DPG Method 19 / 180

34 DPG method, a summary so far Stiffness matrix is always hermitian and positive-definite (it is a generalization of the least squares method...). The method delivers the best approximation error (BAE) in the energy norm : b(u, v) u E := Bu V = sup v V v V J.T.Oden, L.D., T.Strouboulis and Ph. Devloo, Adaptive Methods for Problems in Solid and Fluid Mechanics, in Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley & Sons, London 1986 ETH Zurich, August 2016 DPG Method 19 / 180

35 DPG method, a summary so far Stiffness matrix is always hermitian and positive-definite (it is a generalization of the least squares method...). The method delivers the best approximation error (BAE) in the energy norm : b(u, v) u E := Bu V = sup v V v V The energy norm of the FE error u u h equals the residual and can be computed, u u h E = Bu Bu h V = l Bu h V = R 1 V (l Bu h) V = ψ V where the error representation function ψ comes from { ψ V (ψ, δv) V = l Bu h, δv = l(δv) b(u h, δv), δv V No need for a-posteriori error estimation, note the connection with implicit a-posteriori error estimation techniques J.T.Oden, L.D., T.Strouboulis and Ph. Devloo, Adaptive Methods for Problems in Solid and Fluid Mechanics, in Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley & Sons, London 1986 ETH Zurich, August 2016 DPG Method 19 / 180

36 DPG method, a summary A lot depends upon the choice of the test norm V ; for different test norms, we get get different methods. ETH Zurich, August 2016 DPG Method 20 / 180

37 DPG method, a summary A lot depends upon the choice of the test norm V ; for different test norms, we get get different methods. How to choose the test norm in a systematic way? ETH Zurich, August 2016 DPG Method 20 / 180

38 DPG method, a summary A lot depends upon the choice of the test norm V ; for different test norms, we get get different methods. How to choose the test norm in a systematic way? Is the inversion of Riesz operator (computation of the optimal test functions, energy error) feasible? ETH Zurich, August 2016 DPG Method 20 / 180

39 DPG method, a summary A lot depends upon the choice of the test norm V ; for different test norms, we get get different methods. How to choose the test norm in a systematic way? Is the inversion of Riesz operator (computation of the optimal test functions, energy error) feasible? Being a Ritz method, DPG does not experience any preasymptotic limitations. ETH Zurich, August 2016 DPG Method 20 / 180

40 Digression: Trial to Test Operator of Barret and Morton U B V R U R V U B V DPG trial-to-test operator: Barret-Morton trial-to-test operator: B (B ) 1 R U T 1 V J.W. Barret and K. W. Morton, Comp. Meth. Appl. Mech and Engng., 46, 97 (1984). ETH Zurich, August 2016 DPG Method 21 / 180

41 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 22 / 180

42 Time-harmonic linear acoustics Mathematician s version : iωp +div u = f in Ω iωu + p = g in Ω p = u n on Γ (Strong) operator form: { p H 1 (Ω), u H(div, Ω), p = u n on Γ A(p, u) = (f, g) where A(p, u) = (iωp + div u, iωu + p), f L 2 (Ω), g (L 2 (Ω)) N. ETH Zurich, August 2016 DPG Method 23 / 180

43 Trivial and ultraweak variational formulations Trivial formulation: p H 1 (Ω), u H(div, Ω), p = u n on Γ iω(p, q) + (div u, q) = (f, q) q L 2 (Ω) iω(u, v) + ( p, v) = (g, v) v (L 2 (Ω)) N or, in operator form: Ultraweak formulation: where A = A. p H 1 (Ω), u H(div, Ω), p = u n on Γ (A(p, u), (q, v)) = (f, q) + (g, v) q L 2 (Ω), v (L 2 (Ω)) N p L 2 (Ω), v (L 2 (Ω)) N ((p, u), A (q, v)) = (f, q) + (g, v) q H 1 (Ω), v H(div, Ω), q = v n on Γ ETH Zurich, August 2016 DPG Method 24 / 180

44 Mixed and reduced formulations I Relaxing conservation of mass eqn, p H 1 (Ω), u (L 2 (Ω)) N iω(p, q) (u, q) + p, q Γ = (f, q) q H 1 (Ω) iω(u, v) + ( p, v) = (g, v) v (L 2 (Ω)) N Multiplying the first equation by iω and eliminating velocity, we obtain the standard formulation for Helmholtz eqn, { u H 1 (Ω) ( p, q) ω 2 (p, q) + iω p, q Γ = (iωf + g, v) v H 1 (Ω) I.e. integrating by parts and building the BC in... ETH Zurich, August 2016 DPG Method 25 / 180

45 Mixed and reduced formulations II Relaxing conservation of momentum eqn, we get: p L 2 (Ω), u V iω(p, q) + (div u, q) = (f, q) q L 2 (Ω) iω(u, v) (p, div v) + u n, v n Γ = (g, v) v V The energy space for the velocity has now to incorporate an extra regularity assumption resulting from building in the impedance BC, V := {v H(div, Ω) : v n H 1/2 (Γ)} As before, we can eliminate now the pressure to obtain a variational formulation in terms of the velocity only, { u V (div u, div v) ω 2 (u, v) + iω u n, v n Γ = (f + iωg, v), v V ETH Zurich, August 2016 DPG Method 26 / 180

46 Stability and well posedness All formulations are simultaneously well- or ill-posed. L.D. Various Variational Formulations and Closed Range Theorem, ICES Report 15/03 ETH Zurich, August 2016 DPG Method 27 / 180

47 Time-harmonic Maxwell equations where: E + iωµh = 0 Faraday Law H iωɛe σe = J imp Ampère Law (µh) = 0 Gauss Magnetic Law (ɛe) = ρ imp + ρ Gauss Electric Law iωρ + (σe) = 0 conservation of charge E H ρ µ ɛ σ J imp ρ imp electric field magnetic field free charge permeability permittivity conductivity impressed current impressed charge the unknowns material constants } load data ETH Zurich, August 2016 DPG Method 28 / 180

48 A bit of history Electrostatics: Coulomb s Law (1775) electric field polarization, dielectrics, ɛ Gauss Electric Law (1835) Magnetostatics: Ampère Force Law (1820) (steady currents) magnetic field magnetic polarization, µ Ampère Law Gauss Magnetic Law Faraday Law (1831) Maxwell equations (1856) including correction to Ampère Law. Current formalism due to Heaviside (1884) Inspired Einstein on his way to Special Relativity. ETH Zurich, August 2016 DPG Method 29 / 180

49 Maxwell cavity problem Eliminating the free charge, we get: E + iωµh = 0 Faraday Law H iωɛe σe = J imp Ampère Law (µh) = 0 Gauss Magnetic Law ((σ + iωɛ)e) = iωρ imp continuity equation The equations are linearly dependent: Take curl of the Faraday Law to obtain the Gauss Magnetic Law. Take curl of the Ampère Law to obtain the continuity equation. Notice that J imp, ρ imp must satisfy the compatibility condition: Boundary Conditions (BC): J imp = iωρ imp. n E = n E imp on Γ 1 n H = n H imp =: J imp S }{{} impressed surface current on Γ 2 ETH Zurich, August 2016 DPG Method 30 / 180

50 Different variational formulations To relax or not to relax? n E = n E imp on Γ 1 n H = n H imp on Γ 2 E + iωµh = 0 /φ (1) H (iωɛ + σ)e = J imp /ψ (2) (1) (2) name energy setting 1 no no trivial (strong) E, H H(curl), φ, ψ L 2 2 no yes standard E, ψ H(curl), H, φ L 2 3 yes no standard H, φ H(curl), E, ψ L 2 4 yes yes ultraweak E, H L 2, φ, ψ H(curl) The inf-sup constants for different variational formulations are equal or O(1)-equivalent (Closed Range Theorem at work). Only standard variational formulations are eligible for the Bubnov-Galerkin method (symmetric functional setting) ETH Zurich, August 2016 DPG Method 31 / 180

51 Standard variational formulation Relax the Ampère equation: multiply with factor iω, ( iωh) + ( ω 2 ɛ + iωσ)e = iωj imp, multiply with a test function ψ and integrate by parts the curl term, ( iωh, ψ) + n ( iωh), ψ + (( ω 2 ɛ + iωσ)e, ψ) = iω(j imp, ψ), build the second boundary condition into the formulation and eliminate the rest of the boundary term by not testing on Γ 1, i.e. assuming n ψ = 0 on Γ 1, ( iωh, ψ)+(( ω 2 ɛ+iωσ)e, ψ) = iω(j imp, ψ)+iω J imp S, ψ Γ 2 n ψ = 0 on Γ 1 Use the strong form of the Faraday equation to eliminate H: E H(curl, Ω), n E = n E imp on Γ 1, ( 1 E, ψ) + µ (( ω2 ɛ + iωσ)e, ψ) = iω(j imp, ψ) + iω J imp S, ψ Γ 2 ψ H(curl, Ω) : n ψ = 0 on Γ 1. ETH Zurich, August 2016 DPG Method 32 / 180

52 Ultraweak variational formulation Relax both equations: E, H L 2 (Ω) (E, φ) + (iωµh, φ) = n E inc, φ φ H(curl, Ω), n φ = 0 on Γ 2 (H, ψ) ((iωɛ + σ)e, ψ) = (J imp, ψ) n H inc, ψ ψ H(curl, Ω), n ψ = 0 on Γ 1 You may test on the whole boundary: E L 2 (Ω), Ê H 1/2 t (curl, Γ), n Ê = n Einc on Γ 1 H L 2 (Ω), Ĥ H 1/2 t (curl, Γ), n Ĥ = n Hinc on Γ 2 (E, φ) + (iωµh, φ) + n Ê, φ = 0 φ H(curl, Ω) (H, ψ) ((iωɛ + σ)e, ψ) + n Ĥ, ψ = (J imp, ψ) ψ H(curl, Ω) where H 1/2 t (curl, Γ) := tr ΓH(curl, Ω). ETH Zurich, August 2016 DPG Method 33 / 180

53 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 34 / 180

54 The paradigm of breaking test functions In each of the discussed formulations, we can eliminate BCs on test functions and replace the test spaces with corresponding broken test spaces, H 1 (Ω) H 1 (Ω h ) H(curl, Ω) H(curl, Ω h ) H(div, Ω) H(div, Ω h ), i.e. we can break test functions at the expense of introducing additional unknowns (Lagrange multipliers) that are traces of dual energy spaces to mesh skeleton Γ h := K Th K, H 1 (Ω h ) introduces n ˆv H 1/2 (Γ h ) := tr Γh H(div, Ω) H(curl, Ω h ) introduces n Ĥ H 1/2 (div, Γ h ) := tr Γ h H(curl, Ω) H(div, Ω h ) introduces û H 1/2 (Γ h ) := tr Γh H 1 (Ω) ETH Zurich, August 2016 DPG Method 35 / 180

55 Formulations with broken test spaces for acoustics Classical formulation: { p H 1 (Ω), û n H 1/2 (Γ h ) ( p, h q) + iω p, q Γ û n, q Γh = (iωf + g, q), q H 1 (Ω h ) where H 1/2 (Γ h ) := tr Γh H 0 (div, Ω) In other words, the additional unknown vanishes on domain boundary Γ. Ultraweak formulation: p L 2 (Ω), u (L 2 (Ω)) N ˆp H 1/2 (Γ h ), û n H 1/2 (Γ h ), ˆp = û n on Γ (p, iωq + div h v) (u, iωv + h q)+ ˆp, v n Γh + û n, q Γh = (f, q) + (g, v) q H 1 (Ω h ), v H(div, Ω h ) Above, h, div h denote element-wise defined operators.. ETH Zurich, August 2016 DPG Method 36 / 180

56 Lemma 1 (Brezzi s argument) We can break test functions in any well-posed variational problem, { u U (1) b(u, v) = l(v) v V (Ω) that goes into: u U, t (V (Ω h ) b(u, v) + t, v = l(v) }{{} v V (Ω h ) (2) =:b((u,t),v) If (1) is well posed than so is (2). Sketch of the proof: We already control u : We control t in the dual norm: b((u, t), v) b(u, v) sup sup γ u U v V (Ω h ) v v V (Ω) v t, v = l(v) b(u, v) ( t (V (Ωh )) = sup v l + b u U t,v v 1 + b γ ) l ETH Zurich, August 2016 DPG Method 37 / 180

57 Lemma 2 The dual norm of t can be interpreted as a minimum-energy (quotient) norm. Assume: V C - Hilbert (test) space with graph norm t V. Then t V C v 2 V C := Cv 2 + v 2 t, v := sup = t VC v V C v VC where t is the minimum energy extension of t in V C -norm. Sketch of the proof: Computation of the dual norm reduces to the inversion of the Riesz operator which leads to a Neumann problem: (Cv t, Cδv) + (v t, δv) = t, δv C Cv }{{} t +v t = 0 in Ω /C t Cv t = t on Γ Computation of the minimum-energy extension norm reduces to the solution of a Dirichlet problem; C }{{} C t + t = 0 in Ω /C = t t = t on Γ and Cv t = t, C t = t, t 2 V C = C t 2 + t 2 = v t 2 V c ETH Zurich, August 2016 DPG Method 38 / 180

58 Theorem (C. Carstensen, L.D., J. Gopalakrishnan, 2015) Assume that the original variational problem is well posed. Then the variational problem with broken test spaces is well posed as well with a mesh-independent stability constant γ of order of inf-sup constant for the original problem, and the following stability result holds: b((u, t), v) sup γ( u 2 U + t 2 C v V (Ω h ) v )1/2 V (Ωh ) where the Lagrange multiplier t is measured in the minimum energy extension (quotient) norm implied by the test norm. Comment on the role of norms in U and V. C. Carstensen, L.D. and J. Gopalakrishnan, Breaking Spaces and Forms for the DPG method and Applications Including Maxwell Equations, Comput. Math.Appl., 72(3): , ETH Zurich, August 2016 DPG Method 39 / 180

59 Breaking test functions in Maxwell problems Primal DPG formulation: E H(curl, Ω), n E = n E imp on Γ 1, Ĥ H 1/2 t (curl, Γ h ), n Ĥ = n Hinc on Γ 2 ( 1 µ E, h ψ) + (( ω 2 ɛ + iωσ)e, ψ) + iω Ĥ, ψ = iω(j imp, ψ) ψ H(curl, Ω h ) Ultraweak DPG formulation: E L 2 (Ω), Ê H 1/2 t (curl, Γ h ), n Ê = n Einc on Γ 1 H L 2 (Ω), Ĥ H 1/2 t (curl, Γ h ), n Ĥ = n Hinc on Γ 2 (E, h φ) + (iωµh, φ) + n Ê, φ = 0 φ H(curl, Ω h) (H, h ψ) ((iωɛ + σ)e, ψ) + n Ĥ, ψ = (J imp, ψ) ψ H(curl, Ω h ) ETH Zurich, August 2016 DPG Method 40 / 180

60 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 41 / 180

61 Approximation of optimal test functions (Practical DPG Method) Idea: Replace infinite dimensional test space V with a finite-dimensional enriched space Ṽ V, dim Ṽ >> dim U h. Natural choice: use element of higher order r := p + p. The enriched test space Ṽ =: V r may be determined a-priori or a-posteriori. In all reported experiments, typically, p = 1, 2, 3. Comments: Other approximations of optimal test functions are possible, see Niemi et al for use of subelement Shishkin meshes. A completely different approach has been proposed by Cohen et al. where an additional (inner) adaptive loop is introduced to approximate error representation function ψ. The resulting approximate optimal test space does not guarantee the discrete stability. Discuss the difference between the a-priori and a-posteriori approximation of ψ. A. Niemi, N. Collier and V. Calo, Stable Discontinuous Petrov-Galerkin Methods for Stationary Transport Problems: Quasi-optimal Test Space Norm, Comput. Math. Appl., 66(10): , 2013 A. Cohen, W. Dahmen, and G. Welper, Adaptivity and variational stabilization for convection-diffusion equations, ESAIM Math. Model. Numer. Anal., 46(5): , ETH Zurich, August 2016 DPG Method 42 / 180

62 Petrov-Galerkin method with optimal test functions Ideal DPG method: { uh U h U b(u h, v h ) = l(v h ) v h T (U h ) where the ideal trial-to-test operator T : U h V is defined by (T δu h, δv) V = b(δu h, δv) u h U h, δv V Practical DPG method: { uh U h U b(u h, v h ) = l(v h ) v h T r (U h ) where the practical trial-to-test operator T r : U h V r V is defined by (T r δu h, δv) V = b(δu h, δv) u h U h, δv V r ETH Zurich, August 2016 DPG Method 43 / 180

63 Minimum residual method Ideal DPG method: u h = arg min wh U h l Bu h V Practical DPG method: = arg min wh U h sup v V l(v) b(u h, v) v V u h = arg min wh U h l Bu h (V r ) = arg min l(v) b(u h, v) w h U h sup v V r v V ETH Zurich, August 2016 DPG Method 44 / 180

64 Mixed method Ideal DPG method: ψ V, u h U h (ψ, φ) V b(u h, φ) = l(φ) φ V b(w h, ψ) = 0 w h U h where ψ = l Bu h V is the residual. Practical DPG method: ψ V r, ũ h U h ( ψ, φ) V b(u h, φ) = l(φ) φ V r b(w h, ψ) = 0 w h U h where ψ = l Bu h (V r ) is the computable approximate residual. ETH Zurich, August 2016 DPG Method 45 / 180

65 Accounting for the error in computing test functions and ψ Introduce a Fortin operator Π : V V r, Theorem b(u h, Πv v) = 0 u h U h Πv V C v V u u h U MC γ inf w h U h u w h U Theorem There exists a family of commuting, element-wise defined Fortin operators for standard energy spaces and the enriched spaces for p = N. H 1 (K)/IR grad H(curl, K) curl H(div, K) div L 2 (K) Π grad p+3 Π curl p+3 Π div p+3 Π p+2 P p+3(k)/ir grad N p+3(k) curl R p+3(k) div P p+2(k) J.G. and W. Qiu, An Analysis of the Practical DPG Method, Math. Comp., 83(286): , 2014, C. Carstensen, L.D. and J. G., Breaking Spaces and Forms for the DPG method and Applications Including Maxwell Equations, Comput. Math. Appl., 72(3): , ETH Zurich, August 2016 DPG Method 46 / 180

66 Accounting for the error in computing test functions and ψ The Fortin operator can also be used to assess the error in approximation of error representation function. Theorem Let η = ψ V. Then and γ 2 u u h 2 η 2 + ( Π η + osc(l)) 2 η M u u h U For additional study of Fortin operators, see ICES Report C. Carstensen, L.D., J. Gopalakrishnan, A Posteriori Error Control for DPG Methods, SIAM J. Numer. Anal., 52(3): , ETH Zurich, August 2016 DPG Method 47 / 180

67 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 48 / 180

68 Main point: ψ is condensed out elementwise Group unknown (watch for the overloaded symbol): u h := ( u }{{} h, ˆt h ) }{{} field flux Mixed system: G B 1 B 2 B T B T ψ u ˆt = l 0 0 where B 1, B 2 correspond to ( u h, h v h ) and ˆt h, v h, resp. Eliminate ψ to get the DPG system: ( B T 1 G 1 B 1 B T 1 G 1 B 2 B T 2 G 1 B 1 B T 2 G 1 B 2 ) ( ṷ t ) = ( B T 1 G 1 l B T 2 G 1 l After backsubstitution, ψ V provides the element contribution to the global residual which drives adaptivity. ) ETH Zurich, August 2016 DPG Method 49 / 180

69 Are all variational formulations equal? ETH Zurich, August 2016 DPG Method 50 / 180

70 1D acoustics, ω = 60, 50 elements, p = 2. Spectrum of stiffness matrix after static condensation of interior dof, for classical Galerkin, least squares (strong formulation) primal and ultraweak formulations. Comment on CG convergence ETH Zurich, August 2016 DPG Method 51 / 180

71 Corresponding solutions Both least squares and ultraweak DPG methods are robust, i.e. the stability constants are independent of ω but they converge in different norms: least squares in graph norm, ultraweak in L 2 -norm. Solutions obtained classical Galerkin, least squares (strong formulation) primal and ultraweak formulations. In one space dimension the ultraweak DPG method is pollution free! ETH Zurich, August 2016 DPG Method 52 / 180

72 In 1D the ultraweak DPG is pollution free With the graph test norm, (v, q) ) 2 V := A(v, q) 2 + (v, q) 2 the corresponding UW DPG method is pollution free, ( (u u h, p p h ) 2 + (û û h, ˆp ˆp h ) 2 ) 1/2 1 ( inf (u w γ h, p r h ) 2 + inf (û ŵ h, ˆp ˆr h ) 2 (w h,r h ) (ŵ h,ˆr h ) }{{}}{{} pollution free =0 ) 1/2 Comment on the choice of the enriched space. ETH Zurich, August 2016 DPG Method 53 / 180

73 3D examples All reported 3D examples were coded within hp3d - a three-dimensional hp code supporting: hybrid meshes with elements of all shapes: hexas, tets, prisms and pyramids, first Nedèlèc family of H 1 -,H(curl)-, H(div)-, and L 2 -conforming elements, 1-irregular meshes and anisotropic hp-refinements. The computations used a recently developed suite of orientation embedded shape functions for elements of all shapes and the whole exact sequence that can be downloaded from: Developed with Paolo Gatto and Kyungjoo Kim. F. Fuentes, B. Keith, L. D. and S. Nagaraj, Orientation Embedded High Order Shape Functions for the Exact Sequence Elements of All Shapes, em Comput. Math. Appl., 70: , ETH Zurich, August 2016 DPG Method 54 / 180

74 3D Maxwell cavity problem, h-convergence Unit domain, PEC BCs, µ = ɛ = 1, σ = 0, ω = 1, p = 1, 2, 3, p = 2, manufactured solution: E 1 = sin πx sin πy sin πz, E 2 = E 3 = 0. Primal formulation, hexas, H(curl)-error and residual. ETH Zurich, August 2016 DPG Method 55 / 180

75 3D Maxwell cavity problem, h-convergence Unit domain, PEC BCs, µ = ɛ = 1, σ = 0, ω = 1, p = 1, 2, 3, p = 2, manufactured solution: E 1 = sin πx sin πy sin πz, E 2 = E 3 = 0. Primal formulation, tets, H(curl)-error and residual. ETH Zurich, August 2016 DPG Method 55 / 180

76 3D Maxwell cavity problem, h-convergence Unit domain, PEC BCs, µ = ɛ = 1, σ = 0, ω = 1, p = 1, 2, 3, p = 2, manufactured solution: E 1 = sin πx sin πy sin πz, E 2 = E 3 = 0. Primal formulation, prisms, H(curl)-error and residual. ETH Zurich, August 2016 DPG Method 55 / 180

77 3D Maxwell cavity problem, h-convergence Unit domain, PEC BCs, µ = ɛ = 1, σ = 0, ω = 1, p = 1, 2, 3, p = 2, manufactured solution: E 1 = sin πx sin πy sin πz, E 2 = E 3 = 0. Ultraweak formulation, hexas, L 2 -error and residual. ETH Zurich, August 2016 DPG Method 55 / 180

78 3D Maxwell cavity problem, h-convergence Unit domain, PEC BCs, µ = ɛ = 1, σ = 0, ω = 1, p = 1, 2, 3, p = 2, manufactured solution: E 1 = sin πx sin πy sin πz, E 2 = E 3 = 0. Ultraweak formulation, tets, L 2 -error and residual. ETH Zurich, August 2016 DPG Method 55 / 180

79 3D Maxwell cavity problem, h-convergence Unit domain, PEC BCs, µ = ɛ = 1, σ = 0, ω = 1, p = 1, 2, 3, p = 2, manufactured solution: E 1 = sin πx sin πy sin πz, E 2 = E 3 = 0. Ultraweak formulation, prisms, L 2 -error and residual. ETH Zurich, August 2016 DPG Method 55 / 180

80 DPG does not suffer form any preasymptotic behavior and it enables adaptivity starting with very coarse meshes ETH Zurich, August 2016 DPG Method 56 / 180

81 Example: 2D acoustics: Gaussian beam ETH Zurich, August 2016 DPG Method 57 / 180

82 Mesh 1 and real part of pressure ETH Zurich, August 2016 DPG Method 58 / 180

83 Mesh 2 and real part of pressure ETH Zurich, August 2016 DPG Method 59 / 180

84 Mesh 3 and real part of pressure ETH Zurich, August 2016 DPG Method 60 / 180

85 Mesh 4 and real part of pressure ETH Zurich, August 2016 DPG Method 61 / 180

86 Mesh 5 and real part of pressure ETH Zurich, August 2016 DPG Method 62 / 180

87 Mesh 6 and real part of pressure ETH Zurich, August 2016 DPG Method 63 / 180

88 Mesh 7 and real part of pressure ETH Zurich, August 2016 DPG Method 64 / 180

89 Mesh 8 and real part of pressure ETH Zurich, August 2016 DPG Method 65 / 180

90 Mesh 9 and real part of pressure ETH Zurich, August 2016 DPG Method 66 / 180

91 Mesh 10 and real part of pressure ETH Zurich, August 2016 DPG Method 67 / 180

92 Mesh 11 and real part of pressure ETH Zurich, August 2016 DPG Method 68 / 180

93 Mesh 12 and real part of pressure ETH Zurich, August 2016 DPG Method 69 / 180

94 Mesh 13 and real part of pressure ETH Zurich, August 2016 DPG Method 70 / 180

95 Mesh 14 and real part of pressure ETH Zurich, August 2016 DPG Method 71 / 180

96 Mesh 15 and real part of pressure ETH Zurich, August 2016 DPG Method 72 / 180

97 Mesh 16 and real part of pressure ETH Zurich, August 2016 DPG Method 73 / 180

98 Mesh 17 and real part of pressure ETH Zurich, August 2016 DPG Method 74 / 180

99 Mesh 18 and real part of pressure ETH Zurich, August 2016 DPG Method 75 / 180

100 Mesh 19 and real part of pressure ETH Zurich, August 2016 DPG Method 76 / 180

101 Mesh 20 and real part of pressure ETH Zurich, August 2016 DPG Method 77 / 180

102 Mesh 21 and real part of pressure ETH Zurich, August 2016 DPG Method 78 / 180

103 Mesh 22 and real part of pressure ETH Zurich, August 2016 DPG Method 79 / 180

104 Mesh 23 and real part of pressure ETH Zurich, August 2016 DPG Method 80 / 180

105 Mesh 24 and real part of pressure ETH Zurich, August 2016 DPG Method 81 / 180

106 Mesh 25 and real part of pressure ETH Zurich, August 2016 DPG Method 82 / 180

107 Mesh 26 and real part of pressure ETH Zurich, August 2016 DPG Method 83 / 180

108 Mesh 27 and real part of pressure ETH Zurich, August 2016 DPG Method 84 / 180

109 Mesh 28 and real part of pressure ETH Zurich, August 2016 DPG Method 85 / 180

110 Mesh 29 and real part of pressure ETH Zurich, August 2016 DPG Method 86 / 180

111 Mesh 30 and real part of pressure ETH Zurich, August 2016 DPG Method 87 / 180

112 Mesh 31 and real part of pressure ETH Zurich, August 2016 DPG Method 88 / 180

113 Mesh 32 and real part of pressure ETH Zurich, August 2016 DPG Method 89 / 180

114 Mesh 33 and real part of pressure ETH Zurich, August 2016 DPG Method 90 / 180

115 Mesh 34 and real part of pressure ETH Zurich, August 2016 DPG Method 91 / 180

116 Mesh 35 and real part of pressure ETH Zurich, August 2016 DPG Method 92 / 180

117 Mesh 36 and real part of pressure ETH Zurich, August 2016 DPG Method 93 / 180

118 Mesh 37 and real part of pressure ETH Zurich, August 2016 DPG Method 94 / 180

119 Mesh 38 and real part of pressure ETH Zurich, August 2016 DPG Method 95 / 180

120 Mesh 39 and real part of pressure ETH Zurich, August 2016 DPG Method 96 / 180

121 Mesh 40 and real part of pressure ETH Zurich, August 2016 DPG Method 97 / 180

122 Mesh 41 and real part of pressure ETH Zurich, August 2016 DPG Method 98 / 180

123 Mesh 42 and real part of pressure ETH Zurich, August 2016 DPG Method 99 / 180

124 Mesh 44 and real part of pressure ETH Zurich, August 2016 DPG Method 100 / 180

125 Mesh 46 and real part of pressure ETH Zurich, August 2016 DPG Method 101 / 180

126 Residual and L 2 error convergence history Error (dotted line) decreases monotonically while residual (solid line) decreases monotically only in the p-refinement stage. ETH Zurich, August 2016 DPG Method 102 / 180

127 Fichera corner Divide it into eight smaller cubes and remove one: ETH Zurich, August 2016 DPG Method 103 / 180

128 Fichera corner microwave Attach a waveguide: ɛ = µ = 1, σ = 0 ω = 5(1.6 wavelengths in the cube) Cut the waveguide and use the lowest propagating mode for BC along the cut. ETH Zurich, August 2016 DPG Method 104 / 180

129 Fichera corner microwave Standard variational formulation (Primal DPG method) Standard test norm: ψ 2 V := ψ 2 H(curl,Ω) = ψ 2 + ψ 2 ETH Zurich, August 2016 DPG Method 105 / 180

130 Initial mesh and real part of E 1 ETH Zurich, August 2016 DPG Method 106 / 180

131 Mesh and real part of E 1 after 1st refinement ETH Zurich, August 2016 DPG Method 107 / 180

132 Mesh and real part of E 1 after 2nd refinement ETH Zurich, August 2016 DPG Method 108 / 180

133 Mesh and real part of E 1 after 3rd refinement ETH Zurich, August 2016 DPG Method 109 / 180

134 Mesh and real part of E 1 after 4th refinement ETH Zurich, August 2016 DPG Method 110 / 180

135 Mesh and real part of E 1 after 5th refinement ETH Zurich, August 2016 DPG Method 111 / 180

136 Mesh and real part of E 1 after 6th refinement ETH Zurich, August 2016 DPG Method 112 / 180

137 Mesh and real part of E 1 after 7th refinement ETH Zurich, August 2016 DPG Method 113 / 180

138 Mesh and real part of E 1 after 8th refinement ETH Zurich, August 2016 DPG Method 114 / 180

139 Mesh and real part of E 1 after 9th refinement ETH Zurich, August 2016 DPG Method 115 / 180

140 Mesh and real part of E 1 after 10th refinement ETH Zurich, August 2016 DPG Method 116 / 180

141 Residual history Residual decreases monotonically. ETH Zurich, August 2016 DPG Method 117 / 180

142 Fichera corner microwave Ultraweak variational formulation (Original DPG method) Adjoint graph norm: (φ, ψ) 2 V := φ 2 + ψ 2 + φ + (iωɛ σ)ψ 2 + ψ iωµφ 2 ETH Zurich, August 2016 DPG Method 118 / 180

143 Initial mesh and real part of E 1 ETH Zurich, August 2016 DPG Method 119 / 180

144 Mesh and real part of E 1 after 1st refinement ETH Zurich, August 2016 DPG Method 120 / 180

145 Mesh and real part of E 1 after 2nd refinement ETH Zurich, August 2016 DPG Method 121 / 180

146 Mesh and real part of E 1 after 3rd refinement ETH Zurich, August 2016 DPG Method 122 / 180

147 Mesh and real part of E 1 after 4th refinement ETH Zurich, August 2016 DPG Method 123 / 180

148 Mesh and real part of E 1 after 5th refinement ETH Zurich, August 2016 DPG Method 124 / 180

149 Mesh and real part of E 1 after 6th refinement ETH Zurich, August 2016 DPG Method 125 / 180

150 Mesh and real part of E 1 after 7th refinement ETH Zurich, August 2016 DPG Method 126 / 180

151 Mesh and real part of E 1 after 8th refinement ETH Zurich, August 2016 DPG Method 127 / 180

152 Residual history Residual decreases monotonically. ETH Zurich, August 2016 DPG Method 128 / 180

153 Comparison of primal and UW results Fichera microwave problem. Comparison of results obtained with primal (left) and ultraweak (right) formulations, meshes 3-9. ETH Zurich, August 2016 DPG Method 129 / 180

154 Comparison of primal and UW results Fichera microwave problem. Comparison of results obtained with primal (left) and ultraweak (right) formulations, meshes 3-9. ETH Zurich, August 2016 DPG Method 129 / 180

155 Comparison of primal and UW results Fichera microwave problem. Comparison of results obtained with primal (left) and ultraweak (right) formulations, meshes 3-9. ETH Zurich, August 2016 DPG Method 129 / 180

156 Comparison of primal and UW results Fichera microwave problem. Comparison of results obtained with primal (left) and ultraweak (right) formulations, meshes 3-9. ETH Zurich, August 2016 DPG Method 129 / 180

157 Comparison of primal and UW results Fichera microwave problem. Comparison of results obtained with primal (left) and ultraweak (right) formulations, meshes 3-9. ETH Zurich, August 2016 DPG Method 129 / 180

158 Comparison of primal and UW results Fichera microwave problem. Comparison of results obtained with primal (left) and ultraweak (right) formulations, meshes 3-9. ETH Zurich, August 2016 DPG Method 129 / 180

159 Comparison of primal and UW results Fichera microwave problem. Comparison of results obtained with primal (left) and ultraweak (right) formulations, meshes 3-9. ETH Zurich, August 2016 DPG Method 129 / 180

160 A non-trivial question for Maxwell Primal formulation: ETH Zurich, August 2016 DPG Method 130 / 180

161 A non-trivial question for Maxwell Primal formulation: Does the satisfaction of the weak form of the Ampère equation imply the satisfaction of the continuity equation? ETH Zurich, August 2016 DPG Method 130 / 180

162 A non-trivial question for Maxwell Primal formulation: Does the satisfaction of the weak form of the Ampère equation imply the satisfaction of the continuity equation? More precisely, does the minimization of the residual corresponding to the Ampère eqn imply the control of the residual corresponding to the continuity equation? (Not the case for an explicit a-posteriori error estimation for the standard Galerkin method.) ETH Zurich, August 2016 DPG Method 130 / 180

163 A non-trivial question for Maxwell Primal formulation: Does the satisfaction of the weak form of the Ampère equation imply the satisfaction of the continuity equation? More precisely, does the minimization of the residual corresponding to the Ampère eqn imply the control of the residual corresponding to the continuity equation? (Not the case for an explicit a-posteriori error estimation for the standard Galerkin method.) The answer to the last two questions seems to be positive. Indeed, b((e, sup Ĥ), hq) q q H1 (Ω h ) }{{} controlled sup q since h H 1 (Ω h ) H(curl, Ω h ). b((e, Ĥ), hq) h q b((e, Ĥ), ψ) sup ψ ψ H(curl,Ωh ) }{{} minimized ETH Zurich, August 2016 DPG Method 130 / 180

164 More examples DPG works when standard Galerkin may not ETH Zurich, August 2016 DPG Method 131 / 180

165 Full Wave Form Inversion From Ph.D. Dissertation of Jamie Bramwell: (a) CG µ, low frequency (b) DPG µ, high frequency (c) CG λ, low frequency (d) DPG λ, high frequency Solution of Elastic Marmousi problem: Left: standard Galerkin, Right: DPG Jamie Bramwell, CSEM (co-supervised with O. Ghattas), A Discontinuous Petrov-Galerkin Method for Seismic Tomography Problems, Ph.D. Thesis, University of Texas at Austin, April ETH Zurich, August 2016 DPG Method 132 / 180

166 Metamaterials: Model Problem 1 where Ω = ( 2, 1) (0, 1), and { x1 < 0 a(x) = x 1 > 0 { Mesh and corresponding solution u = 0 on Γ div(a(x) u) = f in Ω f(x) = { 1 x1 < 0 0 x 1 > 0 Work with Patrick Ciarlet ETH Zurich, August 2016 DPG Method 133 / 180

167 Metamaterials: Model Problem 1 Convergence of DPG vs. standard Galerkin method Left: residuals for DPG method. Right: Norm of the difference between fine and coarse mesh solutions ETH Zurich, August 2016 DPG Method 134 / 180

168 Metamaterials: Model Problem 2 - Scattering { iωσ 1 u + p = 0 with impedance BC: iωηp +divu = 0 p u n = g inc := p inc u inc n Coefficients σ, η are negative for the scatterer. Optimal Mesh and real part of u 1 after 10 refinements: ETH Zurich, August 2016 DPG Method 135 / 180

169 Metamaterials: Model Problem 2 Convergence of DPG vs. standard Galerkin method Left: residual for DPG method. Right: Norm of the difference between fine and coarse mesh solutions ETH Zurich, August 2016 DPG Method 136 / 180

170 2D acoustics (electromagnetics) cloaking problem You can solve efficiently cloaking problems if you select the right test norm. Exact solution (pressure or magnetic field) L.D and J. Li, Numerical Simulations of Cloaking Problems using a DPG Method, Comp. Mech., 51(5): , ETH Zurich, August 2016 DPG Method 137 / 180

171 2D acoustics (electromagnetics) cloaking problem You can solve efficiently cloaking problems if you select the right test norm. An hp mesh (4 bilinear elements per wavelength) L.D and J. Li, Numerical Simulations of Cloaking Problems using a DPG Method, Comp. Mech., 51(5): , ETH Zurich, August 2016 DPG Method 137 / 180

172 2D acoustics (electromagnetics) cloaking problem You can solve efficiently cloaking problems if you select the right test norm. Numerical solution (pressure or magnetic field) L.D and J. Li, Numerical Simulations of Cloaking Problems using a DPG Method, Comp. Mech., 51(5): , ETH Zurich, August 2016 DPG Method 137 / 180

173 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. High frequency problems. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 138 / 180

174 High frequency time-harmonic wave propagation problems Linear Acoustics iωu + p = 0, in Ω iωp + div u = 0 p u n = g on Ω where u is the velocity and p is the pressure. DPG Ultra-weak Formulation (Relax both equations) u (L 2 (Ω)) d, p L 2 (Ω) û H 1/2 (Γ h ), ˆp H 1/2 (Γ h ) ˆp û = g, on Ω (iωu, v) (p, div h v) + ˆp, v n = 0, v H(div, Ω h ) (iωp, q) (u, h q) + û, q = 0, q H 1 (Ω h ) where Γ h is the mesh skeleton and H 1/2 (Γ h ) := tr H(div, Ω) on Γ h, H 1/2 (Γ h ) := tr H 1 (Ω) on Γ h ETH Zurich, August 2016 DPG Method 139 / 180

175 High frequency time-harmonic wave propagation problems Remarks on Ultra-weak formulation Denote A ω (u, p) = (iωu + p, iωp + divu). Then A ω = A ω. Ideally if v V = A v then the method delivers L 2 projection in U. u E = sup v V b(u, v) v V = sup v V (u, A v) A v = u In practice, in order to resolve the optimal test functions, we use a weighted adjoint graph norm with the continuation parameter ω = max(ω, 6/h), where h is the element size. where α is O(1). v V = A ωv + α v Field unknowns u and p are condensed out of the final system and we solve only for the traces and fluxes, reducing the size of the final system significantly. ETH Zurich, August 2016 DPG Method 140 / 180

176 Resonating Cavity (300 wavelengths) Refinement 0 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

177 Resonating Cavity (300 wavelengths) Refinement 20 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

178 Resonating Cavity (300 wavelengths) Refinement 40 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

179 Resonating Cavity (300 wavelengths) Refinement 60 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

180 Resonating Cavity (300 wavelengths) Refinement 80 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

181 Resonating Cavity (300 wavelengths) Refinement 100 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

182 Resonating Cavity (300 wavelengths) Refinement 120 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

183 Resonating Cavity (300 wavelengths) Refinement 140 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

184 Resonating Cavity (300 wavelengths) Refinement 160 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

185 Resonating Cavity (300 wavelengths) Refinement 180 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

186 Resonating Cavity (300 wavelengths) Refinement 200 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

187 Resonating Cavity (300 wavelengths) Refinement 220 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

188 Resonating Cavity (300 wavelengths) Refinement 240 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

189 Resonating Cavity (300 wavelengths) Refinement 260 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

190 Resonating Cavity (300 wavelengths) Refinement 280 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

191 Resonating Cavity (300 wavelengths) Refinement 300 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

192 Resonating Cavity (300 wavelengths) Refinement 350 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

193 Resonating Cavity (300 wavelengths) Refinement 400 Mesh Pressure ETH Zurich, August 2016 DPG Method 141 / 180

194 Resonating Cavity (300 wavelengths) Residual convergence 10 1 hp-adaptivity Residual Degrees of Freedom ETH Zurich, August 2016 DPG Method 142 / 180

195 Integration of solvers with adaptivity A multi-frontal solver is faster than any iterative solver for this kind of meshes (1D structure). However restarting the problem 400 times makes the cost prohibitive. This calls for the use of an iterative solver where a partially converged solution could drive adaptivity. Being a minimum residual method, DPG always delivers a Hermitian positive definite stiffness matrix. This makes the Conjugate Gradient ideal but there is still the need for a good preconditioner. Ultimate Goal: Integrate the solver with adaptivity by building a preconditioner for Conjugate Gradient that utilizes information from previous meshes. ETH Zurich, August 2016 DPG Method 143 / 180

196 Two grid-like Preconditioner for Conjugate Gradient Description Step 1: Direct Solver. Use a direct solver to solve the problem at the n th mesh and store the Cholesky decomposition of the matrix. This will be our coarse mesh. ETH Zurich, August 2016 DPG Method 144 / 180

197 Two grid-like Preconditioner for Conjugate Gradient Description Step 1: Direct Solver. Use a direct solver to solve the problem at the n th mesh and store the Cholesky decomposition of the matrix. This will be our coarse mesh. Step 2: Macro grid. n th (Coarse) Grid (n + k) th (Fine) Grid ETH Zurich, August 2016 DPG Method 144 / 180

198 Two grid-like Preconditioner for Conjugate Gradient Description Step 1: Direct Solver. Use a direct solver to solve the problem at the n th mesh and store the Cholesky decomposition of the matrix. This will be our coarse mesh. Step 2: Macro grid. n th (Coarse) Grid (n + k) th (Macro) Grid ETH Zurich, August 2016 DPG Method 144 / 180

199 Two grid-like Preconditioner for Conjugate Gradient Description Step 1: Direct Solver. Use a direct solver to solve the problem at the n th mesh and store the Cholesky decomposition of the matrix. This will be our coarse mesh. Step 2: Macro grid. n th (Coarse) Grid (n + k) th (Macro) Grid On the (n + k) th, 1 k m (fine) mesh we statically condense out all the new degrees of freedom which are not on the skeleton of the n th (coarse) mesh. We name this the macro mesh. ETH Zurich, August 2016 DPG Method 144 / 180

200 Two grid-like Preconditioner for Conjugate Gradient Description Step 3: Additive Schwarz Smoother ETH Zurich, August 2016 DPG Method 145 / 180

201 Two grid-like Preconditioner for Conjugate Gradient Description Step 3: Additive Schwarz Smoother n th (Coarse) Grid (n + k) th (Macro) Grid ETH Zurich, August 2016 DPG Method 145 / 180

202 Two grid-like Preconditioner for Conjugate Gradient Description Step 3: Additive Schwarz Smoother n th (Coarse) Grid (n + k) th (Macro) Grid Given a macro grid we define a patch to be the support of a coarse grid vertex basis function. A block is constructed by the interactions of the macro degrees of freedom within a patch. ETH Zurich, August 2016 DPG Method 145 / 180

203 Two grid-like Preconditioner for Conjugate Gradient Description Step 4: Coarse grid correction ETH Zurich, August 2016 DPG Method 146 / 180

204 Two grid-like Preconditioner for Conjugate Gradient Description Step 4: Coarse grid correction n th (Coarse) Grid (n + k) th (Macro) Grid ETH Zurich, August 2016 DPG Method 146 / 180

205 Two grid-like Preconditioner for Conjugate Gradient Description Step 4: Coarse grid correction n th (Coarse) Grid (n + k) th (Macro) Grid Construct prolongation (I m c ) and restriction (I c m = (I m c ) ) operators between the coarse and the macro mesh. Given that the two meshes have the same geometric topology and also the fact that all the degrees of freedom live on the skeleton of the meshes, the construction of such operators can be done edge-wise. (Face-wise in 3D.) ETH Zurich, August 2016 DPG Method 146 / 180

206 Two grid-like Preconditioner for Conjugate Gradient Description Step 5: PCG with Additive Schwarz and coarse-grid correction. We implement a two-grid cycle between the coarse and the macro grid. Let S be the smoothing operator, C = Ic m A 1 c Im c the coarse grid correction, I the identity operator and let P = I A m C, where A m denotes the macro grid stiffness matrix. Two grid cycle Pre-smooth Post-smooth (Addititive Schwarz) (Addititive Schwarz) Restriction(I c m) Prolongation(I m c ) Then the resulting preconditioner is Hermitian and positive definite and it s given explicitly by Coarse grid solve (Back substitution) M = SP + P S + C SP A m S ETH Zurich, August 2016 DPG Method 147 / 180

207 Two grid-like Preconditioner for Conjugate Gradient Results (40 wavelengths) Additive Schwarz vs Two-grid Preconditioner 400 Two Grid PCG Additive Schwarz PCG Direct Solve Number of Iterations Degrees of freedom ETH Zurich, August 2016 DPG Method 148 / 180

208 Comments on the new solver The coarse grid correction seems to be necessary in order for the iterations to remain bounded. However, the cost of a single iteration increases within every refinement (static condensation, smoothing, coarse grid solve). The use of a relaxation parameter for smoothing is very essential. Hope for some analysis. It is sufficient to use a partially converged solution to drive adaptivity. ETH Zurich, August 2016 DPG Method 149 / 180

209 Current work Convergence Analysis 3D acoustics and Maxwell (OpenMP/MPI implementations may be necessary.) ETH Zurich, August 2016 DPG Method 150 / 180

210 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. High frequency problems. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 151 / 180

211 Motivation: Nonlinear Optics Study of EM-wave propagation in optical fibers Maxwell equations: E = µ 0 H t H = ɛ 0 E t + P t, Polarization vector P is composed of a linear and nonlinear part: P = P L + P NL. P L : linear Lorentz polarization, P NL : nonlinear Kerr and Raman polarization.. P = ɛ 0 (χ 1 E + χ 2 : E E + χ 3. E E E) ETH Zurich, August 2016 DPG Method 152 / 180

212 Non Linear Schrödinger Equation in Optics The principal model of Maxwell equations in fiber optics adopts the slowly varying envelope approximation: E(r, t) = 1 2ˆx (E(r, t) e iω0t + c.c. ) Additional assumptions and approximations imply an approximate solution in (r, θ, z): E(r, θ, z) = 1 2ˆx (F (r, θ) u(z, t) ei(β0z ω0t) + c.c. ) Complexified amplitude function u(z, t) satisfies a nonlinear Schrödinger (NLS) type equation: i u z β 2 2 u 2 t 2 + γ u 2 u = 0 Space and time are reversed ETH Zurich, August 2016 DPG Method 153 / 180

213 The Linear Schrödinger Equation in n Dimensions Ω 0 R n is a bounded, Lipschitz domain and time t [0, T ] with T <. Ω := Ω 0 [0, T ]. Define Γ T = Ω 0 [0, T ] Ω 0 {t = T }. Consider the LSE in n space dimensions: i u u = f in Ω, t u(t, ) = 0 on Ω 0, u = 0 at t = 0. ETH Zurich, August 2016 DPG Method 154 / 180

214 The Linear Schrödinger Equation in n Dimensions Define Lu := u and Schrödinger operator Au := iu t + Lu. We have an orthonormal eigenbasis e k (x) and eigenvalues 0 < ω1 2 ω2 2 ωk 2 that satisfy: and Le k (x) = ω 2 ke k (x). u(t, x) = u k (t)e k (x), k=1 Au = f implies, with f k (t) = (f, e k (x)) L2 (Ω): i u k (t) + ω 2 ku k (t) = f k (t), This leads to an explicit formula for the solution: u(t, x) = ( i k=1 t 0 e iω2 k (t s) f k (s)ds)e k (x). ETH Zurich, August 2016 DPG Method 155 / 180

215 Ill-posedness of 1st Order Formulation The bound u L2 (Ω) C T f L2 (Ω) = C T Au L2 (Ω) holds for some C = C(Ω). However, we cannot control derivatives: u L2 (Ω), u t L2 (Ω) can be unbounded. Consequently, the naive 1st order formulation i u div τ = f, t u τ = g, is ill-posed in the L 2 setting! u(t, ) = 0 on Ω 0, u = 0 at t = 0. Unpleasant reality: We cannot develop a DPG (or any) method for the above first order system formulation! ETH Zurich, August 2016 DPG Method 156 / 180

216 Only 2 Variational Formulations in Space-Time We therefore have only 2 well-posed 2nd order formulations. The strong and ultra-weak: Strong formulation: { u {w L 2 (Ω) : Aw L 2 (Ω) with B.C. s} Mathematical challenges: (Au, v) L2 (Ω) = (f, v) L 2 (Ω) Non-standard graph energy space W = {u L 2 (Ω) : Au L 2 (Ω)} Unlike H 1, H(curl), H(div), we do not have notion of trace for W. Integration by parts gives: (u, A v) L 2 (Ω) + Du, v = (f, v) L 2 (Ω) Need to develop a boundary map D : W (W ) to account for relaxing the strong formulation Need to develop specialized conforming element and best approximation error estimates for the discrete solution ETH Zurich, August 2016 DPG Method 157 / 180

217 The Boundary Operator D The boundary operator D : W (W ) defined as: Du, v = (Au, v) L 2 (Ω) (u, A v) L 2 (Ω) D is continuous with graph norms on W, W. D generalizes notion of trace: action of D can be thought as composition of trace (when it exists) with duality pairing on boundary Using D, we interpret domain of operator A as : V := dom(a) = (DV ), where V := D ΓT W is the space of smooth functions in W that vanish on the boundary Γ T. Standard Friedrichs theory not applicable: A + A not L 2 bounded for linear Schrödinger operator Ern, Guermond, Caplain, An intrisic criterion for the bijectivity of Hilbert operators related to Friedrichs systems, Communications in Partial Differential Equations, 32(2) , Feb 2007 ETH Zurich, August 2016 DPG Method 158 / 180

218 Well-posedness of the Strong and UW Formulations The linear Schrödinger operator A : V L 2 (Ω) is a continuous bijection. Thus, the strong formulation { u V, (Au, v) L 2 (Ω) = (f, v) L 2 (Ω), v L 2 (Ω) corresponding to A is inf-sup stable, i.e., A is bounded below. The UW formulation is obtained by treating Du as an additional unknown from the quotient û (W ) /D(V ) { u L 2 (Ω), û (W ) /D(V ), (u, A v) L2 (Ω) + û, v (W ),W = (f, v) L 2 (Ω), v W is also inf-sup stable. Here, û comes from the quotient space (W ) /D(V ) and must be measured in the quotient norm. ETH Zurich, August 2016 DPG Method 159 / 180

219 UW DPG Method We now have the broken UW variational formulation: { u L 2 (Ω), û (W h ) /D h (V ), (u, A hv) L2 (Ω) + û, v h = (f, v) L2 (Ω), v W where A h is the operator A applied elementwise, W h is the broken graph space and D h is the elementwise auxiliary pairing operator: Wh := W (K) K Ω h D h u, v h := Du K, v K (W (K)),W (K) K Ω h The test norm on W is given as: v 2 W = (v t, v t ) L2 (Ω) + ( v, v) L2 (Ω) + i(v t, v) L2 (Ω) i( v, v t ) L2 (Ω) Notice the fourth order nature of the ( v, v) L 2 (Ω) term ETH Zurich, August 2016 DPG Method 160 / 180

220 Conforming Discretization: 1D Space Case for Optics Stability of the ideal DPG method: ( u u h 2 L 2 (Ω) + û û h Q) 1/2 C( inf w h,ŵ h ( u w h 2 L 2 (Ω) + û ŵ h Q) 1/2 ), where Q is the quotient norm in (W ) /D(V ). We need an A-operator conforming FE space V h and an interpolation operator Π : V V h with a 1D space-time element whose degrees of freedom respect conformity. The conforming element with p = 2 Sufficient conditions for conformity: u t, u xx L 2 (Ω) = Au L 2 (Ω). V h := P p+1 P p+2 ETH Zurich, August 2016 DPG Method 161 / 180

221 Interpolation Error Estimates Applying Cauchy-Schwarz inequality gives: D(u Πu) ( A(u Πu) 2 L 2 (Ω) + u Πu 2 L 2 (Ω) ) 1 2. We estimate u Πu L2 (K), (u Πu) t L2 (K) and (u Πu) xx L2 (K). Overall, u Πu V C(Ω)h p+1 u H p+2. Here, p + 1 is polynomial order in time and p + 2 in space Lowest order element is order 2 in time and 3 in space ETH Zurich, August 2016 DPG Method 162 / 180

222 Rates of Convergence ETH Zurich, August 2016 DPG Method 163 / 180

223 Solution Plots Plots of solutions: exact complex Gaussian (left), and manufactured Gaussian beam solution with ω = 20 (right) ETH Zurich, August 2016 DPG Method 164 / 180

224 Solving 1D NLS Linearize the NLS and apply Newton iterations to the linearized problem to converge to the solution of the NLS Define The linearized scheme is: A(u) = i u t 2 u x + γ u 2 u, 2 L(u) = i u t 2 u x. 2 u 0 : initial guess Solve for u n : L( u n ) + γ(2 u n 2 u n + u 2 n( u n ) ) = A(u n ) u n+1 = u n + u n Convergence criterion: u n < ɛ ETH Zurich, August 2016 DPG Method 165 / 180

225 Summary Similar to other wave propagation problems (first order acoustics and Maxwell equations ), the LSE in space-time domain admits only two variational formulations: strong and ultraweak. However, contrary to acoustics and Maxwell systems, the LSE does not admit a well-posed, 1st order L 2 stable variational formulation. Inversion of Gram matrix involving second derivative leads to conditioning issues Is there a better way to proceed? C.Wieners, The Skeleton Reduction for Finite Element Substructuring Methods, ENUMATH Proceedings 2015 ETH Zurich, August 2016 DPG Method 166 / 180

226 Outline Petrov-Galerkin Method with Optimal Test Functions. Variational formulations. Breaking test functions. Localization. Approximation of optimal test functions. Implementation. Numerical examples. High frequency problems. Preconditioning. Space-time discretizations. Schrödinger equation. Relation with Barret-Morton optimal test functions and weakly conforming least squares. ETH Zurich, August 2016 DPG Method 167 / 180

227 εdpg method Ultraweak variational formulation for the first order system of linear acoustics equations with impedance BC on Γ : u n p = g, (p, iωq + div h v) + (u, iωv + h q) + u n p, q Γ 0 h + ˆp, v n + q Γh = g, q Γ Abstract notation: where (watch for overloaded symbols): (u, A hv) + ŵ, v = (f, v) + û 0, v u = (p, u) v = (q, v) A hv = (ωq + div h v, ωv + h q) ŵ = ( u n p, ˆp) εdpg method minimizes residual in dual norm to the test norm: v 2 V := A hv 2 + ɛ 2 v 2 ETH Zurich, August 2016 DPG Method 168 / 180

228 Things improve with ε 0: Dissipation Simulate a plane wave propagating at θ = π/8 ETH Zurich, August 2016 DPG Method 169 / 180