S PACE –T IME M ETHODS FOR PDE S , 7–11 N OVEMBER 2016, R ICAM , L INZ Space–time Trefftz discontinuous Galerkin methods for wave problems Andrea Moiola D EPARTMENT OF M ATHEMATICS AND S TATISTICS , U NIVERSITY OF R EADING Joint work with I. Perugia
Minimal Trefftz example: Laplace equation Imagine you want to approximate the solution u of Laplace eq. in Ω ⊂ R n , ( +any BCs on ∂ Ω) , ∆ u = 0 using a standard discontinuous Galerkin (DG) method. You seek the approximate solution in � � v hp ∈ L 2 (Ω) : v hp | K ∈ P p ( K ) ∀ K ∈ T h where P p ( K ) is the space of polynomials of degree at most p on the element K of a mesh T h . Why not use only (piecewise) harmonic polynomials � � v hp ∈ L 2 (Ω) : v hp | K ∈ P p ( K ) , ∆ v hp | K = 0 ∀ K ∈ T h ? Comparable accuracy for O ( p n − 1 · # el ) vs O ( p n · # el ) DOFs. (E.g., n =2, p =10: 21 vs 66 DOFs/el.; p =20: 41 vs 231 DOFs/el.) 2
Trefftz methods Consider a linear PDE L u = 0 . Trefftz methods are finite element schemes such that test and trial functions are solutions of the PDE in each element K of the mesh T h . E.g.: piecewise harmonic polynomials if L u = ∆ u . Our main interest is in wave propagation, in: ◮ Frequency domain, Helmholtz eq. − ∆ u − k 2 u = 0 lot of work done, h / p / hp -theory, Maxwell, elasticity. . . (recent survey: Hiptmair, AM, Perugia, arXiv:1506.04521) c 2 ∂ 2 ◮ Time domain, wave equation − ∆ U + 1 ∂ t 2 U = 0 Trefftz methods are in space–time, as opposed to semi-discretisation + time-stepping. 3
Trefftz methods for wave equation Why Trefftz methods? Comparing with standard DG, ◮ better accuracy per DOFs and higher convergence orders; ◮ PDE properties “known” by discrete space, e.g. dispersion; ◮ lower dimensional quadrature needed; ◮ simpler and more flexible, adapted bases and adaptivity. . . No typical drawbacks of time-harmonic Trefftz (ill-cond., quad.). Existing works on Trefftz for time-domain wave equation: ◮ M ACIA ¸ G , S OKALA , W AUER 2005–2011, L IU , K UO 2016, single element Trefftz; ◮ P ETERSEN , F ARHAT , T EZAUR , W ANG 2009&2014, DG with Lagrange multipliers; ◮ E GGER , K RETZSCHMAR , S CHNEPP , T SUKERMAN , W EILAND 3 × 2014–2015, Maxwell equations; K RETZSCHMAR , M OIOLA , P ERUGIA , S CHNEPP 2015, analysis; M OIOLA , P ERUGIA , arXiv:1610.08002. ◮ B ANJAI , G EORGOULIS , L IJOKA 2016, interior penalty-DG (see talk on Wednesday). 4
Simplest Trefftz space: Trefftz polynomials in K ⊂ R n + 1 ( c const). − ∆ U + c − 2 U ′′ = 0 Consider wave eq. Choose Trefftz space of polynomials of deg. ≤ p on element K : � � v ∈ P p ( K ) , − ∆ v + c − 2 v ′′ = 0 U p ( K ) : = . ◮ Basis functions are easily constructed: b j ,ℓ ( x , t ) = ( d j ,ℓ · x − ct ) j for suitable propagation directions d j ,ℓ ( | d j ,ℓ | = 1 ). ◮ Orders of approximation in h are for free, because Taylor polynomial of (smooth) U belongs to U p ( K ) . � � � � U p ( K ) = O p →∞ ( p n ) ≪ dim P p ( K ) = O p →∞ ( p n + 1 ) . ◮ dim 5
Part II Trefftz-DG for acoustic wave equations
Initial–boundary value problem First order initial–boundary value problem (Dirichlet): find ( v , σ ) ∇ v + ∂ σ in Q = Ω × ( 0 , T ) ⊂ R n + 1 , n ∈ N , ∂ t = 0 ∇ · σ + 1 ∂ v in Q , ∂ t = 0 c 2 on Ω , v ( · , 0 ) = v 0 , σ ( · , 0 ) = σ 0 on ∂ Ω × ( 0 , T ) . v ( x , · ) = g From − ∆ U + c − 2 ∂ 2 ∂ t 2 U = 0 , choose v = ∂ U ∂ t and σ = −∇ U . Ω ⊂ R n Lipschitz bounded. Velocity c piecewise constant. ◮ Neumann σ · n = g & Robin ϑ c v − σ · n = g BCs ( � ), ◮ Maxwell equations ( � ), Extensions: ◮ elasticity, ◮ 1 st order hyperbolic systems ( ∼ ), ◮ Maxwell equations in dispersive materials. . . 6
Space–time mesh and assumptions Introduce space–time polytopic mesh T h on Q . Assume: c = c ( x ) constant in elements. Assume: each face F = ∂ K 1 ∩ ∂ K 2 with normal ( n x F ) is either F , n t ◮ space-like: c | n x F , denote F ⊂ F space , or F | < n t h ◮ time-like: n t F = 0 , denote F ⊂ F time . h DG notation: t F T w | K 1 + w | K 2 τ | K 1 + τ | K 2 h T { { w } } := , { { τ } } := , 2 2 ] N := w | K 1 n x K 1 + w | K 2 n x [ [ w ] K 2 , ] N := τ | K 1 · n x K 1 + τ | K 2 · n x [ [ τ ] K 2 , K 2 = ( w − − w + ) n t ] t := w | K 1 n t K 1 + w | K 2 n t [ [ w ] F , K 2 = ( τ − − τ + ) n t ] t := τ | K 1 n t K 1 + τ | K 2 n t [ [ τ ] F , n x K K F 0 F T 0 h := Ω × { 0 } , h := Ω × { T } , x F 0 F space F time h F ∂ h := ∂ Ω × [ 0 , T ] . h h 7
DG elemental equation and numerical fluxes � Trefftz ( w , τ ) ∈ L 2 ( Q ) , ( w | K , τ | K ) ∈ H 1 ( K ) 1 + n , T ( T h ) := space: � ∇ w + ∂ τ ∇ · τ + c − 2 ∂ w ∂ t = 0 , ∂ t = 0 ∀ K ∈ T h . Multiplying PDEs with test ( w , τ ) , integrating by parts in K , using Trefftz property and summing over K ∈ T h : ∀ ( w , τ ) ∈ T ( T h ) � � � � � � σ · τ + 1 ( v τ + σ w ) · n x n t K + c 2 v w d S = 0 . K ∂ K K ∈T h We approximate skeleton traces of ( v , σ ) with numerical fluxes σ hp ) , defined as α, β ∈ L ∞ ( F time ∪ F ∂ ( � v hp , � h ) h v − σ − on F space , hp hp h on F T v hp σ hp h , on F 0 � v hp := σ hp := � h , v 0 σ 0 on F time { { v hp } } + β [ { { σ hp } } + α [ [ σ hp ] ] N [ v hp ] ] N , h on F ∂ σ hp − α ( v − g ) n x g h . Ω α = β = 0 → K RETZSCHMAR –S.–T.–W., αβ ≥ 1 4 → M ONK –R ICHTER . 8
Trefftz-DG formulation Substituting the fluxes in the elemental equation and choosing any finite-dimensional V p ( T h ) ⊂ T ( T h ) , write Trefftz-DG as: Seek ( v hp , σ hp ) ∈ V p ( T h ) s.t. , ∀ ( w , τ ) ∈ V p ( T h ) , where A ( v hp , σ hp ; w , τ ) = ℓ ( w , τ ) � v − � hp [ [ w ] ] t � + σ − ] t + v − ] N + σ − A ( v hp , σ hp ; w , τ ):= hp · [ [ τ ] hp [ [ τ ] hp · [ [ w ] ] N d S c 2 F space h � � � + { { v hp } } [ [ τ ] ] N + { { σ hp } } · [ [ w ] ] N + α [ [ v hp ] ] N · [ [ w ] ] N + β [ [ σ hp ] ] N [ [ τ ] ] N d S F time h � � � � ( c − 2 v hp w + σ hp · τ ) d S + + σ hp · n Ω + α v hp w d S , F T F ∂ h h � � ( c − 2 v 0 w + σ 0 · τ ) d S + ℓ ( w , τ ) := g ( α w − τ · n Ω ) d S . F 0 F ∂ h h 9
Global, implicit and explicit schemes 1 Trefftz-DG formulation is global in space–time domain Q : large linear system! Might be good for adaptivity. t 2 If mesh is partitioned in time-slabs S 3 Ω × ( t j − 1 , t j ) , matrix is block lower-triangular: S 2 for each time-slab a system can be solved S 1 sequentially: implicit method. x 3 If mesh is suitably chosen, Trefftz-DG solution t can be computed with a sequence of local systems: explicit method, allows parallelism! “Tent pitching algorithm” of Ü NGÖR –S HEFFER , x M ONK –R ICHTER , G OPALAKRISHNAN –M ONK –S EPÚLVEDA , G OPALAKRISHNAN –S CHÖBERL –W INTERSTEIGER . . . (See talk tomorrow.) Versions 1–2–3 are algebraically equivalent (on the same mesh). 10
Tent-pitched elements Tent-pitched elements/patches obtained from regular space meshes in 2+1D give parallelepipeds or octahedra+tetrahedra: t t 2 3 2 1 1 1 2 2 2 1 3 2 3 3 3 2 1 3 2 1 1 1 2 2 3 2 2 2 2 1 2 3 3 3 2 3 1 1 1 1 1 3 2 2 1 2 2 2 3 2 3 3 3 3 2 2 3 1 2 1 1 1 2 3 2 2 2 1 3 2 3 3 2 1 3 2 1 1 2 2 3 2 3 3 Trefftz requires quadrature on faces only: only the shape of space elements matters. Simplices around a tent pole can be merged in single element. 11
Part III Trefftz-DG error analysis
Trefftz-DG norms Assume α, β > 0 , F ∈ [ 0 , 1 ) on F space . γ := c | n x F | / n t h Define jump/averages seminorms on H 1 ( T h ) 1 + n : � � � � � DG := 1 2 2 ||| ( w , τ ) ||| 2 � c − 1 w � � � � h ) + � τ � � 2 L 2 ( F 0 h ∪F T L 2 ( F 0 h ∪F T h ) n � � � � � 2 2 � 1 − γ � 1 / 2 � 1 − γ � 1 / 2 � � � � c − 1 [ + [ w ] ] t + [ [ τ ] ] t � � � � n t n t � � � � L 2 ( F space L 2 ( F space F F ) n ) h h � � � � � � 2 2 2 � α 1 / 2 [ � β 1 / 2 [ � α 1 / 2 w � � � � � � + [ w ] ] N ) n + [ τ ] ] N ) + h ) , � � � L 2 ( F time L 2 ( F time L 2 ( F ∂ h h ||| ( w , τ ) ||| 2 DG + := ||| ( w , τ ) ||| 2 DG � n t � n t � � 2 � � 2 � 1 / 2 � 1 / 2 � � � � c − 1 w − τ − F F + 2 + 2 � � � � 1 − γ 1 − γ � � � � L 2 ( F space L 2 ( F space ) ) n h h � � � � � � 2 2 2 � β − 1 / 2 { � α − 1 / 2 { � α − 1 / 2 τ · n � � � � � � + { w } } ) + { τ } } ) n + h ) . � � � L 2 ( F time L 2 ( F time L 2 ( F ∂ h h They are norms on Trefftz space T ( T h ) . 12
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