Space-Time Discontinuous Galerkin Discretizations for Linear - PowerPoint PPT Presentation

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems Willy Dörfler | Stefan Findeisen | Christian Wieners Institut für Angewandte und Numerische Mathematik www.kit.edu

Outline Linear hyperbolic first-order systems M ∂ t u + A u = f in [ 0 , T ] , u ( 0 ) = u 0 well-posedness in a space-time Hilbert space setting Discontinuous Galerkin method in space and Petrov-Galerkin method in time fully implicit adaptive space-time method for L u = f with L = M ∂ t + A inf-sup stability of a discrete space-time Petrov-Galerkin setting Error Control duality based space-time error representation weighted residual error indicator A space-time multigrid strategy combined p − q − h subspace correction preconditioner smoothing in space and in time Applications linear transport equation Maxwell’s equations for polarized waves Acoustic waves (joint work with Daniel Ziegler) Dörfler / Findeisen / Wieners: Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems. Comput. Methods Appl. Math. 2016. We gratefully acknowledge financial support by DFG through RTG 1294 and CRC 1173. 1

Linear Hyperbolic First-order Systems We consider in the bounded domain Ω ⊂ R D and the time interval [ 0 , T ] M ∂ t u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , T ) , u ( 0 ) = u 0 and a flux function F ( v ) = [ B 1 v , . . . , B D v ] with B j ∈ L ∞ (Ω) J × J sym such that D ∂ d ( B d v ) ∈ L 2 (Ω) J , � A v = div F ( v ) = v ∈ D ( A ) . d = 1 Linear transport ∞ (Ω) D with div q = 0 find u such that For ρ ∈ L ∞ (Ω) and q ∈ W 1 ρ∂ t u + div ( u q ) = f . Acoustic waves Find the pressure p and the velocity v (isotropic and homogeneous media) with ∂ t p + div v = 0 , ∂ t v + ∇ p = 0 . Electro-magnetic waves Find an electric field E and magnetic field H such that ε∂ t E − curl H = f , µ∂ t H + curl E = 0 , div ( ε E ) = ρ , div ( µ H ) = 0 for given permeability µ and permittivity ε . 2

A Space-Time Setting Let H ⊆ L 2 (Ω) J be a Hilbert space with ( v , w ) H = ( M v , w ) 0 , Ω . Let D ( A ) ⊂ H be the domain of the operator A with ( A v , v ) 0 , Ω ≥ 0 in D ( A ) . Consider L = M ∂ t + A on the space-time cylinder Q = Ω × ( 0 , T ) . u ∈ C 1 ( 0 , T ; D ( A )): u ( 0 ) = 0 � � Let V = D ( L ) be the closure of with respect to the weighted graph norm � v � 2 V = ( M v , v ) 0 , Q + ( M − 1 L v , L v ) 0 , Q . Let W be the closure of L ( V ) in L 2 ( 0 , T ; H ) with � u � 2 W = ( M u , u ) 2 0 , Q . Lemma For given f ∈ L 2 ( Q ) J a unique solution u ∈ V exists solving the variational problem ( L u , w ) 0 , Q = ( f , w ) 0 , Q , w ∈ W For the proof we check inf-sup stability of b : V × W → R with b ( v , w ) = ( L v , w ) 0 , Q . We observe � v � W ≤ 2 T � M − 1 L v � W for all v ∈ V . This yields b ( v , M − 1 L v ) � M − 1 L v � W b ( v , w ) 1 √ v ∈ V sup inf ≥ inf = inf ≥ 1 + 4 T 2 . � v � V � M − 1 L v � W � v � V � w � W � v ∈ V v ∈ V � v � 2 W + � M − 1 L v � 2 w ∈ W W 3

A Petrov–Galerkin Space-Time Discretization Let Q = � R ∈R R be a decomposition of the space-time cylinder into space-time cells R = K × I with K ⊂ Ω and I = ( t − , t + ) ⊂ ( 0 , T ) . Choose locally V h , R , W h , R ⊂ L 2 ( R ) J with W h , R ⊂ ∂ t V h , R and set � � v h ∈ H 1 ( 0 , T ; H ): v h ( x , 0 ) = 0 for a.a. x ∈ Ω and v h , R = v h | R ∈ V h , R V h = , � � W h = w h ∈ L 2 ( 0 , T ; H ): w h , R = w h | R ∈ W h , R . Tensor-product space-time meshes Consider a fixed mesh K in space and a time series 0 = t 0 < t 1 < · · · < t N = T , � J and � N � i.e., R = � n = 1 K × ( t n − 1 , t n ) . Then, set W h , R = P p ( K ) ⊗ P q − 1 K ∈K � v h ∈ H 1 ( 0 , T ; H ) : v h ( x , 0 ) = 0 and for ( x , t ) ∈ R = K × ( t n − 1 , t n ) V h = t n − t v h ( x , t n − 1 ) + t − t n − 1 � v h ( x , t ) = w R , h ( x , t ) , w h ∈ W h , R . t n − t n − 1 t n − t n − 1 General space-time meshes � J , and define � Set W h , R = P p R ( K ) ⊗ P q R − 1 v h , R ∈ L 2 ( R ) J : v h , R ( x , t ) = t + − t t + − t − v h ( x , t − ) + t − t − � V h , R = t + − t − w h , R ( x , t ) , � v h ∈ V h | [ 0 , t − ] , w h , R ∈ W h , R , ( x , t ) ∈ R = K × ( t − , t + ) . 4

A Discontinuous Galerkin Approximation We define the discontinuous Galerkin operator A h ∈ L ( V h , W h ) by �� � � � A h v h , w h ) 0 , Q = div F ( v h , R ) , w h , R 0 , R R = K × I � � � n K · ( F num � + ( v h ) − F ( v h , R )) , w h , R K 0 , f × I f ∈F K using the upwind flux F num ( v h ) on every face f ⊂ ∂ K . K Let Π h : W → W h be the L 2 -projection, and define d T ( t ) = T − t . Lemma Let L h = M ∂ t + A h and f ∈ L 2 ( Q ) J , and assume that � � � � M ∂ t v h , d T v h 0 , Q ≤ L h v h , d T Π h v h 0 , Q , v h ∈ V h . (1) Then, a unique discrete solution u h ∈ V h exists solving ( L h u h , v h ) 0 , Q = ( f , v h ) 0 , Q , v h ∈ W h . The proof relies on discrete inf-sup stability and the identity for v h ∈ V h � T � v h � 2 � � � � W = − M v h ( t ) , v h ( t ) 0 , Ω ∂ t d T ( t ) d t = 2 M ∂ t v h , d T v h 0 , Q . 0 5

A Discontinuous Galerkin Approximation Theorem Assume that (1) holds and set � v h � 2 V h = � v h � 2 W + � M − 1 h L h v h � 2 W . Then, we have � � � 4 T 2 + 1 � u − u h � V h ≤ 1 + inf � u − v h � V h . v h ∈ V h If in addition the solution is sufficiently smooth, we obtain the a priori error estimate △ t q + △ x p �� � � ∂ q + 1 u � 0 , Q + � D p + 1 u � 0 , Q � � u − u h � V h ≤ C t for △ t, △ x and p , q ≥ 1 with △ t ≥ t + − t − , △ x ≥ diam K, p ≤ p R and q ≤ q R . Lemma In the case of tensor product space-time discretizations, the condition (1) is satisfied, and we have for v h ∈ V h � � � � � � Π h ∂ t v h = ∂ t v h , M h ∂ t v h , d T v h 0 , Q ≤ M h ∂ t v h , d T Π h v h 0 , Q , 0 ≤ A h v h , d T Π h v h 0 , Q . � � 0 = k for orthonormal Legendre polynomials λ k . The proof relies on t ∂ t λ k , λ k 6

A Dual Space-Time Error Representation Let L ∗ = − L be the adjoint operator defined on the adjoint Hilbert space V ∗ = � � w ∈ W : there exists g ∈ W such that ( L v , w ) 0 , Q = ( v , g ) 0 , Q for all v ∈ V . Note that we have w ( T ) = 0 for w ∈ V ∗ . For a given error functional E ∈ W ′ we introduce the dual solution u ∗ ∈ V ∗ with ( w , L ∗ u ∗ ) 0 , Q = � E , w � , w ∈ W . Lemma If the dual solution is sufficiently smooth, we have for all w h ∈ W h �� f − M ∂ t u h − div F ( u h ) , u ∗ − w h � � � E , u − u h � = 0 , R R = K × I ∈R ( u h )) , u ∗ − w h � � � + n K · ( F ( u h ) − F num . K 0 ,∂ K × I Inserting the discrete dual solution u ∗ h ∈ W h with ( v h , L ∗ h u ∗ h ) 0 , Q = � E , v h � for v h ∈ V h and a higher-order recovery I h u ∗ h defines the weighted residual error indicator η R = � f − M ∂ t u h − div F ( u h ) � 0 , R � u ∗ h − I h u ∗ h � 0 , R + � n K · ( F ( u h ) − F num ( u h )) � 0 ,∂ K × I � u ∗ h − I h u ∗ h � 0 ,∂ K × I . 7

A Multigrid Preconditioner Let R 0 , 0 be the coarse space-time mesh, and let R l , k be the mesh obtained by l = 1 , . . . , l max refinements in space and k = 1 , . . . , k max refinements in time. B GS B GS l , k l , k ( l , k ) R l , k P l , k B GS B GS l − 1 , k l − 1 , k l − 1 , k l − 1 , k ( l − 1 , k ) B J B J 0 , k 0 , k ( 0 , k ) R 0 , k P 0 , k B J B J 0 , k − 1 0 , k − 1 0 , k − 1 0 , k − 1 ( 0 , k − 1 ) B ML 0 , 0 ( 0 , 0 ) L l , k ∈ L ( V l , k , W l , k ) approximates L on R l , k the prolongation P l , k l − 1 , k ∈ L ( V l − 1 , k , V l , k ) represents the natural injection the restriction R l , k l − 1 , k ∈ L ( W l , k , W l − 1 , k ) represents the L 2 projection l , k = θ l , k block_diag ( L l , k ) − 1 is the block-Jacobi smoother with damping θ l , k B J � − 1 Gauss-Seidel smoother B GS � l , k = θ l , k block_lower ( L l , k ) + block_diag ( L l , k ) 8

Numerical Test: Rotating Cone ∂ t u + div ( u q ) = 0 in Ω = ( − 0 . 5 , 0 . 5 ) 2 for t ∈ ( 0 , 1 ) with q ( x 1 , x 2 ) = 2 π ( − x 2 , x 1 ) 9