STDGFEM for a model problem Application of the STDFEM to compressible flow Error analysis of the space-time DGFEM for nonstationary nonlinear convection-diffusion problems Miloslav Feistauer and Jan Česenek Charles University Prague, Faculty of Mathematics and Physics, Prague, Czech Republic Workshop Numerical Analysis for Singularly Perturbed Problems Dedicated to the 60th Birthday of Martin Stynes Dresden, 18 Nov. 2011 Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Contents: Contents : Theory of the space-time discontinuous Galerkin finite element method (STDGFEM) for a scalar model problem Application to the solution of compressible flow and FSI Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Continuous model problem Continuous model problem R 2 be a bounded polygonal domain and T > 0. Find Let Ω ⊂ I u : Q T = Ω × ( 0 , T ) → I R such that 2 ∂ u ∂ f s ( u ) � in Q T = Ω × ( 0 , T ) , (1) ∂ t + − div ( β ( u ) ∇ u )) = g ∂ x s s = 1 � (2) u ∂ Ω × ( 0 , T ) = u D , � u ( x , 0 ) = u 0 ( x ) , (3) x ∈ Ω . g , u D , u 0 , f s – given functions, f s ∈ C 1 ( I | f ′ s | ≤ C , s = 1 , 2 R ) , 0 < β 0 < β 1 < ∞ , (4) β : I R → [ β 0 , β 1 ] , (5) | β ( u 1 ) − β ( u 2 ) | ≤ L | u 1 − u 2 | , ∀ u 1 , u 2 ∈ I R . Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Let the exact solution be regular in the following sense: ∂ u u ∈ L 2 ( 0 , T ; H 2 (Ω)) , ∂ t ∈ L 2 ( 0 , T ; H 1 (Ω)) , (6) �∇ u ( t ) � L ∞ (Ω) ≤ C R for a.e. t ∈ ( 0 , T ) . (7) Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Space-time discretization Partition in the time interval [ 0 , T ] : 0 = t 0 < · · · < t M = T denote I m = ( t m − 1 , t m ) , τ m = t m − t m − 1 , τ = max m = 1 ,..., M τ m . For ϕ defined in � M m = 1 I m we put ϕ ± m = ϕ ( t m ± ) = lim t → t m ± ϕ ( t ) (one-sided limits at time t m ) { ϕ } m = ϕ ( t m +) − ϕ ( t m − ) (jump). For each I m consider a partition T h , m of the closure Ω of the domain Ω into a finite number of closed triangles with mutually disjoint interiors. The partitions T h , m are in general different for different m . F h , m – the system of all faces of all elements K ∈ T h , m F I h , m – the set of all inner faces F B h , m – the set of all boundary faces Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Each Γ ∈ F h , m associated with a unit normal vector n Γ , which has the same orientation as the outer normal to ∂ Ω for Γ ∈ F B h , m h K = diam ( K ) for K ∈ T h , m , h m = max K ∈T h , m h K , h = max m = 1 ,..., M h m ρ K – the radius of the largest circle inscribed into K . Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Γ � n Γ K ( R ) Γ K ( L ) Γ Neighbouring elements For each face Γ ∈ F I h , m there exist two neighbours K ( L ) Γ , K ( R ) ∈ T h , m such that Γ ⊂ ∂ K ( L ) ∩ ∂ K ( R ) . Γ Γ Γ n Γ is the outer normal to ∂ K ( L ) and the inner normal to Γ ∂ K ( R ) . Γ h , m , then K ( L ) If Γ ∈ F B will denote the element adjacent to Γ . Γ Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Let C W > 0 be a fixed constant. We set h K ( L ) + h K ( R ) for Γ ∈ F I (8) h (Γ) = Γ Γ h , m , 2 C W h K ( L ) for Γ ∈ F B h (Γ) = Γ h , m . C W Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow DG spaces : Broken Sobolev spaces: H k (Ω , T h , m ) = { v ; v | K ∈ H k ( K ) ∀ K ∈ T h , m } . If v ∈ H 1 (Ω , T h , m ) and Γ ∈ F h , m , then v ( L ) Γ , v ( R ) = the traces of v on Γ from the side of elements Γ K ( L ) Γ , K ( R ) adjacent to Γ Γ If Γ ∈ F I h , m , then � � v ( L ) + v ( R ) , [ v ] Γ = v ( L ) − v ( R ) � v � Γ = 1 . Γ Γ Γ Γ 2 Discrete spaces Let p , q ≥ 1 be integers. For each m = 1 , . . . , M , S p ϕ ∈ L 2 (Ω); ϕ | K ∈ P p ( K ) ∀ K ∈ T h , m � � h , m = . (9) The approximate solution is sought in the space q � t i ϕ i S p , q � ϕ ∈ L 2 ( Q T ); ϕ � h ,τ = I m = (10) � i = 0 � with ϕ i ∈ S p h , m , m = 1 , . . . , M . Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Forms Forms For u , v , ϕ ∈ H 2 (Ω , T h , m ) , we define the folowing forms: Diffusion form � � (11) a h , m ( v , u , ϕ ) = β ( v ) ∇ u · ∇ ϕ d x K K ∈T h , m � � − ( � β ( v ) ∇ u � · n Γ [ ϕ ] + θ � β ( v ) ∇ ϕ � · n Γ [ u ]) d S Γ Γ ∈F I h , m � � − ( β ( v ) ∇ u · n Γ ϕ Γ Γ ∈F B h , m + θ β ( v ) ∇ ϕ · n Γ u − θβ ( v ) ∇ ϕ · n Γ u D ) d S θ = 1 , or θ = 0 or θ = − 1 – the symmetric (SIPG) or incomplete (IIPG) or nonsymmetric (NIPG) variants of the approximation of the diffusion terms, respectively. Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Interior and boundary penalty � � h (Γ) − 1 J h , m ( u , ϕ ) = [ u ] [ ϕ ] d S Γ Γ ∈F I h , m � � h (Γ) − 1 + u ϕ d S Γ Γ ∈F B h , m (12) A h , m = a h , m + β 0 J h , m , Right-hand side form � � h (Γ) − 1 u D ϕ d S (13) ℓ h , m ( ϕ ) = ( g , ϕ ) + β 0 Γ Γ ∈F B h , m Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Convection form 2 � f s ( u ) ∂ϕ � � (14) b h , m ( u , ϕ ) = − d x ∂ x s K K ∈T h , m s = 1 � � � u ( L ) Γ , u ( R ) � + H , n Γ [ ϕ ] d S Γ Γ Γ ∈F I h , m � � � u ( L ) Γ , u ( L ) � + Γ , n Γ ϕ d S H Γ Γ ∈F B h , m H – numerical flux with the following properties: R 2 × B 1 , where H ( u , v , n ) is defined in I R 2 ; | n | = 1 } , and is Lipschitz-continuous with B 1 = { n ∈ I respect to u , v . H ( u , v , n ) is consistent: H ( u , u , n ) = � 2 s = 1 f s ( u ) n s , u ∈ I R , n = ( n 1 , n 2 ) ∈ B 1 . H ( u , v , n ) is conservative: H ( u , v , n ) = − H ( v , u , − n ) , u , v ∈ I R , n ∈ B 1 . Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow ( · , · ) – the scalar product in L 2 (Ω) , � · � – the norm in L 2 (Ω) . � 1 / 2 � � – norm in K ∈T h , m | ϕ | 2 � ϕ � DG , m = H 1 ( K ) + J h , m ( ϕ, ϕ ) H 1 (Ω , T h , m ) Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow notation: U ′ = ∂ U /∂ t , u ′ = ∂ u /∂ t . Approximate solution : U ∈ S p , q h ,τ such that � ( U ′ , ϕ ) + A h , m ( U , U , ϕ ) + b h , m ( U , ϕ ) � � (15) d t I m { U } m − 1 , ϕ + � � + m − 1 � ∀ ϕ ∈ S p , q m = 1 , . . . , M , = ℓ h , m ( ϕ ) d t , h ,τ , I m 0 = L 2 (Ω) − projection of u 0 on S p U − h , 1 . The exact regular solution u satisfies the identity � ( u ′ , ϕ ) + A h , m ( u , u , ϕ ) + b h , m ( u , ϕ ) � � (16) d t I m { u } m − 1 , ϕ + � � + m − 1 � ∀ ϕ ∈ S p , q with u ( 0 − ) = u 0 . = ℓ h , m ( ϕ ) d t h ,τ , I m Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
STDGFEM for a model problem Application of the STDFEM to compressible flow Error analysis Error analysis The main goal: analysis of the estimation of the error e = U − u Π m – the L 2 (Ω) -projection on S p h , m . S p , q h ,τ -interpolation π of functions v ∈ H 1 ( 0 , T ; L 2 (Ω)) : π v ∈ S p , q a) b) (17) h ,τ , ( π v ) ( t m − ) = Π m v ( t m − ) , � ∀ ϕ ∗ ∈ S p , q − 1 ( π v − v , ϕ ∗ ) d t = 0 c) ∀ m = 1 , . . . , M . , h ,τ I m e = U − u = ξ + η , ξ = U − π u ∈ S p , q h ,τ and η = π u − u Miloslav Feistauer and Jan Česenek Error analysis of the space-time DGFEM for nonstationary no
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