Discontinuous Galerkin Methods and Local Time-Stepping For Transient - PowerPoint PPT Presentation

Discontinuous Galerkin Methods and Local Time-Stepping For Transient Wave Propagation Marcus Grote joint work with: J. Diaz, INRIA T. Mitkova, A. Schneebeli, Univ. Basel D. Sch¨ otzau, UBC Marcus Grote RICAM workshop, Linz, 21–25.11.2011

High-Order Explicit LTS Methods Outline : • Introduction • The (damped) wave equation • CG and DG Finite Element discretizations • The damped case: Adams-Bashforth LTS methods • Numerical experiments • The undamped case: Leap-Frog LTS methods • Numerical experiments • Concluding remarks Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Introduction Marcus Grote RICAM workshop, Linz, 21–25.11.2011

The (Damped) Wave Equation Model problem (second-order form) u tt + σu t − ∇ · ( c ∇ u ) = f in Ω × (0 , T ) u = 0 on ∂ Ω × (0 , T ) u | t =0 = u 0 , u t | t =0 = v 0 in Ω • Ω ⊂ R d bounded, σ ( x ) ≥ 0, c ( x ) > 0 Marcus Grote RICAM workshop, Linz, 21–25.11.2011

The (Damped) Wave Equation Model problem (second-order form) u tt + σu t − ∇ · ( c ∇ u ) = f in Ω × (0 , T ) u = 0 on ∂ Ω × (0 , T ) u | t =0 = u 0 , u t | t =0 = v 0 in Ω • Ω ⊂ R d bounded, σ ( x ) ≥ 0, c ( x ) > 0 Weak formulation Find u ∈ C 0 (0 , T ; H 1 0 (Ω)) ∩ C 1 (0 , T ; L 2 (Ω)): ∀ v ∈ H 1 � u tt , v � ( H − 1 ,H 1 0 ) + ( σu t , v ) + a ( u, v ) = ( f, v ) , 0 (Ω) , with a ( u, v ) = ( c ∇ u, ∇ v ) Marcus Grote RICAM workshop, Linz, 21–25.11.2011

The (Damped) Wave Equation Model problem (second-order form) u tt + σu t − ∇ · ( c ∇ u ) = f in Ω × (0 , T ) u = 0 on ∂ Ω × (0 , T ) u | t =0 = u 0 , u t | t =0 = v 0 in Ω • Ω ⊂ R d bounded, σ ( x ) ≥ 0, c ( x ) > 0 Weak formulation Find u ∈ C 0 (0 , T ; H 1 0 (Ω)) ∩ C 1 (0 , T ; L 2 (Ω)): ∀ v ∈ H 1 � u tt , v � ( H − 1 ,H 1 0 ) + ( σu t , v ) + a ( u, v ) = ( f, v ) , 0 (Ω) , with a ( u, v ) = ( c ∇ u, ∇ v ) Energy conservation For σ = 0, f = 0 the energy E [ u ]( t ), � � E [ u ]( t ) := 1 � u t � 2 + a ( u, u ) ≡ const. 2 Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Conforming Galerkin approximation Finite element space V h ⊂ H 1 0 (Ω): V h = { v ∈ H 1 0 (Ω) : v | K ∈ S ℓ ( K ) , K ∈ T h } ⇒ elements continuous across edges / faces Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Conforming Galerkin approximation Finite element space V h ⊂ H 1 0 (Ω): V h = { v ∈ H 1 0 (Ω) : v | K ∈ S ℓ ( K ) , K ∈ T h } ⇒ elements continuous across edges / faces Semi-discrete Galerkin formulation: Find u h : [0 , T ] × V h → R such that ( u h tt , v ) + ( σu h t , v ) + a ( u h , v ) = ( f, v ) ∀ v ∈ V h , t ∈ (0 , T ) Conforming mass-lumped FEM: ( Cohen-Joly-Roberts-Tordjman, SINUM, 2001 ) � � a ( u, v ) := c ∇ u · ∇ v dx K K ∈T h ⇒ The semi-discrete formulation automatically inherits the main properties from the continuous formulation. Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Properties of conforming FEM • For σ = 0 and f = 0, the energy E [ u h ]( t ), � � E [ u h ]( t ) = 1 � ∂ t u h � 2 + a ( u h , u h ) , 2 is conserved. • For sufficiently smooth solutions we have the optimal error estimates: O ( h ℓ ) , � ( ∂ t u − ∂ t u h )( t, . ) � L 2 + �∇ ( u − u h )( t, . ) � L 2 = h → 0 O ( h ℓ +1 ) , � ( u − u h )( t, . ) � L 2 = see Dupont, Baker, SINUM, 1970’s Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Discontinuous Galerkin (DG) methods • Finite element space with no inter-elemental continuity constrains, i.e. V h ⊂ / H 1 0 (Ω): V h = { v ∈ L 2 (Ω) : v | K ∈ S ℓ ( K ) , K ∈ T h } • Local (elementwise) weak formulation • Numerical fluxes weakly enforce inter-elemental continuity Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Discontinuous Galerkin (DG) methods • Finite element space with no inter-elemental continuity constrains, i.e. V h ⊂ / H 1 0 (Ω): V h = { v ∈ L 2 (Ω) : v | K ∈ S ℓ ( K ) , K ∈ T h } • Local (elementwise) weak formulation • Numerical fluxes weakly enforce inter-elemental continuity + Flexibility in mesh-design (non-matching grids) + easily handles varying polynomial degree ( hp -adaptivity) + (Block-)Diagonal mass matrix ⇒ fully explicit time stepping! Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Discontinuous Galerkin (DG) methods • Finite element space with no inter-elemental continuity constrains, i.e. V h ⊂ / H 1 0 (Ω): V h = { v ∈ L 2 (Ω) : v | K ∈ S ℓ ( K ) , K ∈ T h } • Local (elementwise) weak formulation • Numerical fluxes weakly enforce inter-elemental continuity + Flexibility in mesh-design (non-matching grids) + easily handles varying polynomial degree ( hp -adaptivity) + (Block-)Diagonal mass matrix ⇒ fully explicit time stepping! - increase the number of degrees of freedom - increase the condition number of the stiffness matrix Marcus Grote RICAM workshop, Linz, 21–25.11.2011

DG discretization: notation n ! n + T h : shape-regular mesh K ! K + K : tetrahedra or hexahedra F h : faces of K ∈ T h + ∩ K − f ∈ F h , f = K . . n ± : outward normal on ∂K ± v ± : traces of v from K ± Jumps: ] := n + v + + n − v − [ [ v ] Averages: } := ( v + + v − ) / 2 { { v } Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Interior Penalty (IP) formulation � • u, v ∈ V h : a h ( u, v ) := ( c ∇ h u, ∇ h v ) − [ [ v ] ] · { { c ∇ h u } } ds F h u, v ∈ V h := { v ∈ L 2 (Ω) : v | K ∈ S ℓ ( K ) ∀ K ∈ T h } Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Interior Penalty (IP) formulation � • u, v ∈ V h : a h ( u, v ) := ( c ∇ h u, ∇ h v ) − [ [ v ] ] · { { c ∇ h u } } ds F h � − [ [ u ] ] · { { c ∇ h v } } ds F h u, v ∈ V h := { v ∈ L 2 (Ω) : v | K ∈ S ℓ ( K ) ∀ K ∈ T h } Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Interior Penalty (IP) formulation � • u, v ∈ V h : a h ( u, v ) := ( c ∇ h u, ∇ h v ) − [ [ v ] ] · { { c ∇ h u } } ds F h � − [ [ u ] ] · { { c ∇ h v } } ds F h � + a [ [ u ] ] · [ [ v ] ] ds. F h u, v ∈ V h := { v ∈ L 2 (Ω) : v | K ∈ S ℓ ( K ) ∀ K ∈ T h } Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Interior Penalty (IP) formulation � • u, v ∈ V h : a h ( u, v ) := ( c ∇ h u, ∇ h v ) − [ [ v ] ] · { { c ∇ h u } } ds F h � − [ [ u ] ] · { { c ∇ h v } } ds F h � + a [ [ u ] ] · [ [ v ] ] ds. F h u, v ∈ V h := { v ∈ L 2 (Ω) : v | K ∈ S ℓ ( K ) ∀ K ∈ T h } • Interior penalty stabilization function a := α c h − 1 ∈ L ∞ ( F h ) α > 0: IP stabilization parameter h | F := min { h K + , h K − } , h K diameter of element K c | F := max { c | K + ( x ) , c | K − ( x ) } Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Semi-discrete IP-DG approximation Find u h : [0 , T ] × V h → R such that ( u h tt , v ) + ( σu h t , v ) + a h ( u h , v ) = ( f, v ) ∀ v ∈ V h , t ∈ (0 , T ) , u h | t =0 = Π h u 0 , u h t | t =0 = Π h v 0 . Π h : L 2 -projection onto V h The DG-bilinear form a h ( u, v ) is • symmetric, continuous, and coercive, for α ≥ α min , Marcus Grote RICAM workshop, Linz, 21–25.11.2011

Semi-discrete IP-DG approximation Find u h : [0 , T ] × V h → R such that ( u h tt , v ) + ( σu h t , v ) + a h ( u h , v ) = ( f, v ) ∀ v ∈ V h , t ∈ (0 , T ) , u h | t =0 = Π h u 0 , u h t | t =0 = Π h v 0 . Π h : L 2 -projection onto V h The DG-bilinear form a h ( u, v ) is • symmetric, continuous, and coercive, for α ≥ α min , • and the solution conserves the (discrete) energy E h ( t ) := 1 0 + 1 2 � u h t ( t ) � 2 2 a h ( u h ( t ) , u h ( t )) if σ = 0 and f = 0. Marcus Grote RICAM workshop, Linz, 21–25.11.2011

IP-DG for second-order wave equations • Symmetric bilinear form a h ⇒ conservation of discrete energy (G., Schneebeli, Sch¨ otzau: SINUM, 2006; JCAM 2007; IMA J. Num. Anal. 2008) (G. and Sch¨ otzau: J. Sc. Comp., 2009) Marcus Grote RICAM workshop, Linz, 21–25.11.2011

IP-DG for second-order wave equations • Symmetric bilinear form a h ⇒ conservation of discrete energy • mass matrix (block-) diagonal ⇒ fully explicit time integration (G., Schneebeli, Sch¨ otzau: SINUM, 2006; JCAM 2007; IMA J. Num. Anal. 2008) (G. and Sch¨ otzau: J. Sc. Comp., 2009) Marcus Grote RICAM workshop, Linz, 21–25.11.2011

IP-DG for second-order wave equations • Symmetric bilinear form a h ⇒ conservation of discrete energy • mass matrix (block-) diagonal ⇒ fully explicit time integration • semi-discrete convergence analysis: • optimal a-priori error bounds in energy norm • optimal a-priori error bounds in L 2 norm (G., Schneebeli, Sch¨ otzau: SINUM, 2006; JCAM 2007; IMA J. Num. Anal. 2008) (G. and Sch¨ otzau: J. Sc. Comp., 2009) Marcus Grote RICAM workshop, Linz, 21–25.11.2011