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The Continuous Problem DG Approximation Numerical Results Conclusion Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection Daniele A. Di Pietro joint work with Alexandre Ern and Jean-Luc Guermond CERMICS, Ecole des Ponts, ParisTech, 77455 Marne la Vall´ ee Cedex 2, France M´ ethodes num´ eriques pour les fluides Paris, December 20 2006 D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Introduction ◮ We consider advection-diffusion-reaction problems with ◮ discontinuous ◮ anisotropic ◮ semi-definite diffusivity ◮ The mathematical nature of the problem may not be uniform over the domain ◮ Indeed, because of anisotropy, the problem may be hyperbolic in one direction and elliptic in another ◮ The solution may be discontinuous across elliptic-hyperbolic interfaces D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Model Problem ◮ Ω ⊂ R d bounded, open and connected Lipschitz domain def ◮ P Ω = { Ω i } N i =1 partition of Ω into Lipschitz connected subdomains ◮ Consider the following problem: ∇· ( − ν ∇ u + β u ) + µ u = f ◮ ν ∈ [ L ∞ (Ω)] d , d symmetric piecewise constant on P Ω is s.t. ν ≥ 0 ◮ β ∈ [ C 1 (Ω)] d ◮ µ ∈ L ∞ (Ω) is s.t. µ + 1 2 ∇· β ≥ µ 0 with µ 0 > 0 D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion A One-Dimensional Example β = 1 ν = 1 ν = 1  ǫ + u ǫ ) ′ = 0 , ( − ν u ′ in (0 , 1) ,   ν = ǫ u ǫ (0) = 1 ,   0 1 / 3 2 / 3 1 u ǫ (1) = 0 . Ω 1 Ω 2 Ω 3 1 0.8 0.6 0.4 1e+0 0.2 1e-1 1e-2 1e-3 0 0 0.2 0.4 0.6 0.8 1 lim ǫ → 0 u ǫ = I Ω 1 ∪ Ω 2 ( x ) + 3( x − 1) I Ω 3 ( x ), discontinuous at x = 2 / 3 D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Goals ◮ At the continuous level, design suitable interface and BC’s to define a well-posed problem ◮ At the discrete level, design a DG method that ◮ does not require the a priori knowledge of the elliptic-hyperbolic interface ◮ yields optimal error estimates in mesh-size that are robust w.r.t. anisotropy and semi-definiteness of diffusivity D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Outline The Continuous Problem Weak Formulation Well-Posedness Analysis DG Approximation Design of the DG Method Error Analysis Other Amenities Numerical Results Conclusion D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation Interface Conditions I ◮ Let def Γ = { x ∈ Ω; ∃ Ω i 1 , Ω i 2 ∈ P Ω , x ∈ ∂ Ω i 1 ∩ ∂ Ω i 2 } , where i 1 and i 2 are s.t. ( n t ν n ) | Ω i 1 ≥ ( n t ν n ) | Ω i 2 ◮ We define the elliptic-hyperbolic interface as def ( n t ν n )( x ) | Ω i 1 > 0, ( n t ν n )( x ) | Ω i 2 = 0 } I = { x ∈ Γ; ◮ Set, moreover, I + def I − def = { x ∈ I ; β · n 1 > 0 } , = { x ∈ I ; β · n 1 < 0 } D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation Interface Conditions II ◮ For all scalar ϕ with a (possibly two-valued) trace on Γ, define def def = 1 { ϕ } 2 ( ϕ | Ω i 1 + ϕ | Ω i 2 ) , [[ ϕ ]] = ϕ | Ω i 1 − ϕ | Ω i 2 ◮ We require that [[ u ]] = 0, on I + ( E → H ) ◮ Observe that continuity is not enforced on I − ◮ When ν is isotropic the above conditions coincide with those derived in [Gastaldi and Quarteroni, 1989] in the one-dimensional case D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation A Two-Dimensional Exact Solution I ν 1 = π 2 y For a suitable rhs, θ � I + I − ( θ − π ) 2 , if 0 ≤ θ ≤ π, u = x 3 π ( θ − π ) , if π < θ < 2 π. n 1 n 1 β = e θ β = e θ r r ν 2 = 0 D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation A Two-Dimensional Exact Solution II D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation An Example with Strongly Anisotropic Diffusivity � 0 � 0 ν 2 = 0 1 I + I − � � n 1 n 1 1 0 ν 1 = 0 0 . 25 β = ( − 5 , 0) D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation Friedrichs-Like Mixed Formulation I ◮ We want to reformulate the problem so as to recover the symmetry and dissipativity ( L -coercivity) properties of Friedrichs systems [Friedrichs, 1958] ◮ The problem in symmetric mixed formulation reads � σ + κ ∇ u = 0 , in Ω \ I , (mixed) ∇· ( κσ + β u ) + µ u = 0 , in Ω , def = ν 1 / 2 where κ ◮ For y = ( y σ , y u ), the advective-diffusive flux is defined as = κ y σ + β y u def Φ( y ) D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation Friedrichs-Like Mixed Formulation II ◮ The graph space is κ ∇ y u ∈ L σ and ∇· Φ( y ) ∈ L u } def = { y ∈ L ; W with def def def = [ L 2 (Ω \ I )] d = L 2 (Ω) L σ L u L = L σ × L u ◮ The space choice together with condition ( E → H ) yields { Φ( z ) · n } = 0 , on Γ , (cond. Γ) [[ z u ]] = 0 , on Γ \ I − . D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation Friedrichs-Like Mixed Formulation III ◮ Define the zero- and first-order operators L ( L ; L ) ∋ K : z �→ ( z σ , µ z u ) L ( W ; L ) ∋ A : z �→ ( κ ∇ z u , ∇· Φ( z )) ◮ The bilinear form � def I + ( β · n 1 )[[ z u ]][[ y u ]] a 0 ( z , y ) = (( K + A ) z , y ) L + is L -coercive whenever z and y are compactly supported ◮ a 0 will serve as a base for the construction of a weak problem with boundary and interface conditions weakly enforced D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation Boundary Conditions Weakly Enforced I ◮ Define the operators M and D s.t., for all z , y ∈ W × W � � y t D z , y t M z , � Dz , y � W ′ , W = � Mz , y � W ′ , W = ∂ Ω ∂ Ω where, for α ∈ {− 1 , +1 } , � � � � 0 κ n 0 − ακ n D = , M = ( κ n ) t β · n α ( κ n ) t | β · n | ◮ Observe that M ≥ 0 D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection

The Continuous Problem DG Approximation Numerical Results Conclusion Weak Formulation Boundary Conditions Weakly Enforced II � def I + ( β · n 1 )[[ z u ]][[ y u ]] + 1 a ( z , y ) = (( K + A ) z , y ) L + 2 � ( M − D )( z ) , y � W ′ , W � �� � a 0 ( z , y ) ◮ a is L -coercive on W ◮ Let def def = { x ∈ ∂ Ω; ( n t ν n )( x ) > 0 } , ∂ Ω E ∂ Ω H = ∂ Ω \ ∂ Ω E . Then α = +1 Dirichlet on ∂ Ω E /inflow on ∂ Ω H in Ker( M − D ) ◮ α = − 1 Neumann-Robin on ∂ Ω E /inflow on ∂ Ω H in Ker( M − D ) ◮ D. A. Di Pietro – dipietro@cermics.enpc.fr ENPC/CERMICS Discontinuous Galerkin Methods for Anisotropic and Semi-Definite Diffusion with Advection