Introduction
Diffusion
Diffusion-advection-reaction
Discontinuous Galerkin methods
Alexandre Ern
Université Paris-Est, CERMICS, ENPC
Journées numériques, Nice, 17 mai 2016
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Introduction
◮
dG methods were introduced more than 40 years ago
◮
They have sparked extensive interest in the scientific computing and
applied math communities
200
150
100
50
0
1975
1980
1985
1990
1995
2000
2005
2010
dG-related publications/year (Mathscinet)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
A brief historical perspective
◮
Elliptic PDEs
◮
◮
◮
First-order PDEs
◮
◮
◮
◮
boundary penalty methods [Nitsche 71]
interior penalty methods [Babuška 73, Douglas & Dupont 75,
Baker 77, Wheeler 78, Arnold 82]
neutron transport simulation [Reed & Hill 73] (steady, linear)
CV analysis [Lesaint & Raviart 74, Johnson & Pitkäranta 86]
time-dependent conservation laws [Cockburn & Shu 89-]
Friedrichs systems
◮
◮
linear PDE systems with symmetry and L2 -positivity properties
unify mixed elliptic and first-order PDEs [AE & Guermond, 06-]
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Motivations
◮
Discontinuous Galerkin (dG) methods can be viewed as
◮
◮
◮
finite element methods with discontinuous discrete functions
finite volume methods with more than one DOF per mesh cell
Possible motivations to consider dG methods
◮
◮
◮
◮
◮
flexibility in the choice of basis functions
general meshes: non-matching interfaces, polyhedral cells
local discrete formulation using fluxes and local test functions (in
particular, for strongly-contrasted material properties)
block-diagonal mass matrices for time-stepping
easily amenable to variable polynomial order, local time-stepping
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Outline
◮
Part I: Diffusion
◮
Part II: Diffusion-advection-reaction
Main reference for this lecture Di Pietro & AE, Mathematical aspects
of DG methods, Springer 2012
See also forthcoming book on FEM [AE & Guermond 16]
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Diffusion
◮
Discrete setting
◮
Laplacian
◮
Variable diffusion
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Discrete setting
◮
dG methods accommodate fairly general meshes
◮
◮
polyhedral cells (with various shapes)
nonmatching contact between adjacent cells (hanging nodes)
◮
Mesh indexed by h (e.g., maximal meshsize); CV analysis as h → 0
◮
Given a mesh Th of a domain Ω, examples of discrete spaces are the
broken polynomial spaces (k ≥ 0)
Pkd (Th ) = {vh ∈ L∞ (Ω) | vh|T ∈ Pkd (T ), ∀T ∈ Th }
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Faces, mean-values, and jumps
◮
Interface Fhi ∋ F = ∂T1 ∩ ∂T2
◮
◮
oriented by unit normal nF from T1 to T2 (fixed once and for all)
mean-values and jumps at interfaces (vi := v|Ti , i ∈ {1, 2})
{{v }} = 12 (v1 + v2 )
[[v ]] = v1 − v2
T2
T1
F
◮
Boundary face Fhb ∋ F = ∂T ∩ ∂Ω, nF pointing outward Ω
{{v }} = [[v ]] = v|T
◮
Mesh faces are collected in the set Fh = Fhi ∪ Fhb
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Important algebraic identity
◮
Crucial when integrating by parts cellwise
◮
For pcw. smooth functions a (vector-valued) and b (scalar-valued)
X Z
X Z
∇·(ab) =
(ab)·nT (outward unit normal to T )
T ∈Th
T
T ∈Th
=
∂T
X Z
F ∈Fhi
=
X Z
F ∈Fhi
=
X Z
F ∈Fh
Alexandre Ern
Discontinuous Galerkin methods
[[ab]]·nF +
F
X Z
F ∈Fhb
(ab)·nF
({{a}}[[b]] + [[a]]{{b}})·nF +
F
X Z
F ∈Fhb
{{a}}[[b]]·nF +
F
(nF = nT1 = −nT2 )
F
X Z
F ∈Fhi
({{a}}[[b]])·nF
F
[[a]]{{b}}·nF
F
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Some basic facts from functional analysis
◮
Broken Sobolev spaces, e.g.,
H 1 (Th ) := {v ∈ L2 (Ω) | v|T ∈ H 1 (T ), ∀T ∈ Th }
◮
Broken gradient (defined cellwise) ∇h : H 1 (Th ) → [L2 (Ω)]d
(∇h v )|T = ∇(v|T )
∀T ∈ Th
We have ∇h v = ∇v if v ∈ H 1 (Ω)
◮
A function v ∈ H 1 (Th ) belongs to H 1 (Ω) if and only if
[[v ]] = 0
∀F ∈ Fhi
(distributional argument)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Regularity of a mesh sequence {Th }h>0
◮
Described by means of a matching simplicial submesh
◮
◮
shape-regular in the usual sense of Ciarlet
local meshsize comparable to that of Th
T2
T1
◮
F
Geometric properties resulting from mesh regularity
◮
◮
◮
#(subsimplices) of T ∈ Th is uniformly bounded
#(faces) of T ∈ Th is uniformly bounded
h T1 ∼ h F ∼ h T2
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Analysis tools
◮
Local inverse inequality ∀vh ∈ Pkd (T ), ∀T ∈ Th ,
k∇vh k[L2 (T )]d ≤ Cinv hT−1 kvh kL2 (T )
◮
◮
◮
Markov brothers’ inequality in L∞ (−1, 1) (1890)
Cinv ∼ k 2 [Schwab 98]; Cinv computable from eigenvalue pb.
Multiplicative trace inequality ∀v ∈ H 1 (T ), ∀T ∈ Th ,
−1
1
1
kv kL2 (∂T ) ≤ Cmtr hT 2 kv kL2 (T ) + kv kL22 (T ) k∇v k[L2 2 (T )]d
◮
◮
◮
lowest-order Raviart–Thomas functions and divergence formula
[Carstensen & Funken 00; Stephansen 07; Di Pietro & AE 12]
in a polyhedral cell, carve a sub-simplex from each triangular
sub-face with height ∼ hT (allows for some face degeneration)
Discrete trace inequality ∀vh ∈ Pkd (T ), ∀T ∈ Th ,
−1
kvh kL2 (∂T ) ≤ Cdtr hT 2 kvh kL2 (T )
◮
follows from LI and MT inequalities; Cdtr ∼ k
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Polynomial approximation in polyhedral cells
◮
L2 -orthogonal projection πTk : L2 (T ) → Pkd (T )
(πTk (v ) − v , q)L2 (T ) = 0
◮
∀q ∈ Pkd (T )
Poincaré–Steklov inequality ∀v ∈ H 1 (T ), ∀T ∈ Th ,
kv − πh0 (v )kL2 (T ) ≤ CPS hT k∇v k[L2 (T )]d
◮
◮
◮
πh0 (v ) is the mean-value of v over T
CPS = π −1 for convex T (Poincaré (1894) [eigenvalue pb], Steklov (1897)
[d = 1], Payne & Weinberger (60) [d = 2], Bebendorf (03) [d ≥ 3])
For non-convex T , uniform bound on CPS using simplicial sub-cells and
MT inequality [AE & Guermond 16]
◮
PS inequality can be bootstrapped using Bramble–Hilbert polynomial
to |v − πTk (v )|H m (T ) ≤ Capp hTk+1−m |v |H k+1 (T ) for all 0 ≤ m ≤ k + 1
◮
See also [Dupont & Scott 80] for alternate proof using averaged
Taylor polynomials
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Most useful properties
◮
∀v ∈ H k+1 (T ), ∀T ∈ Th ,
kv − πTk v kL2 (T ) ≤ Capp hTk+1 |v |H k+1 (T )
k∇(v − πTk v )k[L2 (T )]d ≤ Capp hTk |v |H k+1 (T )
k+ 12
kv − πTk v kL2 (∂T ) ≤ Capp hT
◮
◮
|v |H k+1 (T )
bounds extend to fractional Sobolev regularity [AE & Guermond 16]
Global L2 -orth. projection πhk : L2 (Ω) → Pkd (Th ) is assembled cellwise
πhk (v )|T = πTk (v|T )
∀T ∈ Th
(global mass matrix is block-diagonal)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
The Laplacian
◮
◮
Let f ∈ L2 (Ω); seek u : Ω → R s.t. −△u = f in Ω and u|∂Ω = 0
Weak formulation: u ∈ V := H01 (Ω) s.t.
Z
Z
a(u, w ) :=
∇u·∇w =
fw =: ℓ(w )
Ω
◮
The exact solution satisfies
[[u]] = 0
◮
∀w ∈ V
Ω
∀F ∈ Fh = Fhi ∪ Fhb
Other BC’s (Neumann, Robin) can be considered as well
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Normal flux
◮
Physically, the normal component of the diffusive flux σ := −∇u is
continuous across interfaces
◮
What is the mathematical meaning of [[σ]]·nF = 0 for F ∈ Fhi ?
◮
If σ ∈ [Lp (Ω)]d , p > 2, and ∇·σ ∈ L2 (Ω) then
1
σ|T ·nF ∈ W − p ,p (F )
◮
◮
this holds provided u ∈ H 1+s (Ω), s > 0, and △u ∈ L2 (Ω)
If σ ∈ [H s (Ω)]d , s > 12 , then σ|∂T ∈ [L2 (∂T )]d
◮
◮
∀T ∈ Th , ∀F ⊂ ∂T
this holds provided u ∈ H 1+s (Ω), s >
1
2
Elliptic regularity theory shows that on a polyhedron, u ∈ H 1+s (Ω),
s > 21 , and s = 1 if Ω is convex
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Symmetric Interior Penalty
◮
Discrete space Vh := Pkd (Th ), k ≥ 1
◮
Seek uh ∈ Vh s.t. ah (uh , wh ) = ℓ(wh ), ∀wh ∈ Vh , with
Z
X Z
ah (vh , wh ) :=
∇h vh ·∇h wh −
{{∇h vh }}·nF [[wh ]]
Ω
F ∈Fh
F
{z
}
consistency
X η Z
X Z
[[vh ]][[wh ]]
−
[[vh ]]{{∇h wh }}·nF +
hF F
F ∈Fh
F ∈Fh F
{z
} |
{z
}
|
symmetry
penalty
|
◮
Main properties of ah
◮
◮
strong consistency: ah (u, wh ) = ℓ(wh ), ∀wh ∈ Vh
coercivity on Vh if η is large enough
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Step-by-step derivation
◮
Starting point: Use broken gradient in exact bilinear form
(0)
ah (vh , wh ) :=
◮
Z
∇h vh ·∇h wh
Ω
R
P
(0)
(ah (u, wh ) = ℓ(wh ) + F ∈Fh F {{∇u}}·nF [[wh ]])
Z
X Z
(1)
{{∇h vh }}·nF [[wh ]]
∇h vh ·∇h wh −
ah (vh , wh ) :=
Restore consistency
Ω
◮
F
Restore symmetry in a consistent way
(2)
ah (vh , wh ) :=
◮
F ∈Fh
Z
∇h vh ·∇h wh −
Ω
X Z
F ∈Fh
{{∇h vh }}·nF [[wh ]]−
F
X Z
F ∈Fh
[[vh ]]{{∇h wh }}·nF
F
Achieve coercivity by penalizing jumps [Arnold 82]
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Stability
◮
dG norm: broken gradient plus jump seminorm
kvh k2dG := k∇h vh k2[L2 (Ω)]d + |vh |2J ,
|vh |2J =
X 1
k[[vh ]]k2L2 (F )
hF
F ∈Fh
◮
k·kdG is a norm on Vh (direct verification)
◮
◮
◮
discrete Sobolev inequality kvh kLq (Ω) ≤ σq kvh kdG , ∀vh ∈ Vh , with
2d
q ∈ [1, d−2
] if d ≥ 3 and q ∈ [1, ∞) if d = 2
see [Brenner 03] (for q = 2), [Eymard, Gallouët & Herbin 10] (for
FV and general q), [Di Pietro & AE 10] (for general q, k)
2
If η > Cdtr
N∂ , where
◮
◮
Cdtr results from discrete trace inequality (recall Cdtr ∼ k)
N∂ is the maximum number of faces a mesh cell can have
then ∃Csta > 0 s.t. ah (vh , vh ) ≥ Csta kvh k2dG , ∀vh ∈ Vh
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Algebraic realization
◮
SPD stiffness matrix
◮
Compact stencil (only neighbors in the sense of faces)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Error analysis: Boundedness
◮
◮
Approximation error (u − uh ) is in V♭ = (H 1+s (Ω) ∩ V ) + Vh , s >
1
2
Boundedness: ah (v , wh ) ≤ Cbnd kv kdG,♯ kwh kdG , ∀(v , wh ) ∈ V♭ × Vh
X
kv k2dG,♯ := kv k2dG +
hT k∇v ·nT k2L2 (∂T )
T ∈Th
◮
The two norms are equivalent on Vh
kvh kdG ≤ kvh kdG,♯ ≤ C♯ kvh kdG
Alexandre Ern
Discontinuous Galerkin methods
∀vh ∈ Vh
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Error analysis: Second Strang’s Lemma
◮
Optimal error estimate in k·kdG,♯ -norm
ku − uh kdG,♯ ≤ C inf ku − yh kdG,♯
yh ∈Vh
◮
Let yh ∈ Vh ; coercivity, consistency, and boundedness imply
kuh − yh kdG,♯ ≤ C♯ kuh − yh kdG
−1
≤ C♯ Csta
−1
= C♯ Csta
ah (uh − yh , wh )
kwh kdG
wh ∈Vh \{0}
sup
sup
wh ∈Vh \{0}
ah (u − yh , wh )
kwh kdG
−1
≤ C♯ Csta
Cbnd ku − yh kdG,♯
and use the triangle inequality
−1
ku − uh kdG,♯ ≤ (1 + C♯ Csta
Cbnd )ku − yh kdG,♯
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Convergence rates
◮
Assume exact solution u is smooth enough
◮
Using polynomial approximation properties in dG spaces yields
ku − uh kdG,♯ ≤ C
X
hT2k |u|2H k+1 (T )
T ∈Th
◮
!1/2
Assuming full elliptic regularity pickup, Aubin–Nitsche’s duality
argument leads to
ku − uh kL2 (Ω) ≤ C h
X
T ∈Th
Alexandre Ern
Discontinuous Galerkin methods
hT2k |u|2H k+1 (T )
!1/2
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Two side-excursions
◮
Lifting the jumps
◮
Mixed dG methods
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Lifting the jumps I
◮
Local lifting Let l ≥ 0, F ∈ Fh ; rlF : L1 (F ) → [Pld (Th )]d is s.t.
Z
Z
rlF (ϕ)·τh = {{τh }}·nF ϕ
∀τh ∈ [Pld (Th )]d
Ω
◮
◮
◮
◮
◮
F
rlF
is vector-valued, collinear to nF
the support of rlF reduces to the (one or two) mesh cells sharing F
rlF is easy to compute (invert local mass matrix)
see [Bassi, Rebay et al 97], [Brezzi et al 00]
Penalizing local liftings of jumps instead of jumps yields coercivity
for η > N∂ with the same stencil, l ∈ {k − 1, k}
Z
X Z
ah (vh , wh ) :=
∇h vh ·∇h wh −
{{∇h vh }}·nF [[wh ]]
Ω
−
F ∈Fh
X Z
F ∈Fh
Alexandre Ern
Discontinuous Galerkin methods
F
[[vh ]]{{∇h wh }}·nF +
F
X
F ∈Fh
η
Z
Ω
rlF ([[vh ]])· rlF ([[wh ]])
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Lifting the jumps II
◮
Global lifting of jumps: For all v ∈ H 1 (Th ),
X
Rlh ([[v ]]) :=
rlF ([[v ]]) ∈ [Pld (Th )]d
F ∈Fh
◮
Discrete gradient Ghl : H 1 (Th ) → [L2 (Ω)]d s.t.
Ghl (v ) := ∇h v − Rlh ([[v ]])
◮
Discrete gradients are important tools in nonlinear problems
◮
◮
nonlinear elasticity [Lew et al. ’04], Leray–Lions [Burman & AE ’08,
Buffa & Ortner ’09], Navier–Stokes [Di Pietro & AE ’10]
asymptotic consistency: Let (vh )h>0 be a sequence in (Vh )h>0
bounded in the k·kdG -norm. Then, ∃v ∈ H01 (Ω) s.t. as h → 0, up to
subseq., vh → v strongly in L2 (Ω) and for all l ≥ 0, Ghl (vh ) ⇀ ∇v
weakly in [L2 (Ω)]d [Di Pietro & AE ’10]
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Lifting the jumps III
◮
Local formulation with numerical fluxes (FV viewpoint)
◮
Let T ∈ Th with faces collected in FT , let ξ ∈ Pkd (T )
◮
For the exact solution
Z
Z
Z
X
∇u·∇ξ +
ΦF (u)ξ =
fξ
ǫT ,F
T
T
F
F ∈FT
with ǫT ,F = nT ·nF and exact flux ΦF (u) = −∇u·nF
◮
For the discrete solution (l ∈ {k − 1, k})
Z
Z
Z
X
(∇uh − Rlh ([[uh ]]))·∇ξ +
φF (uh )ξ =
fξ
ǫT ,F
T
F ∈FT
with numerical flux φF (uh ) = −{{∇h uh }}·nF +
Alexandre Ern
Discontinuous Galerkin methods
T
F
η
hF
[[uh ]]
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Mixed dG methods I
◮
Mixed formulation: σ + ∇u = 0 and ∇·σ = f in Ω
◮
Mixed dG method: Find uh ∈ Pkd (Th ), σh ∈ [Pkd (Th )]d (equal-order)
s.t.
Z
σh ·ζ −
T
−
Z
uh ∇·ζ +
T
Z
T
X
ǫT ,F
X
ǫT ,F
F ∈FT
ûF (ζ·nF ) = 0
Z
(σ̂F ·nF )ξ =
∀ζ ∈ [P kd (T )]d
F
F ∈FT
σh ·∇ξ +
Z
F
Z
fξ
T
∀ξ ∈ P kd (T )
for all T ∈ Th , with numerical fluxes ûF and σ̂F
◮
σh can be eliminated locally whenever ûF does not depend on σh
◮
See [Arnold, Brezzi, Cockburn, and Marini 02] for a unified analysis
of dG methods based on numerical fluxes
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Mixed dG methods II
◮
Numerical fluxes for SIP
(
{{uh }}
ûF =
0
∀F ∈ Fhi
∀F ∈ Fhb
σ̂F = −{{∇h uh }} + ηhF−1 [[uh ]]nF
◮
∀F ∈ Fh
Numerical fluxes for LDG (Local dG [Cockburn & Shu 98])
ûF as for SIP and
σ̂F = {{σh }} + ηhF−1 [[uh ]]nF
◮
◮
◮
◮
σh can be eliminated locally
main advantage: discrete coercivity for η > 0 (e.g. η = 1)
drawback: larger stencil (neighbors of neighbors)
stencil reduction [Castillo, Cockburn, Perugia, and Schötzau 00]
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Mixed dG methods III
◮
Two-field approach [AE & Guermond 06]
(
{{uh }} + ησ [[σh ]]·nF
ûF =
0
σ̂F = {{σh }} + ηu [[uh ]]nF
◮
Drawback σh cannot be eliminated
◮
Advantages
◮
◮
◮
∀F ∈ Fhi
∀F ∈ Fhb
∀F ∈ Fh
a simple choice for penalty is ηu = ησ = 1
the choice k = 0 is possible
quasi-optimal estimate on the diffusive flux
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Mixed dG methods IV
◮
Hybridizable dG (HDG) methods introduce interface DOFs
◮
◮
[Cockburn, Gopalakrishnan, and Lazarov 09]
see also [Causin and Sacco 05], [Droniou and Eymard 06]
L
Pkd−1 (F )
◮
Skeletal discrete space Λh :=
◮
Discrete unknowns (σh , uh , λh ) ∈ Σh × Uh × Λh
◮
◮
◮
F ∈Fhi
σh and uh can be eliminated locally
global problem in λh ∈ Λh with compact stencil
A new viewpoint emerged recently: Hybrid High-Order (HHO)
methods
◮
◮
introduced in [Di Pietro & AE 15], [Di Pietro, AE & Lemaire 14]
see tomorrow’s lecture!
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Variable diffusion
◮
◮
Seek u : Ω → R s.t. −∇·(κ∇u) = f in Ω and u|∂Ω = 0
Weak formulation: For f ∈ L2 (Ω), seek u ∈ V := H01 (Ω) s.t.
Z
Z
a(u, v ) :=
κ∇u·∇v =
fv
∀v ∈ V
Ω
◮
◮
◮
Ω
κ is scalar-valued, bounded, and uniformly positive in Ω
the model problem is well-posed
Application to groundwater flows
◮
◮
u: hydraulic head, σ = −κ∇u: Darcy velocity
κ: highly-contrasted hydraulic conductivity
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Numerical illustration of high contrasts
◮
σ = −κ∇u ∈ H(div; Ω)
◮
◮
◮
the normal component of σ is continuous across any interface
the normal component of ∇u is discontinuous if κ jumps
Ω = (−1, 1) partitioned into Ω1 = (−1, 0) and Ω2 = (0, 1),
κ|Ω1 = α (α = 0.5 on left; α = 0.01 on right) and κ|Ω2 = 1
0.7
14
0.6
12
0.5
10
0.4
8
0.3
6
0.2
4
0.1
2
0
0
-0.1
-2
-1
Alexandre Ern
Discontinuous Galerkin methods
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Discrete setting
◮
κ pcw. constant on a polyhedral partition PΩ = {Ωi }1≤i≤NΩ of Ω
◮
Th compatible with PΩ (κ pcw. constant on Th )
◮
Discrete space Vh := Pkd (Th ), k ≥ 1
◮
SIP bilinear form
Z
X Z
ah (vh , wh ) =
κ∇h vh ·∇h wh −
{{κ∇h vh }}·nF [[wh ]]
Ω
−
F ∈Fh
X Z
F ∈Fh
Alexandre Ern
Discontinuous Galerkin methods
F
[[vh ]]{{κ∇h wh }}·nF +
F
X
F ∈Fh
γκ,F
η
hF
Z
[[vh ]][[wh ]]
F
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Diffusion-dependent penalty
◮
To achieve coercivity, penalty coefficient must depend on κ
◮
◮
◮
◮
γκ,F = {{κ}} [Houston, Schwab & Süli 02]
for high contrasts, γκ,F is controlled by the highest value of κ (the
most permeable layer)
... γκ,F should be controlled by the lowest value (the least permeable
layer) (as in Mixed FE and FV)
One simple choice is harmonic averaging
−1
:= {{κ−1 }}
γκ,F
We need to modify the consistency and symmetry terms to maintain
coercivity
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Symmetric Weighted Interior Penalty (SWIP)
◮
Weighted average {{v }}ω,F := ωT1 ,F v|T1 + ωT2 ,F v|T2
◮
◮
◮
◮
ωT1 ,F = ωT2 ,F = 21 recovers usual arithmetic averages
2
1
diffusion-dependent weights ωT1 ,F := κ1κ+κ
, ωT2 ,F := κ1κ+κ
2
2
(homogeneous diffusion yields back arithmetic averages)
see [Dryja 03] for idea, [Burman & Zunino 06] for mortaring, [AE,
Stephansen & Zunino 09], [Di Pietro, AE & Guermond 08] for
advection-diffusion with locally small or zero diffusion
SWIP bilinear form
Z
X Z
ah (vh , wh ) =
κ∇h vh ·∇h wh −
{{κ∇h vh }}ω ·nF [[wh ]]
Ω
−
F ∈Fh
X Z
F ∈Fh
◮
F
[[vh ]]{{κ∇h wh }}ω ·nF +
F
X
F ∈Fh
γκ,F
η
hF
Z
[[vh ]][[wh ]]
F
Strong consistency still holds
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Error analysis
◮
◮
C denotes a generic constant uniform w.r.t. h and κ
Coercivity: Assuming η > Ctr2 N∂ , ah (vh , vh ) ≥ Csta kvh k2dG with
X γκ,F
1
kvh k2dG := kκ 2 ∇h vh k2[L2 (Ω)]d +
k[[vh ]]k2L2 (F )
hF
F ∈Fh
◮
◮
Boundedness: ah (v , wh ) ≤ Cbnd kv kdG,♯ kwh kdG with
P
1
kv k2dG,♯ := kv k2dG + T ∈Th hT kκ 2 ∇v ·nT k2L2 (∂T )
Error estimate: ku − uh kdG,♯ ≤ C inf yh ∈Vh ku − yh kdG,♯
ku − uh kdG,♯ ≤ C
X
T ∈Th
◮
κT hT2k |u|2H k+1 (T )
!1/2
Extension to anisotropic κ: use normal component for penalty and
1
averages (error estimate mildly depends on anisotropy ratio ∼ ρ 2 )
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Outline
◮
Advection-reaction
◮
Péclet-robust diffusion-advection-reaction
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Model problem
◮
◮
Let Ω be a domain in Rd (open, bounded, connected, strongly
Lipschitz set)
Let β ∈ [W 1,∞ (Ω)]d and µ ∈ L∞ (Ω) be s.t.
1
µ − ∇·β ≥ µ0 > 0
2
◮
a.e. in Ω
Inflow and outflow parts of boundary ∂Ω
∂Ω± = {x ∈ ∂Ω | ± β(x)·n(x) > 0}
◮
Let f ∈ L2 (Ω); the model problem is
(
µu + β·∇u = f
u=0
Alexandre Ern
Discontinuous Galerkin methods
in Ω
on ∂Ω−
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Functional framework
◮
Graph space W = {v ∈ L2 (Ω) | β·∇v ∈ L2 (Ω)}
◮
◮
Hilbert space with norm kv k2W = kv k2L2 (Ω) + kβ·∇v k2L2 (Ω)
If ∂Ω± are well-separated, there is a bounded trace map
γ : W → L2 (|β·n|; ∂Ω) s.t. for all (v , w ) ∈ W × W ,
Z
Z
Z
Z
(∇·β)vw + (β·∇v )w +
v (β·∇w ) =
(β·n)γ(v )γ(w )
Ω
◮
◮
Ω
Ω
∂Ω
see [AE & Guermond 06]
the separation assumption cannot be circumvented for traces in
L2 (|β·n|; ∂Ω)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Weak formulation
◮
Define on W × W the bilinear form
Z
Z
a(v , w ) :=
µvw + (β·∇v )w +
Ω
(β·n)⊖ vw
∂Ω
◮
where for x ∈ R, x ⊕ = 21 (|x| + x) and x ⊖ = 12 (|x| − x)
R
Define on W the linear form ℓ(w ) := Ω fw
◮
BCs are weakly enforced
◮
Seek u ∈ W s.t. a(u, w ) = ℓ(w ), ∀w ∈ W
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Well-posedness
◮
◮
a is L2 -coercive on W : integrating by parts, we infer that
Z Z
Z
1
1
a(v , v ) =
µ − ∇·β v 2 +
(β·n)γ(v )2 +
(β·n)⊖ γ(v )2
2
2 ∂Ω
Ω
∂Ω
Z
1
2
|β·n|γ(v )2
≥ µ0 kv kL2 (Ω) +
2 ∂Ω
The weak problem is well-posed
◮
◮
L2 -coercivity implies uniqueness
existence by inf-sup argument [Ern & Guermond 06]
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Discrete setting
◮
Discrete space Vh := Pkd (Th ), k ≥ 0
◮
Discrete problem: Seek uh ∈ Vh s.t. ah (uh , wh ) = ℓ(wh ), ∀wh ∈ Vh
◮
Main properties of ah : strong consistency and L2 -coercivity on Vh
◮
We assume that u ∈ H s (Ω), s > 12 ; then,
(β·nF )[[u]] = 0
∀F ∈ Fhi
(distributional argument)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Centered fluxes
◮
◮
Use broken gradient in exact bilinear form
Recover L2 -coercivity in a consistent way by setting
Z
X Z
ahcf (vh , wh ) :=
µvh wh + (β·∇h vh )wh +
(β·n)⊖ vh wh
Ω
−
F ∈Fhb
X Z
F ∈Fhi
F
(β·nF )[[vh ]]{{wh }}
F
P
R
1
|β·n|vh2
F 2
◮
ahcf (vh , vh ) ≥ µ0 kvh k2L2 (Ω) +
◮
Error estimate for smooth solution: ku − uh kL2 (Ω) ≤ Chk |u|H k+1 (Ω)
◮
F ∈Fhb
convergence for k ≥ 1 only, and with suboptimal rate
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Local formulation and stencil
◮
Let T ∈ Th , let ξ ∈ Pkd (T ) (FV viewpoint)
Z
Z
Z
X
(µ − ∇·β)uh ξ − uh (β·∇ξ) +
ǫT ,F
φF (uh )ξ =
fξ
T
F ∈FT
F
T
with ǫT ,F := nT ·nF = ±1 and numerical fluxes
(
(β·nF ){{uh }}
∀F ∈ Fhi
φF (uh ) =
⊕
(β·n) uh
∀F ∈ Fhb
◮
Standard dG stencil (neighbors in the sense of faces)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Upwind fluxes
◮
Strengthen discrete stability by penalizing interface jumps in a
least-squares sense [Brezzi, Marini & Süli 04]
ah (vh , wh ) := ahcf (vh , wh ) + sh (vh , wh )
with stabilization bilinear form
sh (vh , wh ) =
X Z
F ∈Fhi
◮
F
1
2 |β·nF |[[vh ]][[wh ]]
Strong consistency is preserved
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Stability
◮
Stability norm (βT := kβk[L∞ (T )]d )
X Z
X Z
2
2
2
1
kvh kdG := µ0 kvh kL2 (Ω) +
2 |β·n|vh +
F ∈Fhb
+
X
F
F ∈Fhi
F
2
1
2 |β·nF |[[vh ]]
βT−1 hT kβ·∇v k2L2 (T )
T ∈Th
◮
Assume for simplicity hT µ0 ≤ cµ,β βT , Lβ,T + kµkL∞ (T ) ≤ cµ,β µ0
◮
◮
◮
we hide cµ,β in the generic constants
−1
general weight on adv. derivative: time-scale τT = min(µ−1
0 , βT h T )
Discrete inf-sup condition [Johnson & Pitkäranta 86]
Csta kvh kdG ≤
◮
◮
ah (vh , wh )
wh ∈Vh \{0} kwh kdG
sup
first three terms controlled by coercivity
bound on advective derivative: test with wh|T = βT−1 hT hβiT ·∇vh
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Error analysis
◮
Boundedness: ah (v , wh ) ≤ Cbnd kv kdG,♯ kwh kdG with
kv k2dG,♯
:=
kv k2dG
+
X
βT
hT−1 kv k2L2 (T )
+
kv k2L2 (∂T )
T ∈Th
◮
!
Error estimate: ku − uh kdG ≤ C inf yh ∈Vh ku − yh kdG,♯
◮
◮
k·kdG,♯ and k·kdG may not be equivalent on Vh , but they lead to the
same decay rates P
of best-approximation errors on smooth functions
ku − uh kdG ≤ C ( T ∈Th βT hT2k+1 |u|H k+1 (T )2 )1/2
1
◮
◮
quasi-optimal L2 -error estimate O(hk+ 2 )
optimal error estimate on advective derivative
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Local formulation and stencil
◮
Let T ∈ Th , let ξ ∈ Pkd (T )
◮
New numerical fluxes
(
(β·nF ){{uh }} + 12 |β·nF |[[uh ]]
φF (uh ) =
(β·n)⊕ uh
◮
∀F ∈ Fhi
∀F ∈ Fhb
Example: F = ∂T1 ∩ ∂T2 , β flows from T1 to T2 so that β·nF ≥ 0
φF (uh ) = (β·nF )({{uh }} + 21 [[uh ]])
= (β·nF ) 12 (uh |T1 + uh |T2 + uh |T1 − uh |T2 )
= (β·nF )uh|T1
◮
Standard dG stencil (neighbors in the sense of faces)
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Further comments
◮
L2 -coercivity can be relaxed to µ − 21 ∇·β ≥ 0
◮
◮
◮
Localized error estimate to avoid global high-order Sobolev norm
◮
◮
◮
assume that there is ζ ∈ W 1,∞ (Ω) s.t. −β·∇ζ ≥ θ0 > 0
reasonable if β has no stationary points or closed curves [Devinatz,
Ellis & Friedman 74]
cut-off functions, exponential decay away from singular layers
see [Johnson, Schatz & Wahlbin 87; Guzmán 06]
Nonlinear conservation laws
◮
◮
upwinding promotes Gibbs phenomenon [AE & Guermond 13]
needs to add nonlinear stabilization mechanism to temper it
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Diffusion-advection-reaction
◮
◮
Model problem
(
in Ω
on ∂Ω
Assumptions on the data
◮
◮
◮
◮
µu + β·∇u − ∇·(κ∇u) = f
u=0
f ∈ L2 (Ω)
β ∈ [W 1,∞ (Ω)]d , µ ∈ L∞ (Ω), µ − 12 ∇·β ≥ µ0 > 0
κ scalar-valued, bounded, uniformly positive
Local Péclet number PeT =
◮
◮
◮
β T hT
κT
for all T ∈ Th
PeT ≤ 1: diffusion-dominated regime
PeT ≥ 1: advection-dominated regime
h2
−1
more generally, PeT = τT TκT with τT = min(µ−1
0 , βT h T )
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Discrete setting
◮
Discrete space Vh := Pkd (Th ), k ≥ 1
◮
Combine SWIP with upwind fluxes
◮
◮
◮
centered fluxes can be used in diffusion-dominated regime
Scharfetter–Gummel-type weights can be used as well
Discrete bilinear form (we drop the symmetry term and integrate by
parts the advective derivative)
Z
ah (vh , wh ) =
(µ − ∇·β)vh wh − vh (β·∇h wh ) + κ∇h vh ·∇h wh
Ω
X Z
−
({{κ∇h vh }}ω + β{{vh }})·nF [[wh ]]
F ∈Fh
+
X Z
F ∈Fh
with γκ,β,F = η
Alexandre Ern
Discontinuous Galerkin methods
γκ,F
hF
F
γκ,β,F [[vh ]][[wh ]]
F
+ 12 |β·nF | if F ∈ Fhi (or γκ,β,F = ... + (β·nF )⊖ if F ∈ Fhb )
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Error analysis
◮
Stability norm
X kvh k2dG :=
µ0 kvh k2L2 (T ) + βT−1 hT kβ·∇vh k2L2 (T ) + κT k∇vh k2[L2 (T )]d
T ∈Th
+
X
γκ,β,F k[[vh ]]k2L2 (F )
F ∈Fh
◮
Main steps of error analysis
◮
◮
◮
◮
strong consistency
discrete inf-sup stability [technical difficulty for anisotropic κ]
boundedness in suitable k·kdG,♯ -norm
Error estimate for smooth solution
ku − uh kdG ≤ C
X
T ∈Th
◮
(µ0 hT2
+ βT h T +
κT )hT2k |u|2H k+1 (T )
!1/2
expected decay in both diffusion- and advection-dominated regimes
Alexandre Ern
Discontinuous Galerkin methods
Université Paris-Est, CERMICS
Introduction
Diffusion
Diffusion-advection-reaction
Numerical illustrations
◮
Rotating advective field [AE, Stephansen & Zunino 09]
◮
◮
strong x- or y -diffusion, anisotropy ratio 106
SIP+upw enforces zero jumps in under-resolved layers
SWIP+upw
SIP+upw
1
1
0.000675
0.000665
0.000607
0.000595
0.00054
0.000525
0.000472
0.000456
0.000405
0.000386
0.000337
0.000316
0.000269
0.000246
0.000202
0.000176
0.000134
0.000107
6.66e−05
−1e−06
0
0
◮
1
3.68e−05
−3.3e−05
0
0
1
Constant advective field with locally zero anisotropic diffusion
[Di Pietro, AE & Guermond 08]
1
β = (−1, 0)
1
0
κ|Ω1 =
0 0.5
0 0
κ|Ω2 =
0 1
Alexandre Ern
Discontinuous Galerkin methods
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
Université Paris-Est, CERMICS