A Multiscale Discontinuous Galerkin Method P. Bochev Computational Mathematics and Algorithms Sandia National Laboratories T.J.R Hughes Institute for Computational Engineering and Science, The University of Texas at Austin G. Scovazzi Computational Physics Department Sandia National Laboratories LSSC 05 Sozopol Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy ’ s National Nuclear Security Administration under contract DE-AC04-94AL85000.
Computational mathematics and algorithms Nomenclature T h � → Partition of Ω into finite elements K ( ) → The set of all vertices of an element or a collection of elements V ( � ) E ( � ) → The set of all edges of an element or a collection of elements E ( T h ( � )) → The set of all edges = � h Q 1 Q 2 → The set of internal edges = E ( T h ( � / � � )) 0 � h T 1 Q 3 → A generic finite element: T or Q K T 2 ˆ → A generic reference element K
Computational mathematics and algorithms The Discontinuous Space Reference element space Q 1 Q 2 K ) ˆ S p ( ˆ ( ) � polynomials of degree ≤ p( ) ˆ K K p=1 p=1 Local element space: T 1 S p ( K ) K Q 3 ( ) � image of the reference space p=2 p=1 p=3 T 2 Discontinuous space: � h ( � ) = � h � L 2 ( � )| � h | K � S p ( K ) ( K ) � K � T h { } Formal union of the local spaces, polynomial Hierarchical basis degrees not constrained by continuity across element � � � ( ) ( ) ( ) v v V v x e e E x b k B k x � h = + + boundaries. v e k
Computational mathematics and algorithms The (Minimal) Continuous Space The continuous space – Required for the additive decomposition step in VMS – Defined with respect to the same partition of Ω into finite elements – Completely unrelated to the discontinuous space The minimal continuous space Q 1 Q 2 – Used here for simplicity p=1 p=1 – Spanned by vertex shape functions – H 1 conforming T 1 h | K � S 1 ( K ) � K � T h { } h � H 1 ( � )| � h ( � ) = � � Q 3 p=1 p=3 p=2 Basis T 2 N i ( v j ) = � ij N v | K = V v
Computational mathematics and algorithms Orientations, Jumps and Averages Orientation – Elements oriented as sources – + – Edges oriented by choosing a normal e 2 + + Upwind (-) & downwind (+) elements e 1 e 3 – – – Determined by shared edge orientation e 4 + e 5 – + – K + K – � + � � Jumps and averages Inflow & outflow element boundary � = 1 2 � + + � � [ ] = � + n + + � � n � ( ) � – Determined by edge orientations u = 1 2 u + + u � [ ] = u + � n + + u � � n � ( ) u ∂ K + ∂ K –
Computational mathematics and algorithms Variational Multiscale DG approach Model BVP L ( x , D ) � = f in � and R ( x , D ) � = g on � Donor DG formulation ( ) = F DG � h ( ) ( ) B DG � h , � h � � h � � h � ( ) DG space � h � Interscale ( ) a � h � ( ) T : � h � operator ( ) = F DG T � ( ) ( ) B DG T � h , T � � � h � � h � ( ) C 0 space � h � h h VMS-DG formulation
Computational mathematics and algorithms Interscale operator The key to VMS-DG is the definition of T – T must reflect PDE’s structure � � � – T must be locally defined (for efficiency) We use VMS to derive the local problems Additive decomposition ( ) a � = � + � ( ) ( ) � � � � h � � � � � h � � � � h � � Coarse scale Fine scale VMS equations ( ) + B DG ( ) = F DG � ( ) ( ) B DG � h , � � h , � � � � h � � h � ← Coarse scale equation h h h ( ) = F DG ← Fine scale equation ( ) + B DG � ( ) ( ) B DG h , � h , � � � � � � � � � � h � � h � h h h
Computational mathematics and algorithms The local problem Assume: � ( ) = ( ) ← element form B DG � h , � h B K � h , � h K � ← boundary form ( ) B � � h , � h + e �� h ( ) � { � , � h + } , � h { � , � h + } ← edge form B e + � h 0 e �� h Element equations coupled through the edge form To derive the local problem 1. Treat coarse scale function as data for the fine scale equation 2. Restrict fine scale equation to an element K h � S p ( K ) K ( ) � R K ( � ( ) = F DG ( ) � B DG � ( ) B DG � h , � � h , � h ) � � � � � � � h h h 3. Uncouple fine scale equation from adjacent elements
Computational mathematics and algorithms Restriction to an element States of the restricted weight function � h for e � � K � � = 0 for e � � K � � � � K=K – + = ( ) ( ) � � � � � � � h = 0 � h h 0 for e � � K + h for e � � K + � � � � ∂ K + h = 0 h � 0 h = 0 � � � � � � ∂ K – + = � � K � � = � � K + ( ) � ( ) � ( ) ( ) � � � � � � h = 0 � h h � h h K=K + States of the coarse scale solution + = � � = � h � C 0 ( ) ( ) � � � h h h Coarse scale residual ( ) ( ) = F DG ( ) � B � � ( ) � � { } , � ( � K + ) � { } ( ) � B K � h , � ( � K � ) � R K � h , � h , � B e h , � � � � � � � � h h h h h h e � E ( K )
Computational mathematics and algorithms Uncoupling from adjacent elements Coupled fine scale equation on K � � + , � { } , � ( � K + ) � � ( ) ( ) ( ) { h , � ( � K � ) � } ( ) + B � ( ) + � � B K � h , � h , � � B e � � = R K � � � � � � � � � h h h h h h � � e � E ( K ) Redefine edge form in terms of the element fine scale function only � � + , � { } , � ( � K + ) � ( ) ( ) ( ) { } { } h , � ( � K � ) � } , � ( � K + ) � h , � ( � K � ) � � � � { B e � � � � � � B � � , � � � � � h h h e h � � Uncoupled fine scale equation on K ( ) � { } ( ) ( ) � B K ( ) + B � ( ) + B h , � h , � h , � B { } , = R K � � � � � � � � � � � � � � � � K h h h e h h h e � E ( K ) Interscale operator ( ) a � h � ( ) T : � h � ( ) a S p ( K ) K ( ) ( ) = R K � B � h , � � ( ) � � T K : � h � K h h T | K = T K
Computational mathematics and algorithms Application to scalar advection-diffusion Boundary value problem ( ) � n = h on � N � ) F a + F d � ( � N ) = f in � ( � � F a + F d � N � N � N � + � = g on � D ) � n = h � on � N � ( F a + F d � D ) � n = h + on � N � D ( + � F d � D + Flux F a = a � advective diffusive F d = � � � � � � � + a Edge orientation By advective direction ⇒ Upwind & downwind elements consistent with physical meaning
Computational mathematics and algorithms A donor DG formulation Residual formulation F K � � � + f � K � ( ) dx ( ) � d � � � + ) F a � n � h � ( ) � K d � 0 = � ( � N F � n � + + K K � N � D � ← Weak Dirichlet condition ( ) W ( � ) d � + � � � g � D F b � + ; � � ) � + � � � ← Flux balance � � ( ( ) d � + � 0 e e �� h � coupling F c � + ; � � ) � + � � � ← Weak continuity � ( ( ) d � � � + terms � 0 e e �� h � ← Least-squares stabilization � � [ ] � [ ] d � + � � 0 e e �� h Numerical fluxes F b � + ; � � h � + ; � � h + s ( ) = F b ( ) � n = s ( h ) � n 11 F a 12 F d ← Approximate true flux h � + ; � � h + s 22 F d F c � + ; � � ( ) = F c ( ) � n = s 21 F a ( ) � n h
Computational mathematics and algorithms Weak form components Equivalent form of flux terms ) � + � � � h � + ; � � F b � + ; � � ( ( ) d � ( ) � � � [ ] d � F b = e e ) � + � � � F c � + ; � � ( ( ) d � ( ) � � � F c � + ; � � [ ] d � = e e DG bilinear form components F K � � � dx � ( ) = � ← volume B K � , � K ( ) � d � � + ) F a � n � � ← boundary ( ) = ( ) � K d � B � � , � � ( � N F � n � W ( � ) d � + + � � N � D � D ( ) = ← edge { } , � + , � � { } � � ( ) � ( ) � � + , � � F b � + ; � � [ ] + F c � + ; � � [ ] + � � [ ] � [ ] d � B e e � E ( K ) e Coupling terms
Computational mathematics and algorithms Numerical flux & weight functions Flux DG-A DG-B h � + ; � � ( ) ( ) + F d � ( ) + F d � � ( ) F a � � F a � � ( ) F b ( ) ( ) F c � + ; � � sF d � � ( ) s F d � ( ) � n ( ) � + sF d � � W � DG-A ( ) + F d � F a � � ← Advective flux is upwinded , diffusive is averaged ( ) ( ) + F d � � ( ) DG-B F a � � ← Total flux is upwinded DG formulation (see D. Arnold at al, SINUM 2000) s=1 skew-symmetric excellent stability properties s=0 neutral requires LS stabilization adjoint consistent, optimal L 2 rates s=-1 symmetric Hughes, Scovazzi, Bochev, Buffa, submitted to CMAME (2005)
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