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VERSION:2.0
PRODID:-//Memento EPFL//
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SUMMARY:Stabilization methods for transient transport problems
DTSTART:20091118T161500
DTSTAMP:20260406T170048Z
UID:295ccba0fc88dde26d360abca336d15d9f17d27b840c968f720fe11b
CATEGORIES:Conferences - Seminars
DESCRIPTION:Prof. Erik Burman\nStabilized finite element methods is by now
  a standard tool for the computation of hyperbolic transport equations or 
 convection dominated convection--diffusion equations. The stationary probl
 em is well understood and\nboth global and local estimates have been prove
 n for several stabilized methods such as the Streamline Upwind Petrov Gale
 rkin method (SUPG)\, the discontinuous Galerkin method (DG) and the Contin
 uous Interior Penalty method (CIP)\, showing that these methods share simi
 lar properties. \n\nIn the transient case the situation is completely diff
 erent. The standard method of lines treatment\, consisting of first discre
 tizing in space with finite elements and then in time using some standard 
 finite dfference scheme\, has resisted analysis when applied to the SUPG s
 cheme\, and it has been questioned if this method is stable. The DG and th
 e CIP method on the other hand both are members of a class of methods with
  symmetric stabilization for which a method of lines treatment is straight
 forward. Several questions\, however have remained open in this case as we
 ll\, such as the unified analysis for diffusion-dominated and convection d
 ominated flow and the analysis of explicit time stepping schemes.\n\nIn th
 is talk we will first consider the case of the SUPG method and give a new 
 analysis showing that\, in the case of hyperbolic transport equations\, th
 e SUPG-method combined with standard A-stable time discretizations is stab
 le and optimally convergent. The so called "small time-step instability" e
 nters the analysis as a special case when smoothness of the exact solution
  is insufficient.\n\nThen we will discuss symmetric stabilization methods 
 and give some recent results showing that these methods allows a seamless 
 transition from convection dominated flow to diffusion dominated flow with
  (quasi) optimality in all regimes both with respect to space and time.\n\
 nFinally we will consider the case of explicit Runge-Kutta methods and pre
 sent a new analysis of the second (RK2) and the third order (RK3) Runge-Ku
 tta methods for linear symmetric hyperbolic systems unifying the DG and th
 e CIP method. The key ingredient\nhere is new energy estimates  for RK2 an
 d RK3. The stabilization operator plays a crucial role for both stability 
 and continuity in the resulting stability and error estimates.
LOCATION:MAA110
STATUS:CONFIRMED
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