11th GAMM-Workshop on

Multigrid and Hierarchic Solution Techniques


  A. Almendral  
  M. Bader  
  R. Bank  
  M. Bebendorf  
  S. Beuchler  
  D. Braess  
  C. Douglas  
  L. Grasedyck  
  B. Khoromskij  
  R. Kornhuber  
  B. Krukier  
  U. Langer  
  C. Oosterlee  
  G. Pöplau  
  A. Reusken  
  J. Schöberl  
  M.A. Schweitzer  
  S. Serra Capizzano  
  B. Seynaeve  
  D. Smits  
  O. Steinbach  
  R. Stevenson  
  M. Wabro  
  R. Wienands  
  Ariel Almendral : On the convergence of multigrid methods for PDEs in mathematical finance

The traditional analysis of multigrid methods for second order elliptic and parabolic problems is based on the assumption that the diffusion term of the equation is uniformly elliptic. This condition is not satisfied by the Black and Scholes equation arising in mathematical finance. The purpose of this presentation is to show that the Multigrid V-cycle for the one-dimensional Black and Scholes equation converges in a time dependent energy norm. We extend the standard multigrid analysis for uniformly elliptic problems to this particular degenerate parabolic equation. In particular, we prove that the Jacobi smoother, used as a point smoother, satisfies suitable ``approximation properties''. Furthermore, the multigrid algorithm is shown to be an order optimal method for this problem.
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