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Wavelet-discretizations of parabolic integro-differential equations

by T. von Petersdorff and Ch. Schwab

(Report number 2001-07)

Abstract
We consider parabolic problems \dot{u} + Au = f in (0,T)xOmega, T < infty, where Omega \subset Rd is a bounded domain and A is a strongly elliptic, classical pseudo-differential operator of order rho in [0,2] in \tilde{H}{rho/2}(Omega). We use a theta-scheme for time discretization and a Galerkin method with N degrees of freedom for space discretization. The full Galerkin matrix for A can be replaced with a sparse matrix using a wavelet basis, and the linear systems for each time step are solved approximatively with GMRES. We prove that the total cost of the algorithm for M time steps is bounded by O(MN (log N)beta) operations and O(N (log N)beta) memory. We show that the algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution with respect to L2 in time and the energy norm in space.

Keywords:

@Techreport{vS01_284,
  author = {T. von Petersdorff and Ch. Schwab},
  title = {Wavelet-discretizations of parabolic integro-differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2001-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2001/2001-07.pdf },
  year = {2001}
}

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