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Space-time hp-approximation of parabolic equations

by D. Devaud and Ch. Schwab

(Report number 2017-50)

Abstract
A new space-time finite elements methods (FEM) approximation for the solution of parabolic partial differential equations (PDE) is introduced. Considering a mesh-dependent norm, it is first shown that the discrete bilinear form is coercive and continuous, yielding existence and uniqueness of the associated solution. In a second step, error estimates in this norm are derived. In particular, we show that combining low-order elements for the space variable together with an $hp$-approximation of the problem with respect to the temporal variable allows us to decrease the optimal convergence rates for the approximation of elliptic problems only by a logarithmic factor. For simultaneous space-time $hp$-discretization in both, the spatial as well as the temporal variable, overall exponential convergence in mesh-degree dependent norms on the space-time cylinder is proved, under analytic regularity assumptions on the solution with respect to the spatial variable. Numerical results for linear model problems confirming exponential convergence are presented.

Keywords:

@Techreport{DS17_746,
  author = {D. Devaud and Ch. Schwab},
  title = {Space-time hp-approximation of parabolic equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-50},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-50.pdf },
  year = {2017}
}

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