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Quasi-optimality in the backward Euler-Galerkin method for linear parabolic problems

  • April 28, 2014
  • 1 p.m.
  • LeConte 312


Galerkin finite element methods are widely used for the numerical solution of parabolic problems. Although a quasi-optimality result like Céa's Lemma is an important step in the derivation of a priori error bounds for linear elliptic problems, it has been rarely invoked in the context of parabolic problems. Our analysis of the backward Euler-Galerkin method for linear parabolic problems aim at quasi-optimality results, and it is based on the framework given by the inf-sup condition.

Concerning the spatial discretization, we prove that the H1-stability of the L2-projection is also a necessary condition for quasi-optimality, both in the H1(H-1)∩L2(H1)-norm and in the L2(H1)-norm. Concerning the discretization in time with backward Euler, we prove that the error in a norm that mimics the H1(H-1)∩L2(H1)-norm is equivalent to the sum of the best errors with piecewise constants for the exact solution and its time derivative, if the time partition is locally quasi-uniform.

Concerning the case when the spatial discretization is allowed to change with time, a remarkable example of non convergence can be found in Dupont '82. The spatial mesh changes every time-step in such a way that, even if the spatial mesh-size h and temporal time step \tau converge to zero, the discrete solution does not converge to the exact one, if h^4/\tau goes to infinity. We provide a bound for the error that includes the best error and an additional term, which vanishes if there are not modifications of the spatial dicretization and it is consistent with Dupont's example. We combine these elements in an analysis of the backward Euler-Galerkin method and derive error estimates in case the spatial discretization is based on finite elements.

It is a joint work with Andreas Veeser, from University of Milan.

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