QUASI-OPTIMALITY IN THE BACKWARD EULER-GALERKIN METHOD FOR LINEAR PARABOLIC PROBLEMS

We analyse the backward Euler-Galerkin method for linear parabolic problems, looking for quasi-optimality results in the sense of Céa's Lemma. We cast the problem into the framework given by the inf-sup theory, and we analyse the spatial discretization, the discretization in time and the topic of varying the spatial discretization separately. Concerning the spatial discretization, we prove the 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, 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 partition is locally quasi-uniform. Turning to the topic of varying the spatial dicretization, 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 which is consistent with the example of non convergence in Dupont '82. 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.