We analyze Euler-Galerkin approximations (conforming finite elements in
space and implicit Euler in time) to
coupled PDE systems in which one dependent
variable, say u, is governed by an elliptic equation and the other,
say p, by a parabolic-like equation. The underlying application is the
poroelasticity system within the quasi-static assumption. Different
polynomial orders are used for the u- and p-components to
obtain optimally convergent a priori bounds for all
the terms in the error energy norm.
Then, a residual-type
a posteriori error analysis is performed. Upon extending the
ideas of Verfürth for the heat equation [Calcolo
an optimally convergent bound is derived for the dominant term in the
error energy norm and an equivalence result between residual and
error is proven. The error bound can be classically split into
time error, space error and data oscillation terms.
Moreover, upon extending the elliptic reconstruction technique
introduced by Makridakis and Nochetto [SIAM J. Numer. Anal.
41 (2003) 1585–1594],
an optimally convergent bound is derived for the remaining terms in the
error energy norm. Numerical results are presented to
illustrate the theoretical analysis.