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We provide new insights into the a priori theory for a time-stepping schemebased on least-squares finite element methods for parabolic first-ordersystems. The elliptic part of the problem is of generalreaction-convection-diffusion type. The new ingredient in the analysis is anelliptic projection operator defined via a non-symmetric bilinear form,although the main bilinear form corresponding to the least-squares functionalis symmetric. This new operator allows to prove optimal error estimates in thenatural norm associated to the problem and, under additional regularityassumptions, in the $L^2$ norm. Numerical experiments are presented whichconfirm our theoretical findings.

New a priori analysis of first-order system least-squares finite element methods for parabolic problems