New a priori analysis of first-order system least-squares finite element methods for parabolic problems
This work provides a theoretical advancement for the analysis of least-squares finite element methods for parabolic problems, but it is incremental as it extends existing techniques to a specific class of problems.
The authors develop a new a priori analysis for least-squares finite element methods applied to parabolic problems, introducing an elliptic projection operator that yields optimal error estimates in both natural and L2 norms, as confirmed by numerical experiments.
We provide new insights into the a priori theory for a time-stepping scheme based on least-squares finite element methods for parabolic first-order systems. The elliptic part of the problem is of general reaction-convection-diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a non-symmetric bilinear form, although the main bilinear form corresponding to the least-squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the $L^2$ norm. Numerical experiments are presented which confirm our theoretical findings.