  We present a space-time least squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space-time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space-time adaptivity. We also present a-priori error analysis on simplicial space-time meshes which are highly structured. Numerical results conclude this work. 
Space-time least-squares finite elements for parabolic equations