A least squares space-time approach for parabolic equations

Abstract We propose a least squares formulation for abstract parabolic equations in the natural $$L^2(0,T;V^\star )\times H$$ L 2 ( 0 , T ; V ⋆ ) × H norm which only relies on natural regularity assumptions on the data of the problem. The resulting bilinear form then is symmetric, coercive, and continuous and contains the $$V^\star $$ V ⋆ -norm. We propose a solution approach that uses a conforming Galerkin discretization of an equivalent saddle point problem to circumvent the evaluation of $$V^\star $$ V ⋆ -norms. We prove strong convergence of discrete solutions towards continuous solutions and provide a preconditioner for efficient numerical solution of the discrete saddle point problem. We illustrate our analytical findings by selected numerical experiments.