A least squares space-time approach for parabolic equations

Abstract

We propose a least squares formulation for abstract parabolic equations in the natural L2(0,T;V⋆)×H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(0,T;V^\star )\times H$$\end{document} 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⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V^\star $$\end{document}-norm. We propose a solution approach that uses a conforming Galerkin discretization of an equivalent saddle point problem to circumvent the evaluation of V⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V^\star $$\end{document}-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.