Two-scale composite finite element method for parabolic problems with smooth and nonsmooth initial data

We consider a new class of finite elements, called composite finite elements, for the discretization of parabolic problems in a two-dimensional convex polygonal domain. More precisely, an effort has been made in this paper to extend two-scale composite finite element method for elliptic problems to parabolic problems. We analyze both semidiscrete and fully discrete composite finite element methods and derive convergence properties in the L∞(L2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (L^2)$$\end{document} and L∞(H1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (H^1)$$\end{document} norms for smooth data. Moreover, nonsmooth data error estimates in the L∞(L2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (L^2)$$\end{document} norm for positive time is established. Compared to the classical finite element method, the composite finite element method can be viewed as a coarse scale generalization which yields a minimal dimension of the approximation space and the asymptotic order of convergence is preserved on coarse scale meshes which do not resolve the boundary. Numerical results are presented to illustrate the theoretical rates of convergence.