Linear Complexity Solution of Parabolic Integro-differential Equations

The numerical solution of parabolic problems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t + \mathcal{A} u = 0$$\end{document} with a pseudo-differential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A}$$\end{document} by wavelet discretization in space and hp discontinuous Galerkin time stepping is analyzed. It is proved that an approximation for u(T) can be obtained in N points with accuracy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{O}(N^{-p-1})$$\end{document} for any integer p ≥ 1 in work and memory which grows logarithmically-linear in N.