Transmission Problems for Parabolic Operators on Polygonal Domains and Applications to the Finite Element Method

We study linear parabolic equations ∂tu+Lu=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t u+Lu=f$$\end{document}, where L=-div(A∇)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L = -{\text {div}}(A \nabla )$$\end{document} is a second-order strongly elliptic operator, on non-smooth two-dimensional bounded domains. The domain is polygonal and not assumed to be convex. The coefficient matrix A is piecewise smooth and exhibits jump discontinuities across a finite number of piecewise smooth curves, collectively denoted the interface. We assume mixed Dirichlet–Neumann boundary conditions and standard transmission conditions at the interface. Under some additional assumptions, we establish well-posedness of the initial-value problem using suitable weighted Sobolev spaces. The solution admits a decomposition u=ureg+ws\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u = u_\mathrm{{reg}} + w_\mathrm{{s}}$$\end{document}, into a function ureg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\mathrm{{reg}}$$\end{document} that belongs to the weighted Sobolev space and a function ws\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\mathrm{{s}}$$\end{document} that is locally constant near the vertices, thus proving well-posedness in an augmented space. We use the theoretical analysis to devise graded meshes that give quasi-optimal rates of convergence for a fully discrete scheme that utilizes finite elements on a space grid and finite differences in time.