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

We study linear parabolic equations ∂_tu+Lu=f, where L=-div(A∇) 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=u_reg+w_s, into a function u_reg that belongs to the weighted Sobolev space and a function w_s 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.