TY - JOUR

T1 - Transmission Problems for Parabolic Operators on Polygonal Domains and Applications to the Finite Element Method

AU - Zhang, Yajie

AU - Mazzucato, Anna L.

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2021.

PY - 2022/3

Y1 - 2022/3

N2 - 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=ureg+ws, into a function ureg that belongs to the weighted Sobolev space and a function ws 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.

AB - 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=ureg+ws, into a function ureg that belongs to the weighted Sobolev space and a function ws 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.

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U2 - 10.1007/s44007-021-00013-8

DO - 10.1007/s44007-021-00013-8

M3 - Article

AN - SCOPUS:85153748995

SN - 2730-9657

VL - 1

SP - 225

EP - 262

JO - Matematica

JF - Matematica

IS - 1

ER -