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Finite difference schemes on grids with local refinement in time and space for parabolic problems. I: Derivation, stability, and error analysis. (English) Zbl 0721.65047
A finite-volume method with mesh refinement is proposed for the parabolic equation t u= x (a x u)+ y (a y u)+f on a set Ω×[0,T], where Ω is a domain in the plane. It is assumed that refinement is done by adding points to a given coarse grid. Furthermore, the temporal grid is refined wherever the spatial grid is refined. Finally, the method is used implicitly, so that the matrix equation for a single coarse time step includes a number of fine time steps. The boundary condition between the coarse and fine grids is implemented so as to be conservative. An energy inequality is used to prove stability and convergence of the method.

MSC:
65M06Finite difference methods (IVP of PDE)
65M50Mesh generation and refinement (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35K15Second order parabolic equations, initial value problems
65M15Error bounds (IVP of PDE)
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