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Zbl 0721.65047
Ewing, R.E.; Lazarov, R.D.; Vassilevski, P.S.
Finite difference schemes on grids with local refinement in time and space for parabolic problems. I: Derivation, stability, and error analysis. (English)
[J] Computing 45, No.3, 193-215 (1990). ISSN 0010-485X; ISSN 1436-5057/e

A finite-volume method with mesh refinement is proposed for the parabolic equation $\partial\sb tu=\partial\sb x(a\partial\sb xu)+\partial\sb y(a\partial\sb yu)+f$ on a set $\Omega\times [0,T]$, where $\Omega$ 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.
[G.Hedstrom (Livermore)]

MSC 2000:: *65M06 Finite difference methods (IVP of PDE)
65M50 Mesh generation and refinement (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
35K15 Second order parabolic equations, initial value problems
65M15 Error bounds (IVP of PDE)

Keywords: finite difference schemes; error analysis; finite-volume method; mesh refinement; parabolic equation; energy inequality; stability; convergence

Cited in: Zbl 0853.65085

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