Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

Authors:
T. P. Mathew and G. Russo

Journal:
Math. Comp. **72** (2003), 619-656

MSC (2000):
Primary 65N20, 65F10

DOI:
https://doi.org/10.1090/S0025-5718-02-01462-X

Published electronically:
November 4, 2002

MathSciNet review:
1954959

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Abstract: In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain which is decomposed into an overlapping collection of cylindrical subregions of the form , for . Each of the space-time domains are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters and . In particular, the different space-time grids need not match on the regions of overlap, and the time steps can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit -scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument.

Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and *heterogeneous* hyperbolic-parabolic approximations of parabolic equations.

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Additional Information

**T. P. Mathew**

Affiliation:
115 Seal Rock Drive, San Francisco, California 94121

Email:
tmathew@mindspring.com

**G. Russo**

Affiliation:
Dipartimento di Matematica ed Informatica, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy

Email:
russo@dmi.unict.it

DOI:
https://doi.org/10.1090/S0025-5718-02-01462-X

Keywords:
Nonmatching overset space-time grids,
maximum norm stability,
composite grids,
parallel {S}chwarz alternating method,
parabolic equations,
discrete maximum principle,
discrete barrier functions

Received by editor(s):
July 25, 2000

Published electronically:
November 4, 2002

Article copyright:
© Copyright 2002
American Mathematical Society