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

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 $\Omega \times [0,T]$ which is decomposed into an overlapping collection of cylindrical subregions of the form $\Omega_{l}^{\ast} \times [0,T]$, for $l=1, \dotsc, p$. Each of the space-time domains $\Omega_{l}^{\ast} \times [0,T]$ 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 $h_{l}$ and $\tau_{l}$. In particular, the different space-time grids need not match on the regions of overlap, and the time steps $\tau_{l}$ can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit $\theta$-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.