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Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework
We analyze temporal approximation schemes based on overlapping domain
decompositions. As such schemes enable computations on parallel and distributed
hardware, they are commonly used when integrating large-scale parabolic
systems. Our analysis is conducted by first casting the domain decomposition
procedure into a variational framework based on weighted Sobolev spaces. The
time integration of a parabolic system can then be interpreted as an operator
splitting scheme applied to an abstract evolution equation governed by a
maximal dissipative vector field. By utilizing this abstract setting, we derive
an optimal temporal error analysis for the two most common choices of domain
decomposition based integrators. Namely, alternating direction implicit schemes
and additive splitting schemes of first and second order. For the standard
first-order additive splitting scheme we also extend the error analysis to
semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which
also contains numerical experiments. Version 3 and 4: Only comments added.
Version 2, page 2: Clarified statement on stability issues for ADI schemes
with more than two operator
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