Operator decomposition methods are commonly used to solve multiscale and multiphysics models since they decompose complex systems into individual components with relatively simple physics and with behaviors that occur on a relatively narrow range of scales. In some scientific communities, operator decomposition is referred to as a "segregated" as opposed to a "monolithic" approach. Obtaining an approximation of the full problem typically requires iteration of the solutions of the individual components, introducing transfer and projection errors as well as errors due to iteration.
These new errors compound the discretization errors of individual components in a complicated fashion that is typically poorly understood. The ability to identify and estimate each of these new error contributions is essential when designing adaptive strategies. We have developed adjoint-based a posteriori analyses for a number of common operator decomposition techniques. These analyses address the global effects of operator decomposition as well as the accuracy with which individual components are solved. We present illustrative examples of the application of these analyses to multi-rate differential equations, coupled elliptic and parabolic problems and discuss the challenges (and some ideas) for large scale implementation.
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