A posteriori analysis of Multi-scale, Multi-physics Systems and
Parallel-in-time Integrators
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Numerical simulation of a multi-scale system inevitably involves
significant discretization error. Quantification of this error is
an important facet of predictive science. In particular, science
and engineering applications are primarily interested in
quantifying the error in particular low-dimensional quantities
of interest (QoI) computed from a solution corresponding to
observable data. Some examples include computing the lift on a
wing, conservation of mass or energy, or a spatial average of the
solution. Adjoint based a posteriori techniques have been applied
successfully to estimating errors in quantities of interest for
a wide variety of applications.

This talk is concerned with the accurate computational error
estimation of numerical solutions of multi-scale, multi-physics
systems of reaction-diffusion equations. Such systems present
significantly different temporal and spatial scales within the
components of the model. Hence it is impractical to solve the
system at the finest resolution. The choice of numerical method
is often dictated by high performance computing issues,
indicating the use of independent discretizations for different
components as well as iterative, parallel-in-time and
implicit/explicit (IMEX) schemes for time integration. However,
multi-discretization has significant effects on accuracy and
stability, and traditional adjoint based analysis does not apply
to such systems. In this talk, we extend adjoint-based analysis
to derive accurate a posteriori error estimates for user-defined
quantities of interest. These estimates account for leading order
contributions to the error arising from the numerical solution
of each component as well as errors due to incomplete iteration,
linearization, projection of solution components between
different spatial meshes and parallel-in-time integration.
Several numerical examples with various settings are given to
demonstrate the performance of the error estimators.