  We consider adaptive finite element methods for solving a multiscale system
consisting of a macroscale model comprising a system of reaction-diffusion
partial differential equations coupled to a microscale model comprising a
system of nonlinear ordinary differential equations. A motivating example is
modeling the electrical activity of the heart taking into account the chemistry
inside cells in the heart. Such multiscale models pose extremely
computationally challenging problems due to the multiple scales in time and
space that are involved.
  We describe a mathematically consistent approach to couple the microscale and
macroscale models based on introducing an intermediate "coupling scale". Since
the ordinary differential equations are defined on a much finer spatial scale
than the finite element discretization for the partial differential equation,
we introduce a Monte Carlo approach to sampling the fine scale ordinary
differential equations. We derive goal-oriented a posteriori error estimates
for quantities of interest computed from the solution of the multiscale model
using adjoint problems and computable residuals. We distinguish the errors in
time and space for the partial differential equation and the ordinary
differential equations separately and include errors due to the transfer of the
solutions between the equations. The estimate also includes terms reflecting
the sampling of the microscale model. Based on the accurate error estimates, we
devise an adaptive solution method using a "blockwise" approach. The method and
estimates are illustrated using a realistic problem.
