## An *A Posteriori-A Priori* Analysis of Multiscale Operator Splitting

**Abstract.**
In this paper, we analyze a multiscale operator splitting method for solving systems
of ordinary differential equations such as those that result upon space discretization of a reaction-
diffusion equation. Our goal is to analyze and accurately estimate the error of the numerical solution,
including the effects of any instabilities that can result from multiscale operator splitting. We present
both an *a priori* error analysis and a new type of hybrid *a priori - a posteriori*
error analysis for an operator splitting discontinuous Galerkin finite element method. Both analyses
clearly distinguish between the effects of the operator splitting and the discretization of each component
of the decomposed problem. The hybrid analysis has the form of a computable *a posteriori*
leading order expression and a provably-higher order *a priori* expression.
The hybrid analysis takes into account the
fact that the adjoint problems for the original problem and a multiscale operator splitting
discretization differ in significant ways. In particular, this provides the means to monitor global instabilities
that can arise from operator splitting.

**Keywords.**
*a posteriori* error analysis, adjoint problem, discontinuous
Galerkin method, generalized Green's function, goal oriented error
estimates, multiscale method, operator decomposition, operator splitting,
reaction-diffusion equations, residual