A POSTERIORI ERROR ANALYSIS OF TWO-STAGE COMPUTATION METHODS WITH APPLICATION TO EFFICIENT DISCRETIZATION AND THE PARAREAL ALGORITHM

摘要

We consider numerical methods for initial value problems that employ a two-stage approach consisting of solution on a relatively coarse discretization followed by solution on a relatively fine discretization. Examples include adaptive error control, parallel-in-time solution schemes, and efficient solution of adjoint problems for computing a posteriori error estimates. We describe a general formulation of two-stage computations and then perform a general a posteriori error analysis based on computable residuals and solution of an adjoint problem. The analysis accommodates various variations in the two-stage computation and in the formulation of the adjoint problems. We apply the analysis to computing "dual-weighted" a posteriori error estimates, developing novel algorithms for efficient solution that take into account cancellation of error, and to the Parareal algorithm. We test the various results using several numerical examples.