A posteriori error analysis for domain decomposition
Domain decomposition methods are iterative techniques that are widely used for the numerical solution of partial differential equations. I will present an adjoint-based a posteriori error analysis for one specific domain decomposition method, multiplicative overlapping Schwarz. Numerical computations are commonly performed to determine a particular functional of the solution or ``quantity of interest’’. The error in a quantity of interest obtained using a domain decomposition method can be decomposed into contributions that arise from the spatial discretizations within individual subdomains, and the contribution that arises due to finite iteration. The ability to accurately estimate these two distinct sources of error supports a two-stage solution strategy to reduce the error in the QoI. Depending upon the dominant sources of error, the finite element mesh in specific subdomains may be refined or the number of iterations increased. Since intuition can be difficult to develop in the context of partial differential equations, I first illustrate the key concepts in the familiar framework of linear algebraic systems.