An Optimization-Based Approach for the Design of PDE Solution Algorithms

We develop and analyze an optimization-based approach for the robust and efficient solution of PDE problems consisting of multiple physics operators with fundamentally different mathematical properties. Our approach relies on three essential steps: decomposition of the original problem into subproblems for which robust solution algorithms are available; integration of the subproblems into an equivalent PDE-constrained optimization problem; and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the PDE constraints. This strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components.