The design ideology of this research is such that a particular structure of solution or a set of structures can be preserved simultaneously by transforming the problem into a convex- minimization problem. The solution is then obtained using an iterative approach based on a novel geometric intuition that involves conversion from the original solution space, which does not preserve the structure, to a corresponding feasible space where the structure is preserved. The entire process can be summarized as applying a filter to a structurally non-conforming intermediate solution and obtaining a solution that preserves the desired structure. Experi- mental data from the applications to discontinuous and continuous Galerkin finite element methods show the efficacy of this method to preserve the underlying structure during the simulation.

The major contributions of this research are the robust formulation of the structure-preservation problem, establishing convexity and solvability of the general formulation, developing elegant optimization algorithms to solve it, and applying the filtering approach to real-world problems. This is achieved while retaining the accuracy and the convergence properties of the numerical method in a non-intrusive way. Since this work is agnostic of the type of problems, polynomial order (space of the solution), types of structural constraints, or the numerical method used to solve the problem, the algorithm developed can be applied to ubiquitous polynomial-based solutions of various systems of PDEs.

Host: Svetlana Tokareva