Trade‐off between information and computability: a technique for automated topological computations

Abstract

Finite element‐based PDE solver software systems are typically method‐driven. The user has to supply the data in a particular form required by a numerical method. The method refuses to start if the data is in incorrect format, and breaks down if correctly formatted data is insufficient or inconsistent. However, software can be made more flexible with data‐driven approach. The decisions on existence and uniqueness of the solution, as well as the choice of suitable computing methods are based on the data. This calls for a new stage of data processing for a solver, which is not essentially an expert system. The questions are formalizable and their solution must be based on efficient and robust computational techniques. We present an elementary computational technique for automatic treatment of topological problems arising from potential theory, boundary condition inspection, and coupled problems. The approach is based on computing Smith normal form of the non‐oriented boundary operator matrices, whose elements are from the ring N mod 2, i.e. only 0s and 1s, instead of the integers. This approach obviates the problems of excessive computation time and risk of overflow in integer computations.