Solving Geometric Two‐Point Boundary Value Problems

Presented here is a case study on overcoming difficulties encountered in solving an important class of nonlinear advection‐dominated two‐point boundary value problems. The problems under consideration are ones whose solution is a “curve”, i.e., unique only up to an arbitrary transformation of the independent variable. Of particular interest is the calculation of a maximum flux transition path (“finite temperature” minimum energy path). It is complicated by the presence of an exponential factor having a great range of values. The method proposed for solving the problem includes (i) a suitable finite element discretization and (ii) a robust, efficient, and relatively simple minimization method for solving the resulting system of nonlinear equations.