  In this paper, we propose a numerical method for turning point problems in
one dimension based on Petrov-Galerkin finite element method (PGFEM). We first
give a priori estimate for the turning point problem with a single boundary
turning point. Then we use PGFEM to solve it, where test functions are the
solutions to piecewise approximate dual problems. We prove that our method has
a first-order convergence rate in both $L^\infty$ norm and an energy norm when
we select the exact solutions to dual problems as test functions. Numerical
results show that our scheme is efficient for turning point problems with
different types of singularities, and the convergency coincides with our
theoretical results.
