The accurate determination of stresses at two-dimensional (2D) stress risers is both an important and a challenging problem in engineering. Finite element analysis (FEA) has become the method of choice in making such determinations when new configurations with unknown stress concentrations are encountered in practice. For such FEA to be useful, discretization errors in peak stresses have to be sufficiently controlled. Convergence checks and companion error estimates offer a means of exerting such control. Here, we report some new convergence checks to this end. These checks are designed to promote conservative error estimation. They are applied to seven benchmark problems that have exact solutions for their peak stresses. Associated stress concentration factors span a range that is larger than that normally experienced in engineering. Error levels in the FEA of these peak stresses are classified in accordance with: 1–5%, satisfactory; 1/5–1%, good; and <1/5%, excellent. The present convergence checks result in 91 error assessments for the benchmark problems. For these 91, errors are assessed as being at the same level as true exact errors on 83 occasions and one level worse for the other 8. Thus, stress error estimation is largely accurate (91%) and otherwise modestly conservative (9%).