The ANSYS code offers stress analysts a variety of contact element options: point-to-surface or surface-tosurface and low-order or high-order elements, in concert with any one of five contact algorithms (augmented Lagrangian, penalty method, etc.). This raises questions as to what option performs best under what circumstances. Here we offer some answers to these questions by examining performance in some numerical experiments. The numerical experiments focus on frictionless contact with a rigid indenter; even so, the number of experiments involved is quite large. The experiments use a battery of test problems with known analytical solutions: contact patch tests with nodes matching and nonmatching, Hertzian contact of a cylinder and a sphere, and Steuermann contact of a strip punch with three different edge radii. For each of these test problems, three successively systematically-refined meshes are used to examine convergence. All told, over five hundred finite element analyses are run. Results for this class of problems are the same for the augmented Lagrangian (AL, the default) and the penalty method (PM) algorithms. There is also no difference between results found with the two Lagrange multiplier (LM) algorithms. Thus together with results for the fifth contact algorithm (IMC-internal multipoint constraint), we have but three distinct sets of results. The results for the AL and PM algorithms are good for all problems provided they are used with surfaceto- surface elements. The results for the LM algorithms can be quite sensitive to matching of the nodes on the indented material with those on the indenter. When nodes matched, these algorithms also gave good results. The IMC algorithm, while the fastest, did not give good results for the problems examined. When algorithms worked well, there was little difference in accuracy between low-order and high-order elements.