The basic algorithmic structure and performance characteristics of the
p
‐version of the finite element method are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the
p
‐version is fully established, and a very substantial amount of engineering experience covering linear and nonlinear applications is available. It is shown that
p
‐extensions on properly designed meshes make realization of exponential rates of convergence in practical computations possible and provide for the estimation and control of relative errors in terms of any data of interest. Given the growing demand for verified numerical solutions, the
p
‐version is expected to play an increasingly important role in the development of future finite element software products.