Summary: Significant increases in apparent flow strength are observed when non-uniform plastic deformation of metals occurs at the scale ranging from roughly one to ten microns. Several basic plane strain problems are analysed numerically in this paper based on a new formulation of strain gradient plasticity. The problems are the tangential and normal loading of a finite rectangular block of material bonded to rigid platens and having traction-free ends, and the normal loading of a half-space by a flat rigid punch. The solutions illustrate fundamental features of plasticity at the micron scale that are not captured by conventional plasticity theory. These include the role of material length parameters in establishing the size dependence of strength, and the elevation of resistance to plastic flow resulting from constraint on plastic flow at boundaries. Details of the finite element method employed in the numerical analysis of the higher-order gradient theory will be discussed and related to prior formulations having some of the same features.