Summary: A residual correction method based upon an extension of the finite calculus concept is presented. The method is described and applied to the solution of a scalar convection-diffusion problem and the problem of elasticity in the incompressible or quasi-incompressible limit. The formulation permits the use of equal interpolation for displacements and pressure on linear triangles and tetrahedra as well as any low order element type. To add additional stability in the solution, pressure gradient corrections are introduced as suggested from developments of sub-scale methods. Numerical examples are included to demonstrate the performance of the method when applied to typical test problems.