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A stable, direct solver for the gradient equation

A new finite element discretization of the equation grad p=g is introduced. This discretization gives rise to an invertible system that can be solved directly, requiring a number of operations proportional to the number of unknowns. We prove an optimal error estimate, and furthermore show that the method is stable with respect to perturbations of the right-hand side g. We discuss a number of applications related to the Stokes equations.