A Posteriori Error Estimates for the Stokes Equations: a Comparison

When numerically solving a set of partial differential equations through a finite element strategy associated with a weak formulation, one usually faces the problem of increasing the accuracy of the solution without adding unnecessary degrees of freedom in non-critical parts of the computational domain. In order to identify these regions, indicators were created which allow their automatic determination by computing some function of the characteristic features of the solution, such as indicators based on the gradient of the Mach number in Computational Fluid Dynamics (CFD) [1, 2], indicators derived from a priori error estimates, or indicators involving residuals of the discretized equations [3, 4]. More recently the trend has been to derive a posteriori error estimates based on more mathematical criteria, by solving small local problems resembling the original global one, but involving higher order finite elements [5-8]. In this paper we compare a few of these estimates obtained for the Stokes problem. The finite element scheme used is the classical mini-element formulation, which is recalled in Section 2. Three estimates based on the resolution of a local Stokes problem along with one based only on residuals are presented in Section 3. In Section 4 a few comparison inequalities are stated, and Section 5 examines their numerical behavior on test problems for which an exact solution is known, and on typical examples of CFD as well.