A posteriori estimates for a~generalized Stokes problem

S. Repin ; R. Stenberg

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 39, Tome 362 (2008), pp. 272-302

Résumé

The paper is concerned with a three-field formulation of a generalized Stokes problem related to viscous flow problems for fluids with polymeric chains. For the case of homogeneous Dirichlét boundary conditions, this model and the respective numerical methods were studied in [15]. In the present paper, we consider a generalized Stokes problem with variable viscosity and nonhomogeneous Dirichlét or mixed Dirichlét/Neumann boundary conditions and derive functional a posteriori error estimates for the velocity, pressure, and stress fields. The estimates are practically computable, sharp (i.e., have no gap between the left- and right-hand sides), and are valid for arbitrary functions from the respective functional classes. The estimates are derived by transformations of the integral identity that assigns the generalized solution (this method was suggested and used in [41,42] for certain classes of elliptic type problems). Error majorants are weighted sums of the terms penalizing violations of the constitutive, equilibrium, and divergence relations with weights defined by the constants in the Friederichs inequality and inf-sup (LBB) condition. Bibl. – 53 titles.