Stokes Equations

Abstract

Besides differential equations of second or higher order there are systems of q differential equations for q scalar functions. In Section 12.1 we present the systems of the Stokes and Lamé equations as particular examples and define the ellipticity of such systems. Section 12.2 starts with the variational formulation of Stokes’ equations. The saddle-point structure is discussed in §12.2.2. Solvability of general saddle-point problems is analysed in §12.2.3. The corresponding conditions are verified for the Stokes equations. A reinterpretation in §12.2.5 leads to a V ₀-elliptic problem in a special subspace V ₀. In Section 12.3 the finite-element discretisation is studied. Special inf-sup conditions are to be satisfied since otherwise the problem is not solvable or unstable. Examples of stable finite elements are presented in §12.3.3.