The Stokes Equations
The key juncture in this process is the identification of the energy balances for the first-order systems. The Agmon–Douglis–Nirenberg (ADN) and the vector-operator settings, already encountered in Chapters 5 and 6, are used here as well. For CLSPs defined in the ADN setting, transition to DLSPs relies on the abstract least-squares theory for ADN systems formulated in Chapter 4. Accordingly, error analysis of the resulting LSFEMs is simply a matter of specializing that theory to the first-order Stokes systems. LSFEMs in the ADN setting for the Stokes system are considered in Sections 7.2–7.6. Formulation and analysis of LSFEMs in the vector-operator setting is somewhat more involved because this setting results in non-conforming methods similar to those encountered in Section 6.3.3. However, the vector-operator setting is well worth the effort because it allows us to formulate, for some sets of non-standard boundary conditions (but, unfortunately, not for the velocity boundary condition), locally conservative LSFEMs for the Stokes equations, a feat that is very difficult, if not impossible, to accomplish in the ADN setting. LSFEMs in the vector-operator setting for the Stokes system are considered in Section 7.7.
KeywordsStokes Equation Normal Velocity Element Space Stokes System Minimization Space
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