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First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity. (English) Zbl 0856.76010
Summary: We study the least-squares method for the generalized Stokes equations (including linear elasticity) based on the velocity-vorticity-pressure formulation in \(d=2\) or 3 dimensions. The least-squares functional is defined in terms of the sum of the \(L^2\)- and \(H^{-1}\)-norms of the residual equations, but weighted appropriately by the Reynolds number (Poisson ratio). Our approach for establishing ellipticity of the functional does not use Agmon-Douglis-Nirenberg theory, but is founded more on basic principles. We also analyze the case where the \(H^{-1}\)-norm in the functional is replaced by a discrete \(H^{-1}\)-norm to make the computation feasible. We show that the resulting algebraic equations can be preconditioned by well-known techniques uniformly well in the Reynolds number (Poisson ratio).

76D07 Stokes and related (Oseen, etc.) flows
74B05 Classical linear elasticity
35Q30 Navier-Stokes equations
35Q72 Other PDE from mechanics (MSC2000)
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