Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations. (English) Zbl 1067.76562

Summary: We formulate and study finite element methods for the solution of the incompressible Stokes equations based on the application of a least-squares minimization principle to an equivalent first order velocity-pressure-stress system. Our least-squares functional involves the \(L^ 2\)-norms of the residuals of each equation multiplied by a mesh-dependent weight. Each weight is determined according to the Agmon-Douglis-Nirenberg index of the corresponding equation. As a result, the approximating spaces are not subject to the LBB condition and conforming discretizations are possible with merely continuous finite element spaces. Moreover, the resulting discrete problems involve only symmetric, positive definite systems of linear equations, i.e. assembly of the discretization matrix is not required even at the element level. We prove that the least-squares approximations converge to the solutions of the Stokes problem at the best possible rate and then present some numerical examples illustrating our theoretical results. Among other things, these numerical examples indicate that the method is not optimal without the weights in the least-squares functional.


76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
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