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Abstract: In this article we investigate the LBB condition for axisymmetric flow problems. Specifically, the sufficiency condition for approximating pairs to satisfy the LBB condition established by Stenberg in the Cartesian coordinate setting is presented for the cylindrical coordinate setting. For the cylindrical coordinate setting, the Taylor–Hood () and conforming Crouzeix–Raviart elements are shown to be LBB stable. A priori error bounds for approximations to the axisymmetric Stokes flow problem using Taylor–Hood and Crouzeix–Raviart elements are given. The computed numerical convergence rates for the error for an axisymmetric Stokes flow problem support the theoretical results. [Copyright &y& Elsevier]

Abstract: We analyze a stabilization technique for degenerate transport equations. Of particular interest are coupled parabolic/hyperbolic problems, when the diffusion coefficient is zero in part of the domain. The unstabilized, computed approximations of these problems are highly oscillatory, and several techniques have been proposed and analyzed to mitigate the effects of the sub-grid errors that contribute to the oscillatory behavior. In this paper, we modify a time-relaxation algorithm proposed in and further studied in . Our modification introduces the relaxation operator as a post-processing step. The operator is not time-dependent, so the discrete (relaxation) system need only be factored once. We provide convergence analysis for our algorithm along with numerical results for several model problems. [Copyright &y& Elsevier]

Abstract: A fractional step -method for the approximation of time-dependent viscoelastic fluid flow equations is described and analyzed in this article. The algorithm uses substeps within a time step to sequentially update velocity, pressure, and stress. This lagged approach to temporal integration requires a resolution of smaller systems than a fully implicit approach while achieving a second order temporal accuracy. We establish a priori error estimates for our scheme, and provide numerical computations to support the theoretical results and demonstrate the capability of this method. [Copyright &y& Elsevier]