2019/01/14 by Jay Gopalakrishnan, Philip L. Lederer, Gopalakrishnan, Jay +3 · 3 citations
Engineering · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #Elasticity and Material Modeling
paper · pdf · doi:10.48550/arxiv.1901.04648
We introduce a new discretization of a mixed formulation of the\nincompressible Stokes equations that includes symmetric viscous stresses. The\nmethod is built upon a mass conserving mixed formulation that we recently\nstudied. The improvement in this work is a new method that directly\napproximates the viscous fluid stress \σ, enforcing its symmetry weakly.\nThe finite element space in which the stress is approximated consists of\nmatrix-valued functions having continuous "normal-tangential" components across\nelement interfaces. Stability is achieved by adding certain matrix bubbles that\nwere introduced earlier in the literature on finite elements for linear\nelasticity. Like the earlier work, the new method here approximates the fluid\nvelocity u using H(÷)-conforming finite elements, thus\nproviding exact mass conservation. Our error analysis shows optimal convergence\nrates for the pressure and the stress variables. An additional post processing\nyields an optimally convergent velocity satisfying exact mass conservation. The\nmethod is also pressure robust.\n