2019/12/16 by Mathias Anselmann, Anselmann, Mathias, Markus Bause +1
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1912.07426
openalex publication_date 2019/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose and study numerically the implicit approximation in time of the\nNavier-Stokes equations by a Galerkin-collocation method in time combined with\ninf-sup stable finite element methods in space. The conceptual basis of the\nGalerkin-collocation approach is the establishment of a direct connection\nbetween the Galerkin method and the classical collocation methods, with the\nperspective of achieving the accuracy of the former with reduced computational\ncosts in terms of less complex algebraic systems of the latter. Regularity of\nhigher order in time of the discrete solution is ensured further. As an\nadditional ingredient, we employ Nitsche's method to impose all boundary\nconditions in weak form with the perspective that evolving domains become\nfeasible in the future. We carefully compare the performance poroperties of the\nGalerkin-collocation approach with a standard continuous Galerkin-Petrov method\nusing piecewise linear polynomials in time, that is algebraically equivalent to\nthe popular Crank-Nicholson scheme. The condition number of the arising linear\nsystems after Newton linearization as well as the reliable approximation of the\ndrag and lift coefficient for laminar flow around a cylinder (DFG flow\nbenchmark with Re=100) are investigated. The superiority of the\nGalerkin-collocation approach over the linear in time, continuous\nGalerkin-Petrov method is demonstrated therein.\n