2019/12/30 by Herbert Egger, Egger, Herbert, Mania Sabouri +1
Engineering · #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1912.13086
openalex publication_date 2019/12/30 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We consider the systematic numerical approximation of Biot's quasistatic\nmodel for the consolidation of a poroelastic medium. Various discretization\nschemes have been analysed for this problem and inf-sup stable finite elements\nhave been found suitable to avoid spurios pressure oscillations in the initial\nphase of the evolution. In this paper, we first clarify the role of the inf-sup\ncondition for the well-posedness of the continuous problem and discuss the\nchoice of appropriate initial conditions. We then develop an abstract error\nanalysis that allows us to analyse some approximation schemes discussed in the\nliterature in a unified manner. In addition, we propose and analyse the\nhigh-order time discretization by a scheme that can be interpreted as a variant\nof continuous-Galerkin or particular Runge-Kutta methods applied to a modified\nsystem. The scheme is designed to preserve both, the underlying\ndifferential-algebraic structure and energy-dissipation property of the\nproblem. In summary, we obtain high-order Galerkin approximations with respect\nto space and time and derive order-optimal convergence rates. The numerical\nanalysis is carried out in detail for the discretization of the two-field\nformulation by Taylor-Hood elements and a variant of a Runge-Kutta time\ndiscretization. Our arguments can however be extended to three- and four field\nformulations and other time discretization strategies.\n