Local and parallel finite element algorithms for the time-dependent Oseen equations
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Local and parallel finite element algorithms for the time-dependent Oseen equations Qi Ding1 · Bo Zheng1 · Yueqiang Shang1 Received: 28 August 2018 / Accepted: 29 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Based on two-grid discretizations, local and parallel finite element algorithms are proposed and analyzed for the time-dependent Oseen equations. Using conforming finite element pairs for the spatial discretization and backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Oseen equations using a coarse grid on the entire domain, and then correct the resulted residual using a fine grid on overlapped subdomains by some local and parallel procedures at each time step. By the theoretical tool of local a priori estimate for the fully discrete finite element solution, error bounds of the approximate solutions from the algorithms are estimated. Numerical results are also given to demonstrate the efficiency of the algorithms. Keywords Oseen equations · Finite element · Local and parallel algorithms · Two-grid method Mathematics Subject Classification (2010) 35Q30 · 65M15 · 65M55 · 65M60 · 76D05
1 Introduction Let Ω be a bounded domain with Lipschitz continuous boundary ∂Ω in R2 and (0, T ) be a time interval with T < ∞. We consider the following time-dependent Oseen equations: ut − νu + (b · ∇)u + ∇p = f, in Ω × (0, T ], ∇ · u = 0, in Ω × (0, T ], u = 0, on ∂Ω × (0, T ], u = u0 , on Ω × {0},
Yueqiang Shang
[email protected]
Extended author information available on the last page of the article.
(1)
Numerical Algorithms
where u : (0, T ) × Ω → R2 represents the velocity vector, p : (0, T ) × Ω → R the pressure, f : (0, T ) × Ω → R2 the external body force, ν > 0 the kinematic viscosity, u0 : Ω → R2 the initial velocity satisfying ∇ · u0 = 0, ut = ∂u ∂t , and b : (0, T ) × Ω → R2 a solenoidal vector field satisfying ∇ · b = 0. Our ultimate goal is to study local and parallel finite element discretization algorithms for the time-dependent incompressible Navier-Stokes equations in which b is substituted by the velocity u. By linearizing the incompressible Navier-Stokes equations ( e.g., by a semi-implicit iteration), we find it is reasonable to study the Oseen problem first. Indeed, for the time-dependent Navier-Stokes equations, u is the velocity at the current time, and b may be a finite element approximation of u from a previous time step or linearization step. In the last decades, based on the ideas of Xu and Zhou [31–33] for local finite element discretizations, some local and parallel algorithms have been proposed for the elliptic eigenvalue problems [25, 33], the steady Stokes equations [11], the steady Navier-Stokes equations [10, 19, 35], and the stream function form of Navier-Stokes equations [20]. These algorithms have low communication complexity, and allow existing sequential PDE codes to run in a parallel environment without a large investment in recodin
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