Analysis of fully discrete finite element methods for 2D Navier-Stokes equations with critical initial data. (English) Zbl 1503.65238
Summary: First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier-Stokes equations with \(L^2\) initial data in convex polygonal domains, without extra regularity assumptions or grid-ratio conditions. The proof utilises the smoothing properties of the Navier-Stokes equations in the analysis of the consistency errors, an appropriate duality argument, and the smallness of the numerical solution in the discrete \(L^2(0, t_m; H^1)\) norm when \(t_m\) is smaller than some constant. Numerical examples are provided to support the theoretical analysis.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 35Q30 | Navier-Stokes equations |
Keywords:
Navier-Stokes equations; \(L^2\) initial data; semi-implicit Euler scheme; finite element method; error estimateReferences:
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