Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

2019 ◽  
Vol 36 (1) ◽  
pp. 86-117
Author(s):  
Jules K. Djoko ◽  
Virginie S. Konlack ◽  
Mohamed Mbehou
Keyword(s):  
Error Analysis ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Stokes Equations ◽  
A Priori ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
A Priori Error Analysis ◽  
Nonlinear Slip Boundary Conditions

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kangrui Zhou ◽  
Yueqiang Shang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Parallel Algorithms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Parallel Iterative ◽  
Nonlinear Slip Boundary Conditions

AbstractBased on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS

2013 ◽  
Vol 23 (08) ◽  
pp. 1421-1478 ◽  
Author(s):  
JOHN A. EVANS ◽  
THOMAS J. R. HUGHES
Keyword(s):  
Boundary Conditions ◽  
Numerical Experiments ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
A Priori ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Discrete Velocity ◽  
Slip Boundary

We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.

Local and Parallel Finite Element Algorithms for the Stokes Equations with Nonlinear Slip Boundary Conditions

2019 ◽  
Vol 17 (08) ◽  
pp. 1950050
Author(s):  
Kangrui Zhou ◽  
Yueqiang Shang
Keyword(s):  
Finite Element ◽  
Variational Inequality ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Slip Boundary Conditions ◽  
Fine Grid ◽  
Low Frequencies ◽  
Slip Boundary ◽  
Parallel Finite Element ◽  
Nonlinear Slip Boundary Conditions

Based on two-grid discretizations, local and parallel finite element algorithms are studied for the Stokes equations with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Thereafter, the variational inequality can be transform into the variational identity as a multiplier in a convex set. The main idea of our algorithms is to approximate the low frequencies of the finite element solution using a coarse grid and use a fine grid to correct the resulted residual (that includes mostly high frequencies of the solution) by some local and parallel procedures. Error bounds for the approximate solutions are estimated. Numerical results are also given to demonstrate the effectiveness of the algorithms.

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