On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations

2016 ◽  
Vol 16 (3) ◽  
pp. 507-522 ◽  
Author(s):  
Yanhui Su ◽  
Lizhen Chen ◽  
Xianjuan Li ◽  
Chuanju Xu
Keyword(s):  
Lower Bound ◽  
Error Estimate ◽  
Spectral Method ◽  
Stokes Equations ◽  
Necessary Condition ◽  
Well Posedness ◽  
Saddle Point Problems ◽  
Numerical Examples ◽  
Lbb Condition ◽  
Optimal Lower Bound

AbstractThe Ladyženskaja–Babuška–Brezzi (LBB) condition is a necessary condition for the well-posedness of discrete saddle point problems stemming from discretizing the Stokes equations. In this paper, we prove the LBB condition and provide the (optimal) lower bound for this condition for the triangular spectral method proposed by L. Chen, J. Shen, and C. Xu in [3]. Then this lower bound is used to derive an error estimate for the pressure. Some numerical examples are provided to confirm the theoretical estimates.

scholarly journals A Discontinuous Finite Volume Method for the Darcy-Stokes Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zhe Yin ◽  
Ziwen Jiang ◽  
Qiang Xu
Keyword(s):  
Error Estimate ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Stokes Equations ◽  
Optimal Error Estimate ◽  
Optimal Error ◽  
Numerical Examples ◽  
First Order ◽  
Theoretical Predictions ◽  
Volume Method

This paper proposes a discontinuous finite volume method for the Darcy-Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh-dependent norm. First-orderL2-error estimates are derived for the approximations of both velocity and pressure. Some numerical examples verifying the theoretical predictions are presented.

scholarly journals A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiu Ye ◽  
Shangyou Zhang
Keyword(s):  
Error Estimate ◽  
Convergence Rates ◽  
Stokes Equations ◽  
Optimal Order ◽  
Order Error ◽  
Numerical Examples ◽  
Pressure Formulation ◽  
Primary Velocity ◽  
2D And 3D ◽  
Velocity Pressure

<p style='text-indent:20px;'>A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norm. Optimal order error estimate for pressure in the <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.</p>

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

scholarly journals Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

2020 ◽  
Author(s):  
David Maltese ◽  
Antonín Novotný
Keyword(s):  
Mach Number ◽  
Initial Data ◽  
Error Estimate ◽  
Strong Solution ◽  
Stokes Equations ◽  
Main Tool ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Mac Scheme

Abstract We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

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