scholarly journals A UNIFORMLY STABLE SOLVABILITY OF NLBVP FOR PARAMETERIZED ODE

2021 ◽  
pp. 50-69
Author(s):  
Dovlet DOVLETOV
Keyword(s):  
Uniformly Stable

A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations

2017 ◽  
Vol 10 (1) ◽  
pp. 22-43 ◽  
Author(s):  
Peiqi Huang ◽  
Zhilin Li
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Discontinuous Coefficients ◽  
Brinkman Equation ◽  
Optimal Error Estimates ◽  
Discrete Problem ◽  
Brinkman Equations ◽  
Approximation Errors ◽  
Interior Penalties ◽  
Uniformly Stable

AbstractA nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

A uniformly stable Fortin operator for the Taylor–Hood element

2012 ◽  
Vol 123 (3) ◽  
pp. 537-551 ◽  
Author(s):  
Kent-Andre Mardal ◽  
Joachim Schöberl ◽  
Ragnar Winther
Keyword(s):  
Uniformly Stable

scholarly journals Stability for a Class of Differential Equations with Nonconstant Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jin Liang ◽  
Tzon-Tzer Lu ◽  
Yashan Xu
Keyword(s):  
Differential Equations ◽  
Positive Constant ◽  
Zero Solution ◽  
Continuous Functions ◽  
Delay Equation ◽  
Asymptotically Stable ◽  
Uniformly Stable

Stability is investigated for the following differential equations with nonconstant delayx't=qtFxt-ptfxt-τt,wherep:[0,+∞)→[0,+∞),q:[0,+∞)→R,τ:[0,+∞)→[0,r], andFandf:R→Rwithxfx>0   for   x≠0   and   x≤a(ais a positive constant) are continuous functions. A criterion is given for the zero solution of this delay equation being uniformly stable and asymptotically stable.

scholarly journals HP DISCONTINUOUS GALERKIN APPROXIMATIONS FOR THE STOKES PROBLEM

2002 ◽  
Vol 12 (11) ◽  
pp. 1565-1597 ◽  
Author(s):  
ANDREA TOSELLI
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Order Approximation ◽  
Stokes Problem ◽  
Galerkin Approximation ◽  
Mesh Size ◽  
Numerical Results ◽  
Optimal Error Estimates ◽  
Polynomial Degree ◽  
Uniformly Stable

We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability.

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