arXiv subjects

Cătălin Liviu Bichir

Publications and source records attributed to Cătălin Liviu Bichir.

5 recordsLinked to original sources

arXiv2024

On the construction of a family of well-posed approximate formulations for the stationary Stokes problem using an extended system

We introduce an exact parameterized extended system such that, under adequate data, between the components of its solution, there is the solution of the weak formulation of the homogeneous Dirichlet problem for the stationary Stokes equations. In the extended system, we introduce the momentum equation together with two other forms of this one. This allows us to reformulate, for the stationary case, the consistent pressure Poisson equation of Sani, Shen, Pironneau, Gresho [\textit{Int. J. Numer. Meth. Fluids}, \textbf{50} (2006), pp. 673-682], from the unsteady case. In this way, we can retain the information we need for the approximate pressure on the boundary. We obtain a parameterized perturbed pressure Poisson equation for the stationary Stokes problem. We prove that to solve the stationary Stokes problem is equivalent to solve a problem for the momentum equation, the parameterized equation and the equation that defines the Laplace operator acting on velocity. The approximation of this last problem give a family of well - posed approximate formulations. The solution of each element of this family approximates the solution of the stationary Stokes problem. Some necessary variants of existing results on general Banach spaces are also developed. These concern the density of subspaces related to isomorphisms and the connection between the exact and the approximate problems. First, the well - posedness of the exact extended system is proved in some dense subspaces. The well posed approximate problem does not necessitate the discrete inf-sup condition. The paper is also related to the existing discussions on the boundary conditions for the pressure and on the imposing of the incompressibility constraint on the boundary.

math.NA↗

arXiv2024

On the existence of approximate problems that preserve the type of a bifurcation point of a nonlinear problem. Application to the stationary Navier-Stokes equations

We consider a nonlinear problem $F(λ,u)=0$ on infinite-dimensional Banach spaces that correspond to the steady-state bifurcation case. In the literature, it is found again a bifurcation point of the approximate problem $F_{h}(λ_{h},u_{h})=0$ only in some cases. We prove that, in every situation, given $F_{h}$ that approximates $F$, there exists an approximate problem $F_{h}(λ_{h},u_{h})-\varrho_{h} = 0$ that has a bifurcation point with the same properties as the bifurcation point of $F(λ,u)=0$. First, we formulate, for a function $\widehat{F}$ defined on general Banach spaces, some sufficient conditions for the existence of an equation that has a bifurcation point of certain type. For the proof of this result, we use some methods from variational analysis, Graves' theorem, one of its consequences and the contraction mapping principle for set-valued mappings. These techniques allow us to prove the existence of a solution with some desired components that equal zero of an overdetermined extended system. We then obtain the existence of a constant (or a function) $\widehat{\varrho}$ so that the equation $\widehat{F}(λ,u)-\widehat{\varrho} = 0$ has a bifurcation point of certain type. This equation has $\widehat{F}(λ,u) = 0$ as a perturbation. It is also made evident a class of maps $C^{p}$ - equivalent (right equivalent) at the bifurcation point to $\widehat{F}(λ,u)-\widehat{\varrho}$ at the bifurcation point. Then, for the study of the approximation of $F(λ,u)=0$, we give conditions that relate the exact and the approximate functions. As an application of the theorem on general Banach spaces, we formulate conditions in order to obtain the existence of the approximate equation $F_{h}(λ_{h},u_{h})-\varrho_{h} = 0$.

math.NA↗

arXiv2013

Limit points and Hopf bifurcation points for a one - parameter dynamical system associated to the Luo - Rudy I model

A one - parameter dynamical system is associated to the mathematical problem governing the membrane excitability of a ventricular cardiomyocyte, according to the Luo-Rudy I model. An algorithm used to construct the equilibrium curve is presented. Some test functions are used in order to locate limit points and Hopf bifurcation points. Two extended systems allow to calculate these points. The numerical results are presented in a bifurcation diagram.

math.NA↗

arXiv2011

Approximation of pressure perturbations by FEM

In the mathematical problem of linear hydrodynamic stability for shear flows against Tollmien-Schlichting perturbations, the continuity equation for the perturbation of the velocity is replaced by a Poisson equation for the pressure perturbation. The resulting eigenvalue problem, an alternative form for the two-point eigenvalue problem for the Orr-Sommerfeld equation, is formulated in a variational form and this one is approximated by finite element method (FEM). Possible applications to concrete cases are revealed.

math.NA↗

arXiv2011

Limit cycles by FEM for a one-parameter dynamical system associated to the Luo-Rudy I model

An one-parameter dynamical system is associated to the mathematical problem governing the membrane excitability of a ventricular cardiomyocyte, according to the Luo-Rudy I model. Limit cycles are described by the solutions of an extended system. A finite element method time approximation (FEM) is used in order to formulate the approximate problem. Starting from a Hopf bifurcation point, approximate limit cycles are obtained, step by step, using an arc-length-continuation method and Newton's method. Some numerical results are presented.

math.NA↗