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

TL;DR

This paper proves that for nonlinear problems in Banach spaces, one can construct approximate problems that preserve bifurcation points' properties, with applications to stationary Navier-Stokes equations.

Contribution

It introduces a method to create approximate problems maintaining bifurcation characteristics, extending the understanding of bifurcation preservation in numerical approximations.

Findings

Existence of approximate problems with preserved bifurcation points.

Conditions under which approximate equations replicate bifurcation behavior.

Application to stationary Navier-Stokes equations.

Abstract

We consider a nonlinear problem $F(\lambda,u)=0$ 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 $F_{h}(\lambda_{h},u_{h})=0$ only in some cases. We prove that, in every situation, given $F_{h}$ that approximates $F$, there exists an approximate problem $F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$ that has a bifurcation point with the same properties as the bifurcation point of $F(\lambda,u)=0$. 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 $\widehat{F}(\lambda,u)-\widehat{\varrho} = 0$ has a bifurcation point of certain type. This equation has $\widehat{F}(\lambda,u) = 0$ as a perturbation. It is also made evident a class of maps $C^{p}$ - equivalent (right equivalent) at the bifurcation point to $\widehat{F}(\lambda,u)-\widehat{\varrho}$ at the bifurcation point. Then, for the study of the approximation of $F(\lambda,u)=0$, 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 $F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$.