Stabilization arising from PGEM : a review and further developments

The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condensation. The resulting stabilized method is shown to lead to optimal convergences, and afterward, it is numerically validated

A symmetric nodal conservative finite element method for the Darcy equation

This work introduces and analyzes novel stable Petrov-Galerkin EnrichedMethods (PGEM) for the Darcy problem based on the simplest but unstable continuous P1/P0 pair. Stability is recovered inside a Petrov-Galerkin framework where element-wise dependent residual functions, named multi-scale functions, enrich both velocity and pressure trial spaces. Unlike the velocity test space that is augmented with bubble-like functions, multi-scale functions correct edge residuals as well. The multi-scale functions turn out to be the well-known lowest order Raviart-Thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure. The enrichment strategy suggests the way to recover the local mass conservation property for nodal-based interpolation spaces. We prove that the method and its symmetric version are well-posed and achieve optimal error estimates in natural norms. Numerical validations confirm claimed theoretical results

Emergent electrodynamics from the Nambu model for spontaneous Lorentz symmetry breaking

After imposing the Gauss law constraint as an initial condition upon the Hilbert space of the Nambu model, in all its generic realizations, we recover QED in the corresponding non-linear gauge A_{\mu}A^{\mu}=n^{2}M^{2}. Our result is non-perturbative in the parameter M for n^{2}\neq 0 and can be extended to the n^{2}=0 case. This shows that in the Nambu model, spontaneous Lorentz symmetry breaking dynamically generates gauge invariance, provided the Gauss law is imposed as an initial condition. In this way electrodynamics is recovered, with the photon being realized as the Nambu-Goldstone modes of the spontaneously broken symmetry, which finally turns out to be non-observableComment: 17 page

Ordering properties of radial ground states and singular ground states of quasilinear elliptic equations

none2noIn this paper we discuss the ordering properties of positive radial solutions of the equation Δpu(x)+k|x|δuq-1(x)=0where x∈ Rn, n> p> 1 , k> 0 , δ> - p, q> p. We are interested both in regular ground states u (GS), defined and positive in the whole of Rn, and in singular ground states v (SGS), defined and positive in Rn { 0 } and such that lim |x|→v(x) = + ∞. A key role in this analysis is played by two bifurcation parameters pJL(δ) and pjl(δ) , such that pJL(δ) > p∗(δ) > pjl(δ) > p: pJL(δ) generalizes the classical Joseph–Lundgren exponent, and pjl(δ) its dual. We show that GS are well ordered, i.e. they cannot cross each other if and only if q≥ pJL(δ) ; this way we extend to the p> 1 case the result proved in Miyamoto (Nonlinear Differ Equ Appl 23(2):24, 2016), Miyamoto and Takahashi (Arch Math Basel 108(1):71–83, 2017) for the p≥ 2 case. Analogously we show that SGS are well ordered, if and only if q≤ pjl(δ) ; this latter result seems to be known just in the classical p= 2 and δ= 0 case, and also the expression of pjl(δ) has not appeared in literature previously.openColucci R.; Franca M.Colucci, R.; Franca, M

On the non-autonomous hopf bifurcation problem: Systems with rapidly varying coefficients

We consider the Cauchy-problem for the parabolic equation $$u_t = Delta u+ f(u,|x|),$$ where $x in mathbb R^n$, n >2, and $f(u,|x|)$ is either critical or supercritical with respect to the Joseph-Lundgren exponent. In particular, we improve and generalize some known results concerning stability and weak asymptotic stability of positive ground states