We analyze the stability properties of so-called triple deck model, a classical refinement Prandtl equation to describe boundary layer separation. Combining methodology introduced in [A.-L. Dalibard et al., SIAM J. Math. Anal., 50 (2018), pp. 4203--4245], based on complex analysis tools, and estimates inspired from Dietert Gérard-Varet [Anal. PDE, 5 (2019), 8], we exhibit unstable linearization...

a r t i c l e i n f o a b s t r a c t Adaptive mesh refinement Digital topology Volume-of-fluid Multi-phase simulation In numerical simulations of multi-scale, multi-phase flows, grid refinement is required to resolve regions with small scales. A notable example is liquid-jet atomization and subsequent droplet dynamics. It is essential to characterize the detailed flow physics with variable len...

We have developed a structured adaptive mesh refinement (SAMR) method for parabolic partial differential equation (PDE) systems. Solutions are calculated using the finite-difference or finite-volume method in space and backward differentiation formula (BDF) integration in time. The combination of SAMR in space and BDF in time is designed for problems where the fine-scale profile of sharp fronts...

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant...

We consider solutions of a refinement equation written in the form as

We build wavelet-like functions based on a parametrized family of pseudo-differential operators L~ν that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of L~ν , generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequenc...

We analyse and further develop a hierarchical multiscale method for the numerical simulation of two-phase flow in highly heterogeneous porous media. The method is based upon a mixed finite-element formulation, where fine-scale features are incorporated into a set of coarse-grid basis functions for the flow velocities. By using the multiscale basis functions, we can retain the efficiency of an u...

In this work we introduce and analyse a new adaptive PetrovGalerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. In a general heterogeneous setting we prove convergence of the HMM approximations to the solution of a macroscopic limit equation. The major new contribution of this work is an a-posteriori error estimate for the L2-err...

This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. However, current methods of solution involve discretizing the differential equation directly by finite elements or finite differences, and consequently suffer from the...

In this paper we present an example of a refinement equation such that up to a multiplicative constant it has a unique compactly supported distribution solution while it can simultaneously have a compactly supported componentwise constant function solution that is not locally integrable. This leads to the conclusion that in general the componentwise polynomial solution cannot be globally identi...