Numerical analysis of the nonlinear subgrid scale method

Abstract

Abstract. This paper presents the numerical analysis of the Nonlinear Subgrid Scale (NSGS) model for approximating singularly perturbed transport models. The NSGS is a free parameter subgrid stabilizing method that introduces an extra stability only onto the subgrid scales. This new feature comes from the local control yielded by decomposing the velocity field into the resolved and unresolved scales. Such decomposition is determined by requiring the minimum of the kinetic energy associated to the unresolved scales and the satisfaction of the resolved scale model problem at element level. The developed method is robust for a wide scope of singularly perturbed problems. Here, we establish the existence and uniqueness of the solution, and provide an a priori error estimate. Convergence tests on two-dimensional examples are reported.Mathematical subject classification: Primary: 65N12; Secondary: 74S05.

A Linearized Adaptive Dynamic Diffusion Finite Element Method for Convection-Diffusion-Reaction Equations

“…where τ is an appropriate constant, e.g., τ = f 0,T . This modification of the artificial diffusion allows for the upper and lower bounds of the corresponding local nonlinear operator D T (•;•,•), hence the existence, stability, and convergence of discrete solutions follow by using similar approaches in the analysis of nonlinear finite element methods; see [3,11,39,42,49]. We note that the choice of the artificial diffusion (ξ T (u h ) on each element) is determined by the artificial dissipation term β b , and β b is obtained by minimizing the kinetic energy E k .…”

“…To get rid of the local oscillations, many nonlinear stabilized finite element methods have been developed by adding an extra diffusivity term to the formulation to recover the monotonicity of the continuous problem [1,11,20,25,31,37,[48][49][50][51]. Such methods lead to nonlinear systems due to the introduction of nonlinear artificial dissipation dynamically adjusted in terms of the notion of scale separation.…”

Multiscale Finite Element Formulation for the 3D Diffusion-Convection Equation

“…Over the last years, some nonlinear two-scale variational methods have been developed, such as the Nonlinear Subgrid Stabilization (NSGS) [ 23 , 24 ] and Dynamic Diffusion (DD) methods [ 11 , 25 ] to solve convection-dominated problems; the Nonlinear Multiscale Viscosity (NMV) method [ 26 , 27 ] to solve the system of compressible Euler equations. In [ 28 ] the NSGS method was applied to solve incompressible Navier-Stokes equations.…”