Isaac P. Santos

A Nonlinear Multiscale Viscosity Method to Solve Compressible Flow Problems

In this work we present a nonlinear multiscale viscosity method to solve inviscid compressible flow problems in conservative variables. The basic idea of the method consists of adding artificial viscosity adaptively in all scales of the discretization. The amount of viscosity added to the numerical model is based on the YZ\(\beta \) shock-capturing parameter, which has the property of being mesh and numerical solution dependent. The subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. This new numerical formulation can be considered a free parameter and self adaptive method. Performance and accuracy comparisons with the well known method SUPG combined with shock capturing operators are conducted based on benchmark 2D problems.

Numerical analysis of the nonlinear subgrid scale method

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. Thisnew 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 Nonlinear Finite Element Formulation Based on Multiscale Approach to Solve Compressible Euler Equations

This work presents a nonlinear finite element method for solving compressible Euler equations. The formulation is based on the strategy of separating scales – the core of the variational multiscale (finite element) methodology. The proposed method adds a nonlinear artificial viscosity operator that acts only on the unresolved mesh scales. The numerical model is completed by adding the YZ\(\beta \) shock-capturing operator to the resolved scale, taking into account the Mach number. We evaluate the efficiency of the new formulation through numerical studies, comparing it with other methodologies such as the SUPG combined with a shock-capturing operator.

Error estimates for transport problems with high Péclet number using a continuous dependence assumption

In this paper we discuss the behavior of stabilized finite element methods for the transient advection-diffusion problem with dominant advection and rough data. We show that provided a certain continuous dependence result holds for the quantity of interest, independent of the Peclet number, this quantity may be computed using a stabilized finite element method in all flow regimes. As an example of a stable quantity we consider the parameterized weak norm introduced in Burman (2014). The same results may not be obtained using a standard Galerkin method. We consider the following stabilized methods: Continuous Interior Penalty (CIP) and Streamline Upwind Petrov-Galerkin (SUPG). The theoretical results are illustrated by computations on a scalar transport equation with no diffusion term, rough data and strongly varying velocity field.

A parameter-free dynamic diffusion method for advection–diffusion–reaction problems

Abstract In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.

A Multiscale Finite Element Formulation for the Incompressible Navier-Stokes Equations

In this work we present a variational multiscale finite element method for solving the incompressible Navier-Stokes equations. The method is based on a two-level decomposition of the approximation space and consists of adding a residual-based nonlinear operator to the enriched Galerkin formulation, following a similar strategy of the method presented in [1, 2] for scalar advection-diffusion equation. The artificial viscosity acts adaptively only onto the unresolved mesh scales of the discretization. In order to reduce the computational cost typical of two-scale methods, the subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Accuracy comparisons with the streamline-upwind/Petrov-Galerkin (SUPG) formulation combined with the pressure stabilizing/Petrov-Galerkin (PSPG) method are conducted based on 2D benchmark problems.

Comparative Studies to Solve Compressible Problems by Multiscale Finite Element Methods

In this work we evaluate two multiscale methodologies to solve compressible flow problems, named, Dynamic Diffusion (DD) and Nonlinear Multiscale Viscosity (NMV), using the well know predictor-multicorrector time integration scheme. The subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. The time integration schemes assume that the resolved coarse scale advances in time by second order approximation and the unresolved scale can advance by first and second order approximations. Performance and accuracy comparisons are conducted based on benchmark 2D problems.

Predictor-Multicorrector Scheme for the Dynamic Diffusion Method

In this work we evaluate a predictor-multicorrector integration scheme for transient advection-diffusion-reaction problems using the Dynamic Diffusion method (DD). This multiscale finite element formulation results in a free parameter method in which the subgrid scale space is defined using bubble functions whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. The time advancing scheme assumes that the subscales change in time. The formulation is compared with the Consistent Upwind Petrov-Galerkin (CAU) method using the same predictor-multicorrector scheme. Numerical experiments based on benchmark 2D problems were conducted to illustrate the behavior of this new algorithm applied to advection-diffusion-reaction equations.