A. M. P. Valli

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.

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.