A connection between subgrid scale eddy viscosity and mixed methods

Abstract

We consider a new mixed method (related to the `EVSS' method in computational viscoelasticity) for the convection-dominated, convection–diffusion equation in which stabilization is added then removed through the extra `mixed' variables. This consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in ∇u^h. By suitable choice of the mixed spaces, a method of Guermond is recovered exactly. We show that for a different, natural choice a new method results with global error estimates similar to both Guermond's method and the streamline diffusion/SUPG method.

Introduction

Consider the solution of convection-dominated convection diffusion problems such as

$$\text{−ϵΔu+b·∇u+gu=f}\text{in}\text{Ω,}\text{u=0,}\text{on}\text{∂}\text{Ω.}$$

This is interesting as a test-bed for studying numerical methods for more complex problems such as turbulent or viscoelastic flow. There are various methods available for the approximate solution of (1.1). Among finite element procedures upwind finite element methods and streamline upwind (SUPG) type methods are currently popular choices, see the surveys in [4], [14], [18], for example. The alternative stabilization procedure is `artificial viscosity' (AV) methods. Due to the poor performance of the most straightforward AV method, these have been somewhat less popular.

There is recently, however, a resurgence of interest in better AV discretizations with three main threads. The first is AV with antidiffusion via defect correction [5]. The second is nonlinear AV motivated by work on subgrid scale modeling for turbulent flows, see, e.g., [12], [13], [15], [19] for examples of this thread. The third is an idea of Guermond [8] (see also [7], [11] for recent developments) which, although quite general, can be interpreted as to begin with a Galerkin discretization in a classical finite element space augmented by bubble functions and then to insert AV acting only on the smallest refined mesh scales modeled by extra bubble function degrees of freedom.

The theory of this approach is only at its beginning. Nevertheless, it is quite comparable to that of that SUPG method at the same stage of development. Furthermore, there is no incompatability between Guermond's AV expression and SUPG methods or defect correction methods. Thus, there are interesting possibilities of improving the computational performance of all with some future synthesis of these ideas. However, such a synthesis requires exploring further the connections and other realizations of finest scale AV concept of Guermond.

To present the finest scale AV idea in its clearest form, consider first the variational formulation of (1.1) in

$$\text{X≔}\text{H}{}^1\text{∘}\text{(Ω)≔{v∈L}{}^2\text{(Ω):∇v∈L}{}^2\text{(Ω)}\text{and}\text{v=0}\text{on}\text{Γ=}\text{∂}\text{Ω}.}$$

The usual variational formulation of (1.1) is to find u∈X satisfying

$$\text{a(u,v)≔ϵ(∇u,∇v)+b(u,v)=(f,v)}\text{for}\text{all}\text{v∈X,}$$

where (·,·) is the $\text{L}{}^2\text{(Ω)}$ inner product with norm ∥·∥ and b(u,v)≔(b·∇u+gu,v). It is well-known that for sufficiently smooth, bounded coefficients if

$$\text{g(x)−}\text{1}\text{2}\text{∇·}\text{b}\text{→}\text{(x)⩾g}{}_{\mathrm{\circ }}\text{>0,}$$

then a(u,v) is coercive in the H¹ norm, ∥u∥²₁≔∥∇u∥²+∥u∥², and (1.1) has a unique weak solution for any $\text{f∈H}{}^{\mathrm{-1}}\text{(Ω):=(}\text{H}{}^1\text{∘}\text{(Ω))}{}^{\mathrm{*}}$. The usual conforming, finite element method is to select a finite element space X^h⊂X and calculate u^h∈X^h by solving the linear system:

$$\text{ϵ(∇u}{}^{\mathrm{h}}\text{,∇u}{}^{\mathrm{h}}\text{)+b(u}{}^{\mathrm{h}}\text{,v}{}^{\mathrm{h}}\text{)=(f,v}{}^{\mathrm{h}}\text{)}\text{∀v}{}^{\mathrm{h}}\text{∈X}{}^{\mathrm{h}}\text{.}$$

Since this generically produces a poor quality approximate solution, the modifications mentioned above were developed.

For Guermond's fine scale artificial viscosity operator, a usual finite element space X^H (representing the “large scale”) is constructed. This is augmented with bubble functions to produce X^h. (Note that both spaces utilize essentially the same meshwidth.) For any v^h∈X^h decompose v^h into resolved and fine scales via:

$$\text{v}{}^{\mathrm{h}}\text{=v}{}^{\mathrm{H}}\text{+(v′)}{}^{\mathrm{h}}\text{,}\text{v}{}^{\mathrm{H}}\text{∈X}{}^{\mathrm{H}}\text{,}\text{(v′)}{}^{\mathrm{h}}\text{∈X}{}^{\mathrm{Hh}}$$

corresponding to the decomposition of X^h:

$$\text{X}{}^{\mathrm{h}}\text{=X}{}^{\mathrm{H}}\text{⊕X}{}^{\mathrm{Hh}}\text{.}$$

Guermond's idea consists of adding an AV term acting only on those finest scales in X^H. It calculates u^h∈X^h by solving the linear system:

$$\text{α(h)(∇(u}{}^{\mathrm{\prime }}\text{)}{}^{\mathrm{h}}\text{,∇(v}{}^{\mathrm{\prime }}\text{)}{}^{\mathrm{h}}\text{)+ϵ(∇u}{}^{\mathrm{h}}\text{,∇u}{}^{\mathrm{h}}\text{)+b(u}{}^{\mathrm{h}}\text{,v}{}^{\mathrm{h}}\text{)=(f,v}{}^{\mathrm{h}}\text{)}\text{∀v}{}^{\mathrm{h}}\text{∈X}{}^{\mathrm{h}}\text{.}$$

This report studies artificial viscosity stabilizations of the type exhibited in (1.4) added to a usual Galerkin formulation. In fact, we next show one method of adding such operator in a consistent manner through a mixed formulation allowing possibly greater accuracy in the limit of full resolution of boundary and/or interior layers (as occurs, e.g., with the use of Shiskin meshes [17]). On the other hand, trivially eliminating the extra variables shows that the mixed formulation implicitly introduces a subgrid scale artificial viscosity operator in the u-equation. Thus, depending upon point of view, the simple method we introduce next is either a nonconforming, multiscale eddy viscosity method for u or a conforming, mixed method for u and $\text{q}\text{⇀}\text{=∇u}$.

The method consists of stabilization by adding terms augmenting the coercivity of the problem. These terms are also subtracted for consistency but the subtracted terms are treated as extra variables in a mixed method. This simple idea is universally useful in other problems (such as porous media flow in a fractured reservoir). The same general idea has been used in viscoelastic simulations in the EVSS method [6].