A wide variety of finite element methods (FEM) exist to stably approximate partial differential equations with higher-order accuracy. A key issue is how to achieve this stability for problems in which convection is a significant effect. Typical approaches include discontinuous Galerkin (DG) methods (which achieve stability through the Riemann solvers employed at element boundaries) and stabilized continuous finite element methods which generally achieve stability through the addition of a term that penalizes the strong form residual. SUPG, GLS, and VMS being the most common examples of stabilized continuous FEM (a review of these methods is included in Hughes et. al. 2018). Specifically, for governing equations with the following strong form (which may be nonlinear),
these stabilized methods all have the following residual form,
The first term is the standard weak form of the governing equation while the sum over element volumes is the stabilization term. The pre-multiplier operator P which acts on the weight functions vh is a differential operator which for a nonlinear governing equation will be a function of the solution uh. The tau term will be positive definite and have a dependence on the mesh size as well as the solution (if the problem is nonlinear). The specific forms of the pre-multiplier operator for the common stabilized methods are,
where the ()’ denotes the Frechet derivative and ()* is the adjoint operator. The potential advantage of a continuous FEM is a significant reduction in the number of globally-coupled degrees of freedom (DOF) compared to DG methods. Hybridized DG methods (HDG, EDG) have more recently been developed which allow elimination of elemental DOF and hence a reduction of the number of globally-coupled DOF. In fact, EDG can produce the same matrix size as stabilized continuous FEM methods.
Another approach to stable FEM discretizations is to use a variational multiscale method in which the solution is decomposed into a resolved (coarse) scale and unresolved (fine) scale:
We will consider a multiscale approach in which (1) the coarse scale is approximated using a continuous FEM basis, and (2) the fine scale is approximated in a manner that it can be eliminated from the global problem and hence produce a discrete system which would be the size of the coarse scale problem. A specific variational multiscale approach we have recently been exploring follows from the work of Coley & Evans (2018). This approach uses a discontinuous representation of the fine scale problem (as shown in the figure below) and DG weak form is used for the elemental fine scale residual. We refer to this approach as Variational MultiScale with Discontinuous subscales (VMSD).
For example, we have applied the VMSD discretization with MOESS adaptation to the turbulent flow over a hemisphere-cylinder. The turbulent flow was modeled using the compressible RANS equations with the Spalart-Allmaras turbulence model. The output chosen for adaptation was the drag coefficient. The Mach number and the adaptive mesh for VMSD using linear (p=p’=1) elements with 64K DOF are shown below.
The convergence of VMSD is compared to DG and SUPG (using the NASA Langley SFE discretization in the FUN3D software suite) in the figure below. All discretizations were run with linear elements (p=1). The results include a set of fixed (non-adapted) meshes as well as MOESS drag adapted meshes. Also included are results using VMSDL, a variation of VMSD in which the fine scale is solved using p’ = p-1 which for this example is p’ = 0 The concept of VMSDL is that the order of accuracy of VMSD is determine by the coarse scale and stability can be achieved using a fine scale which is a polynomial degree less than the coarse scale.
In summary, these results show that for this example:
- On the fixed meshes, VMSD achieves similar accuracy to the SUPG (SFE) results.
- On the adapted meshes, VMSD requires about a factor of 10 fewer DOF to achieve similar accuracy as DG.
- VMSD and VMSDL achieve similar accuracy.
Additional information about VMSD, VMSDL, and these results can be found in these referencesHuang_2020_PhDHuang_2020_Variational. A work which discusses the relationship between hybridized DG (in particular EDG) and stabilized continuous FEM and influenced our thinking on the VMSD approach can be found here.