# In what regime do the continuous and discontinuous Galerkin method become unstable for advection-diffusion systems?

I know that the finite volume method (based around a central different stencil) is unstable for advection dominated advection-diffusion problems. This leads to different adaptive schemes to can be applied to stabilise the basic approach (such as exponential fitting).

However, I don't have broad understanding of stability issue with finite elements method, in particular I am interesting in comparing the continuous and discontinuous Galerkin method. From what I have read the continuous Galerkin method can have stability problems for advection dominated problems. But I haven't managed to find an exact criteria for stability, is there one?

Is the discontinuous Galerkin method fundamentally more stable in these situations, or does it also have a stability criteria?

• Unless you add a penalty term (or say stablization term) to penalize the unsatisfied continuity condition, DG is not fundamentally more stable. It really depends on what bilinear form you are using, and under what kind of norm is this bilinear form is coercive (stability). Feb 27 '14 at 20:59
• Thanks Shuhao, I guess I'm interested in a general overview of the pros/cons of the DG and CG, and little bit of the historical reasons for the development of the two approaches. Feb 27 '14 at 23:46

In the advection dominated case the problem can develop physics which is invisible to a computational mesh that is too coarse (say by having elements which contain many wavelengths). This leads to instability because neighboring elements could disagree considerably on what they believe are the right values to represent this physics (think interpolating high frequency waves with low order polynomials, you can get wildly different interpolants depending on how you sample.). This is problematic to continuous Galerkin because despite this potential disagreement, it forces continuity between elements. This is why you have mesh refinement restrictions, basically you only see stability once you have correctly resolved the advection (or add a bit of magic to the formulation).

DG resolves this by not requiring continuity and instead adding a term to penalize jump discontinuities, this penalty term essentially adds artificial dissipation to the method if the solution develops large jumps. This typically yields unconditionally stable methods (unconditional on the mesh size,polynomial order, and advection speed). Whether or not this is a desirable property depends on your application though, I think. Although the formulations are stable, it does not mean they are accurate. One could be stuck in the preasymptotic part of the convergence theory until they refine the mesh (resp. increase polynomial order), and then the level of mesh refinement used for convergence to kick in often ends up being exactly the same amount of refinement used for stability in the continuous cousin. Also be very careful about trusting stability results that depend on integrals being evaluated exactly.

• That's helpful thanks. So there are two pieces to the puzzle. 1) The CG method needs to have a mesh in which the Peclet number is <2 for stability. This requires mesh refinement techniques. 2) In the DG method discontinuous test functions (or basis functions) are used, when this is combined with penalty (artificial diffusion) term it gives an unconditional stable discretisation in terms of the Peclet number. However, you warning is that stability $\neq$ accuracy. Such that DG will also need to employ mesh refinement for accuracy and convergence of the solution. Mar 1 '14 at 1:48

The stability criteria for continuous Galerkin FEM with piecewise linear basis functions in 1-D is that the cell Peclet number should be less than 2. This is the same as the central difference finite-difference formulation and probably the same as some finite volume formulations of similar order.

FYI, this half of the question was answered similarly in your previous question on the topic.

I'll let a DG person speak to the criteria there.

• Thanks Bill, yes it is a bit similar to my last question. Here I'm interested in understanding why the discontinuous method is said to be more stable. Basically where might I want to use a discontinuous approach over the continuous method and visa-versa. But thanks for clarifying the Peclet number condition, that's important which I wasn't too sure about. Feb 27 '14 at 15:14