Impact of adaptation on convergence of higher-order methods

As a mesh is refined, solution approximation errors for higher-order methods have the potential to converge more rapidly than lower-order methods.  However, this potential is limited by the solution regularity (see here for a discussion).  For errors in outputs, define the convergence rate r in the following manner

|| J(uh)-J(u) ||   =  O(hr)      as    h→ 0

where J(u) is an output (e.g. lift, drag, efficiency, etc).  For finite element discretization of PDEs with diffusion and using polynomials with order p (e.g. p=1 are linear polynomials in an element), the optimal output convergence rate is r = 2p.  To achieve this optimal rate, the solution (and the solution to the adjoint problem for the output) must be sufficiently smooth.  Note that the fact that the output converges faster than the solution itself is known as super convergence and is one of the benefits of finite element methods compared to finite difference and finite volume which generally do not have that property.

We know illustrate how solution irregularity can impact output convergence and how adaptivity can reduce the impact of the irregularity.  The essential idea behind the success of adaptive methods is that they can refine regions which exhibit solution irregularity faster than regions where the solution is regular.

As an example, consider the laminar compressible Navier-Stokes flow over a delta wing[1] which was used as a test case for the 1st International Workshop on High-Order CFD Methods (HOW). All of the results in the figure below are from discontinuous Galerkin FEM discretizations. This example demonstrates the following:

  • When using the HOW-provided meshes which were nested (uniformally coarsened from the finest mesh), the rate of convergence for linear (p=1) and quadratic (p=2) elements is nearly the same.  The quadratic elements do have a lower error compared to the linear elements, but the rate is not equal to the optimal h2p (which in terms of DOF would be DOF-2p/3).
  • The MOESS adaptive results performed by Yano achieve convergence rates (somewhat better than) the expected optimal rates.  Note: the faster than optimal rates are likely a combination of: not knowing having the exact drag coefficient; mesh adaptation not conforming sufficiently to the requested metric on the coarser meshes; and an output is not a normed quantity and hence it is possible for the output error to "cross through zero".
  • Also on the plot are results from Leicht & Hartmann (Journal of Computational Physics, 2010).  Leicht & Hartmann (L&H) utilize an anisotropic adaptative process based upon nested refinements from an initial coarse grid.  While the method is adaptive and anisotropic, the direction of the anisotropic refinement is limited by the initial coarse mesh topology. Regardless, the L&H adaptive results are a clear improvement compared to the uniformly refined nested meshes from the HOW workshop.  However, the metric-based adaptive results using MOESS are able to achieve less error per DOF.

DeltaWing_NS_HOW.png image is not rendering


References

  1. Yano M. An Optimization Framework for Adaptive Higher-Order Discretizations of Partial Differential Equations on Anisotropic Simplex Meshes [PhD]. Department of Aeronautics and Astronautics: Massachusetts Institute of Technology; 2012. Available from: http://hdl.handle.net/1721.1/96928