I'm using the cell-centered finite-volume method (for example Morton, Numerical solution of convection-diffusion problems, Chapman&Hall, 1996) to discretize the advection-diffusion equation and the incompressible Navier-Stokes equations. To check the implementation, I'm using the method of manufactured solutions (Roache, Fundamentals of verification and validation, Hermosa publishers, 2009). So I need to know the theoretical order of accuracy to compare with the observed order in my numerical experiments.

The paper by Weiser and Wheeler (SIAM Journal on Numerical Analysis, 25(2), 1988) proves 2nd order of accuracy for the advection-diffusion equation, both for the solution and its gradient on Cartesian but not necessarily uniform grids. I do get 2nd order for Cartesian grids including stretching, also for the incompressible Navier-Stokes equations.

For non-Cartesian grids however, the velocity is still 2nd order, but the velocity gradient drops to 1st order and so does the pressure. In some cases, it even drops to half-order. Is there any theoretical analysis that covers this situation? I know that there are many papers proposing all kinds of corrections for non-Cartesian grids based on numerical evidence, but I'm looking specifically for analysis. The nicest thing would be an overview paper that summarizes a number of propositions and analyzes whether they actually improve the order of accuracy instead of reducing the constant.


1 Answer 1


You are looking for the term "superconvergence". One frequently observes (and can sometimes prove) that on regular enough meshes the convergence order is higher than it should really be (e.g. that the gradient converges at the same order as the values, at least at certain points). This property is typically lost if you destroy certain symmetry properties of the mesh.

  • $\begingroup$ So having 2nd order for the velocity and 1st order for the velocity gradient on irregular grids does not indicate an implementation issue (but order half does)? And what about the pressure? $\endgroup$
    – chris
    Sep 4, 2012 at 7:16
  • $\begingroup$ Typically, the gradient of a numerical solution is one order less accurate than the solution itself. For example, if $u$ is the exact solution of the Laplace equation and $u_h$ the finite element approximation, then you have with the usual finite elements of polynomial degree $p$ that $\|u-u_h\| \le C h^{p+1} \|\nabla^{p+1} u\|$ but $\|\nabla(u-u_h)\| \le C h^{p} \|\nabla^{p+1} u\|$. There are cases where you get half order but that let's frequent (e.g., using most schemes for the linear advection equation give something like $h^{1/2}$). I don't know about your specific case. $\endgroup$ Sep 4, 2012 at 11:51
  • $\begingroup$ The main difference between linear finite-elements and finite-volume seems to be that in finite-volume the gradient is obtained by reconstruction from the cell averages (e.g. least-squares) while in finite-elements the cell average and the gradient are both unknowns for which to solve. This makes me wonder whether error-estimation from finite-elements can be translated to finite-volume. $\endgroup$
    – chris
    Sep 7, 2012 at 6:29
  • $\begingroup$ Yes, and the one it's typically done is to reinterpret finite volume or finite difference schemes as particular variants of finite element schemes. Then apply the usual finite element machinery to get an error estimate. Many FVM schemes have been analyzed this way. $\endgroup$ Sep 7, 2012 at 21:33
  • $\begingroup$ That's what's being done in the paper by Weiser and Wheeler. But I haven't found anaything on incompressible Navier-Stokes for non-Cartesian grids... By the way shouldn't we write $\| \nabla u - \nabla_h u_h\|$? $\endgroup$
    – chris
    Sep 14, 2012 at 20:01

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