# a priori error analysis of cell-centered finite-volume methods

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.

• 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. – Wolfgang Bangerth Sep 4 '12 at 11:51
• 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 $\| \nabla u - \nabla_h u_h\|$? – chris Sep 14 '12 at 20:01