Dear Friends,

I am looking at the option of using Defect correction procedure for obtaining second order accurate gradients. This is achieved by an iterative procedure where the first order accurate gradients I obtain from a linear reconstruction are corrected to second order accuracy.

However, for a simple 2D problem involving a smooth function such as sin(x+y) on a [0,1]*[0,1] domain, I find that the order of accuracy of the gradients relies heavily on the convergence of the procedure. As the grids become finer, the convergence is not only hampered, but a more stringent convergence criterion becomes essential to obtain second order accuracy. For instance while a convergence to 1e-4 is sufficient on a 200 cells mesh, it becomes 1e-8 for a 12000 cells mesh. I solve the iterative problem by a simple Regula-falsi kind of procedure.

I believe that in general the low frequency errors on the finer mesh takes time to damp out, causing convergence troubles. Is it that DeC is used in conjuction mostly with multigrid, becuase the projection converst low freq. errors on the finer mesh to high frequency errors on coarser mesh, which get damped out improving the convergence ? If DeC is used stand alone for obtaining hihger order gradients for a steady--state problem only, is it posible to obtain fast convergence in absence of multigrid ?

Any suggestions/comments are most welcom

Regards,

Ganesh

I am curious, can you describe the way of computing the gradients a bit more?

From what I remember, the following was called defect correction:

Assume you want to solve a dicretized (non-linear) equation: O2(q) = 0 To solve it you use a less accurate descritization of the same problem O1 and then you iterate: O1(q_k+1) = O1(q_k) - O2(q_k) for a few steps and then you update the right hand side and iterate again. Eventually you'll reach: O1(q_k+1) = O1(q_k) and consequently also: O2(q_k) = 0 I came across defect correction while implementing an FAS on node centred unstructered grids (agglomeration). On the coarse levels it is virtually impossible to have anything better than first order - at least for me. Actually FAS multi grid is a defect correction method itself!

I don't quite understand how you apply this to your gradients.

Dear O,

The IDeC procedure you just explained is what I also make use of in obtaining higher order gradients. I have first order gradients obtained from a linear reconstruction procedure, which is a lower order approximation. I attempt to get a higher order accurate gradients. The equation I iteratively solve is similar to the expression for the gradients obtained using a linear reconstruction procedure, with a slight modification which is essentially to correct the first order gradients to second order accuracy. The initial guess to the iterative solution procedure is a lower order gradient and the fixed point of the iterative procedure yields me the second oreder acurate gradients, much the way you have described.

Regards,

Ganesh

I tried similar things with a grid-free method but without much success. I was trying to do the defect correction iterations along with the time-integration. Both the defect correction and time-stepping will converge together, rather than waiting for the defect correction in each time-step to converge. I had stability problems with this and I havent pursued it further. But it seems to be a good approach if you are only interested in steady-state computations.

Dear Praveen,

Thanks for your comments. I am presently looking at DeC for steady state computations only. If you are looking at unsteady computations with DeC, you may possibly intersted in the following report:

A second order Defect correction scheme for unsteady problems, Matin and Gulliard, N2447, INRIA Rep, 1994

Regards,

Ganesh