A Residual of PDEs as convergence criteria of numerical solution

[feynman1]

Use a numerical method to solve a PDE f[u(x),u'(x),...]=0, where f is an operator, e.g. u'(x)+u(x)=0, and obtain a numerical solution v(x). Define f[v(x),v'(x),...] as the residual of the original PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution v(x)?

 

[BvU]

No

 

[pasmith]

The numerical method either forces the residuals to be zero at the points where the approximate solution is obtained, or forces the residual to be orthogonal to some finite space of basis functions.

In neither case is it used to estimate the error.

 

[feynman1]

The numerical method either forces the residuals to be zero at the points where the approximate solution is obtained, or forces the residual to be orthogonal to some finite space of basis functions.

In neither case is it used to estimate the error.

then why not a good estimate of the error?

 

[bigfooted]

Define f[v(x),v'(x),...] as the residual of the original PDE

You do not have the residual of the original PDE. What you do have is the residual of the discretized PDE.

Now, if you use iterative solvers like Krylov subspace methods (for instance GMRES https://en.wikipedia.org/wiki/Generalized_minimal_residual_method) then the Euclidian norm of the residual at iteration i can be used to monitor the convergence towards the numerical solution in this discretized subspace. These residuals are reported by numerical PDE solver software like Comsol or OpenFOAM.

However, if your residual is zero, it only means that you have found the solution of the discretized PDE to within machine precision. You still do not know how large the error is between your discretized solution and the solution of the original PDE. For this you then need to do a mesh refinement study. For example you can take your exact solution of the discretized problem on mesh N and project it onto a larger/finer mesh 2N and recompute the residual. The residual will now not be zero anymore and you have to re-iterate until convergence. The hope is that if you make your mesh larger and larger, these residual changes on the new mesh will become smaller and your discrete solution will converge to the continuous solution.

then why not a good estimate of the error?

In practice, people look at residuals of iterative methods and associate this with 'the error'. There are relationships (from functional analysis) between the norm of the residual and the discretization error, and the actual continuous solution of the original PDE. They depend on the PDE (+boundary conditions) and the numerical method used and they give you information about the order of the convergence (next to the proof that the numerical method actually converges).
You can find this kind of detailed information for instance in the book of Babuska and Strouboulis 'The Finite Element Method and its Reliability' .

 

[feynman1]

You do not have the residual of the original PDE. What you do have is the residual of the discretized PDE.

Thanks, but why don't we have the residual of the original PDE? Unable to take derivatives?

 

[bigfooted]

Thanks, but why don't we have the residual of the original PDE? Unable to take derivatives?

In the discretized problem you are working with a finite number of points. In your original problem $u'(x) + u(x)$ you could take a continuous solution $f(x)$ and substitute it into the ODE and determine a residual. But as soon as you construct some kind of solution vector $x=(x_1,x_2,...,x_N)$ then you are working in a discrete subspace, not in the continuous function space. For instance for first order methods, you assume that the solution is actually in a Hilbert Space $H_1$ of once locally differentiable functions. If your are solving a shock wave problem with the Euler equations, there is a discontinuity at the location of the shock. The solution of this problem is not in $H_1$, at least not in the neighborhood of the discontinuity.

So these discrete problems have analytical solutions that are different from the continuous solution and you only ever get an approximation of the continuous solution unless your solution can be exactly represented by the discrete problem (when the solution is linear and the discretization is also linear for example).

 

[feynman1]

Define f[v(x),v'(x),...] as the residual of the discretized PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution v(x)? How small should it be for an oscillatory solution?

 

[feynman1]

Define f[v(x),v'(x),...] as the residual of the discretized PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution v(x)? How small should it be for an oscillatory solution?

?

 

[feynman1]

how large is the residual of a discrete solution of a PDE from 0 in order of magnitude?