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feynman1

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feynman1

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BvU

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No

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pasmith

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In neither case is it used to estimate the error.

- #4

feynman1

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then why not a good estimate of the error?

In neither case is it used to estimate the error.

- #5

bigfooted

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You do not have the residual of the original PDE. What you do have is the residual of the discretized PDE.Define f[v(x),v'(x),...] as the residual of the original 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

However, if your residual is zero, it only means that you have found the solution of the

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).then why not a good estimate of the error?

You can find this kind of detailed information for instance in the book of Babuska and Strouboulis 'The Finite Element Method and its Reliability' .

- #6

feynman1

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Thanks, but why don't we have the residual of the original PDE? Unable to take derivatives?You do not have the residual of the original PDE. What you do have is the residual of the discretized PDE.

- #7

bigfooted

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

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).

- #8

feynman1

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- #9

feynman1

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?

- #10

feynman1

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

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