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Nope. It depends on the solver. Here in the OP we are talking about residual remaining after the solution is obtained and descretisation is applied. Accuracy is with respect to exact or benchmark solution. So it is possible for solver to produce a solution that has higher residual but is more accurate. This is the whole point. The residual in this context is different than what you have in mind. |
Yes very much.
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There is no sense in this discussion if we do not formalize the object of the issue. - The numerical solution, by definition it satisfies the discrete set of equations A_d(fnum)=q - The exact solution, by definition it satisfies the original PDE A_pde(fex)=q. - Order of accuracy is the main term in the local truncation error of the whole equations. It seems you are talking about point #1, that is you select a numerical solution f_num_app (that is not fnum) and does not satisfy the discrete set of equations A_d(f_num_app)-q=res. And you are stating that f_num_app is a better approximation of fex than fnum. And that a solution f_num_app2, producing a res2>res would be still better. Therefore, we are talking about the discretization error, not about the local truncation error. Am I right? But that has nothing to do with the accuracy order. |
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