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14 votes
Accepted

$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)

Yes, this is the standard Aubin-Nitsche (or duality) trick. The idea is to use the fact that $L^2$ is its own dual space to write the $L^2$-norm as an operator norm $$\|u\|_{L^2} = \sup_{\phi\in L^2\...
Christian Clason's user avatar
11 votes

Convergence rate vs convergence order

$\|x^{(k)}-x^*\|$ is the error in the $k$th term, call it $E_k$. For a "good" numerical method, we want the approximation to get closer and closer to the desired result so $E_k$ has to decrease to ...
Raziman T V's user avatar
10 votes
Accepted

Analytical convergent sequence and numerical divergent sequence

Jean-Michel Muller, et. al., "Handbook of Floating-Point Arithmetic 2nd ed.", Birkhäuser 2018, gives the following example due to Muller, specifically constructed to deliver incorrect results with ...
njuffa's user avatar
  • 1,895
10 votes
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How does gmres method iteration behave for this non-diagonalizable matrix?

Unfortunately, convergence of GMRES does not have a clear dependence on the distribution of eigenvalues. It was proved by Greenbaum, Ptak and Strakos in 1996 that you can construct examples with an ...
Federico Poloni's user avatar
9 votes
Accepted

Lack of quadratic convergence in Newton's method

Yes: See Higham's book "Accuracy And Stability of Numerical Algorithms", second edition, chapter 25: Nonlinear Systems and Newton's Method. In particular, see the section on the "limiting residual" ...
user14717's user avatar
  • 2,155
9 votes
Accepted

Accurate computation of Gauss-Kuzmin entropy

It's fairly easy to evaluate, to do this expand the logs in Taylor series in $x=(k+1)^{-2}$: $$ \log_2(1-x) = \frac{-1}{\log 2}\sum_{m\geq1}\frac{x^{m}}{m}$$ $$ \log_2(-\log_2(1-x)) = \frac{\log x}{\...
Kirill's user avatar
  • 11.4k
9 votes

Why in scientific papers convergence of finite difference and finite volume schemes is tested using multiple norms ($l_1$, $l_2$ and $l_{\infty}$)?

In finite dimensional spaces (say, in $\mathbb R^n$), all norms are equivalent and as a consequence, if something converges with a specific rate in one norm, it also converges with the same rate in ...
Wolfgang Bangerth's user avatar
7 votes
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Global convergence in trust region algorithm

Standard terminology in nonlinear (i.e., derivative-based) optimization is that "global convergence" means "convergence to a stationary point no matter where you start from" (where the limit may in ...
Christian Clason's user avatar
7 votes

Which absolute and/or relative stopping criteria do use for Newton's method?

Some of the confusion comes about because mathematicians often consider artificial functions that happen to have size $O(1)$ whereas practitioners use functions and variables that just happen to be ...
Wolfgang Bangerth's user avatar
7 votes
Accepted

Convergence-test for ODE approximates wrong limit

The Euler method has global error order 1, not 2 To get step size $h=1/N$, you need $N$ steps, which gives $N+1$ nodes in the time subdivision. Currently you compare sequences with step sizes $\frac1{...
Lutz Lehmann's user avatar
  • 6,109
6 votes

Scaling/nondimensionalization for numerical optimization

One thing that makes your non-dimensionalized ODE confusing is that you use the same symbols for dimensionalized and nondimensionalized variables, even though they are different variables. Consider ...
Kirill's user avatar
  • 11.4k
6 votes

Analytical convergent sequence and numerical divergent sequence

In my opinion, an example could be the calculation of the minimal solution of a three term recurrence relation (TTRR) $$y_{n+1} +a_{n}y_{n}+b_{n}y_{n-1} = 0, \quad n=1,2,3,\ldots$$. For example, the ...
GertVdE's user avatar
  • 6,159
6 votes

Accuracy issues with Arpack in Julia for eigenvalues of smallest magnitude

Looks like everything is working relatively OK? Your matrix is of order 1e10, so residuals of 1e-4 are actually close to machine precision. The convergence criterion is indeed violated, but not by ...
Antoine Levitt's user avatar
6 votes

Why in scientific papers convergence of finite difference and finite volume schemes is tested using multiple norms ($l_1$, $l_2$ and $l_{\infty}$)?

Both your questions have been addressed in the answer of @WolfgangBangerth. Here is an applied example of the behaviour of the different norms in the context of polynomial approximation (which is for ...
davidhigh's user avatar
  • 3,177
6 votes

Stability of Euler forward method

The solution of $\frac{du}{dt} = Au$ is $u(t) = \exp(tA)u(0)$, and explicit Euler approximates $\exp(tA)$ using $\lim_{n\to\infty} \left(I+\frac{t}{n}A\right)^n$. Of course in practice you cannot ...
lightxbulb's user avatar
  • 2,197
5 votes
Accepted

First-order ODE scheme implementation giving less than first-order convergence?

Let me modify slighly your ODE to see the essence of the problem. Let $$\left\{\begin{array}{l}y'=-y \\ y(0) = 1 \end{array} \right. \qquad t\in[0,T]\tag{*}$$ the ODE to be solved. Everyone knows that ...
HBR's user avatar
  • 1,648
5 votes
Accepted

Convergence rate Jacobi/Gauss-Seidel with mesh resolution

The convergence rate that is mentioned here is in the sense that the error in iteration $k$ and $k-1$ are related by $$ \| x^{(k)} - x^\ast\| \le r \| x^{(k-1)} - x^\ast\|, $$ which implies that $$ ...
Wolfgang Bangerth's user avatar
5 votes

How does gmres method iteration behave for this non-diagonalizable matrix?

Estimates for GMRES convergence based on eigenvalue distribution often implicitly assume that the matrix is normal. Sometimes the convergence rate is still provable in an asymptotic sense in the non-...
Reid.Atcheson's user avatar
5 votes
Accepted

Why FEM for incompressible materials is ill-posed?

For incompressible materials, as the extent of incompressibility increases, the bulk modulus approaches infinity (for the isotropic case). This is what causes ill-conditioning. Consider the case of ...
Chenna K's user avatar
  • 944
5 votes

Why is the definition of convergence different for root finding algorithms as compared to sequences?

This is not the definition of convergence. It is the definition of the convergence rate -- that is, how fast the sequence converges to a limit. In other words, the definition you quote can only be ...
Wolfgang Bangerth's user avatar
4 votes

What causes periodic humps in residual plots?

These humps occur in the context of optimization when the solution is approached but we then go past it. This frequently occurs in gradient descent with momentum, where they appear regularly on plots ...
Simply Beautiful Art's user avatar
4 votes

Does the convergence of finite element have limit?

The overall error of a finite element solution often has many components. For example, these may include (i) the discretization error due to a finite mesh and/or finite polynomial degree, (ii) the ...
Wolfgang Bangerth's user avatar
4 votes
Accepted

$O(h^2)$ convergence for Elliptic PDE

You appear (and correct me if I am wrong, i do not wish to presume) to be confused on the notion of what is an order of convergence. The order of convergence is the order at which your approximate ...
BlaB's user avatar
  • 1,157
4 votes

First-order ODE scheme implementation giving less than first-order convergence?

I cannot tell you whether that is the cause for your diminished order, but it is asking for trouble to add other approximations than just the one you want to do on purpose. In the current context, ...
Wolfgang Bangerth's user avatar
4 votes

Can I convert CUDA core to CPU core and use it as cpu core while running any program?

I assume, that you have a code that works on a standard CPU. I am not particularly familiar with MQL and Metatrader, but I don't think the answer will be different. For compilable languages, the ...
Anton Menshov's user avatar
  • 8,692
4 votes
Accepted

Asymptotic error of forward Euler

When we say that Euler method is first order accurate, it means that for a class of ode with sufficiently smooth solutions, the error will be at most $O(h)$ and there is at least one ode for which it ...
cfdlab's user avatar
  • 3,028
4 votes

$H^1$-convergence rate of finite element method for Poisson equation, depending on element order

There is a bug in your code :-) First, going down in each column, you have cases where the error becomes larger again, and this should clearly not happen because the finite element spaces are nested ...
Wolfgang Bangerth's user avatar
4 votes

Testing the time dependent Schrodinger Equation with an analytical solution?

It depends what you consider as an analytical solution. If you consider a Fourier series as an analytical solution, you have this: $$i \hbar \frac{\partial \Psi (\mathbf{r},t)}{\partial t} = -\frac{\...
Mithridates the Great's user avatar
4 votes
Accepted

Which absolute and/or relative stopping criteria do use for Newton's method?

This is not a definite answer, but I highly recommend the work of Deuflhard on Newton methods (e.g. this report, this book, and another report I cannot find just yet). The basic idea is the following: ...
Laurent90's user avatar
  • 1,943
4 votes
Accepted

Step3 in deal.II - Convergence of the mean

$h$ is a measure of the mesh size. In the example, they are using rectangular elements. For which a commonly used measure of the mesh size is the length of the largest diagonal. Looking at the table ...
Abdullah Ali Sivas's user avatar

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