# Tag Info

Accepted

• 11.4k

### 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 ...
• 52.4k
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" ...
• 2,055
Accepted

• 52.4k

### 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-...
• 3,243
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 ...
• 810

### Error in result of finite-difference approximation when refining

You've chosen a very specific example where the function in question has a single extremely narrow peak and is almost zero everywhere else in the domain. This is an example of a stiff problem where ...
• 4,571

### 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 ...
Accepted

### What should I put on the paper to show the correctness and convergence of my solution?

You should definitely do a mesh refinement study (some conferences and journals require this). You should compare the results of this study to a known, analytical solution. You can either get this ...
• 10.9k
Accepted

What about $$F(x,y) = \frac{x^2}{y} + \frac{\epsilon}{x^2}, \qquad \nabla F(x,y) = \Big( \frac{2x}{y}-\frac{2\epsilon}{x^3}, -\frac{x^2}{y^2} \Big), \qquad x,y>0$$ with the sequence $(x_k,y_k) = (... • 11.4k 4 votes Accepted ###$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 ...
• 1,137

### 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 ...
• 52.4k

### 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, ...
• 52.4k
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 ...