Questions tagged [convergence]

Questions related to whether the sequence of iterates generated by an iterative method has one or more limit points, and if those limit points have the correct properties.

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4
votes
1answer
163 views

What is the origin of the preasymptotic convergence behaviour in FEM?

When you have fine-scale features (e.g. boundary layers) in the solution, its FEM approximation on coarse meshes converge at strange apparent rates. Looking at Cea's lemma, is this behaviour because ...
1
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3answers
126 views

Guess the final term of a converging series [closed]

I have a non-linear equation that converges, and reaches suitable accuracy after around 20 steps, however each step is very expensive to calculate. The series are never quite the same, but they are ...
3
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0answers
92 views

analytic or numeric integral of diverging function

I'm trying to carry out the following integral numerically $$\int_{r_\mathrm{in}}^{r_\mathrm{out}} \Sigma\left(r'\right) \frac{r'}{r} \left( \frac{1}{r-r'}\, E(L) + \frac{1}{r+r'}\, K(L) \right) \...
2
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1answer
997 views

Error norm in finite difference calculation

I've used an explicit finite difference scheme to model the 1D time dependent temperature distribution in a friction weld. I want to now verify the consistency and convergence of my algorithm. I have ...
6
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3answers
1k views

Significance of p-convergence studies

Consider a method (e.g., FEM) with variable approximation order $p$. Now, we know that the optimal order of convergence is given by $$e = C h^{p+1},$$ where $h$ denotes the mesh size and a constant $C$...
5
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0answers
353 views

Understanding the meaning of Computational Order of Convergence

I am a postgraduate student with interest in numerical methods for solving nonlinear systems of equations. I have read some papers that discussed about 'computational order of convergence' for some ...
7
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2answers
1k views

Convergence/stagnation of BiCGStab(l)

I am solving 3D time-harmonic Maxwell FDFD problems (which result in huge sparse linear systems) using BiCGStab(l). I have tried out a bunch of different methods and for my specific use case, it seems ...
2
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2answers
122 views

Normalizing error when data passes through zero

Given a time-dependent measure of error, $e(t)$, and a time-dependent function to normalize it with, $f(t)$, one would report the normalized error as $$E(t) = \left|\frac{e(t)}{f(t)}\right|.$$ ...
2
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2answers
2k views

Does energy decrease with basis set size in density functional theory?

Based on the variational principle, one might expect that the ground state energy of a density functional theory (DFT) calculation will decrease as the basis set size increases. (As I understand it, ...
1
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0answers
108 views

Hatree-Fock, reasons for convergence/ non-convergence

I'm new here so please forgive me if I lack proper stack exchange etiquette. So, I was wondering if anyone here could provide insight on a problem that I am running into with with a Hartree-Fock ...
5
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0answers
84 views

Non-convergance when calculating temperature/heat flows through a section of rock

I am attempting to calculate temperature of section of rock in the earth as a function of vertical position in the rock and time. Along with it I am calculating the heat flow through the rock as a ...
16
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1answer
1k views

Convergence rate of FFT Poisson solver

What is the theoretical convergence rate for an FFT Poison solver? I am solving a Poisson equation: $$\nabla^2 V_H(x, y, z) = -4\pi n(x, y, z)$$ with $$n(x, y, z) = {3\over\pi} ((x-1)^2 + (y-1)^2 + (...
4
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1answer
644 views

Is it possible to ensure global convergence of a fixed point iteration?

Suppose I have a fixed point iteration of the form $$x_{n+1}=f(x_n).$$ Suppose further that after some initial testing, I found that it does not converge to an a priori known fixed point $x^*$. I ...
2
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1answer
1k views

Convergence of GMRES

From what I understand the GMRES method is (using Arnoldi Iterations/Modified Gram-Schmidt): The first vector of the Krylov subspace span of A is the normalized vector $\frac{\vec b - A\vec x_0} {|| \...
2
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0answers
966 views

Newton Iteration method convergence

I wrote a Python code which solves a second degree nonlinear differential equation using the Newton iteration method. The code converges to a 2-cycle within 50 or so iterations. The cycle only ...
0
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0answers
3k views

Stationary solution converge but time dependent doesn't

I've coupled a COMSOL model for fluid dynamics with a very simple pde that model the transport of humidity in air. When I solve it for the stationary case, the solution converge easily, but when I ...
3
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4answers
2k views

What is the meaning of “preasymptotic” and “superconvergent”?

Precisely the title of the question. I have encountered these terms in two areas: conjugate gradient method, and adaptive finite elements.
1
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3answers
870 views

necessary and sufficient tests to show order of convergence for the numerical method

I would like to know what are the necessary and sufficient tests one has to perform in order to show the convergence of the algorithm. I have not found a good reference to state for that as I am ...
8
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1answer
323 views

Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde

Suppose I have the parabolic system $$u_t=\nabla\cdot(k(x)\nabla u)+f,\quad (x,t)\in\Omega\times I$$ with Dirichlet boundary conditions $$u=g, \quad x\in\partial\Omega$$ and initial condition $$u(x,t)...
12
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2answers
5k views

Strategies for Newton's Method when the Jacobian at the solution is singular

I'm trying to solve the following system of equations for the variables $P,x_1$ and $x_2$ (all else are constants): $$\frac{A(1-P)}{2}-k_1x_1=0 \\ \frac{AP}{2}-k_2x_2=0 \\ \frac{(1-P)(r_1+x_1)^4}{L_1}...
13
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3answers
9k views

Understanding the “rate of convergence” for iterative methods

According to Wikipedia the rate of convergence is expressed as a specific ratio of vector norms. I'm trying to understand the difference between "linear" and "quadratic" rates, at different points of ...
8
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1answer
595 views

Demonstrating that the time step size is small enough in a code with automatic step size selection

I recently inherited a large body of legacy code that solves a very stiff, transient problem. I would like to demonstrate that the spatial and temporal step sizes are small enough that the ...
2
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1answer
200 views

Root Convergence rate of Iterative Scheme

I have an iterative sequence for optimizing an EM (Expectation Maximization) algorithm based loss function $L(X)$ with $t$ being the iteration number as: $X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is ...
6
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2answers
4k views

When to stop Gauss-Seidel-iterations?

I want to have an estimation, that my solution has an error, let's say less than 1e-8. Usually, I stop the Gauss-Seidel algorithm, when the residual is "small enough" and this is already the problem. ...
9
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3answers
1k views

Basin of attraction for Newton's method

Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution. What is "sufficiently close"? Is there literature ...
2
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2answers
466 views

Finite volume solution of electrostatics using magnetic vector potential

I would like to solve for the electric potential and magnetic vector potential using the finite volume method (collocated grid). My equations are: $\nabla\cdot(\sigma\nabla\phi)=0$ $\nabla \cdot \...
3
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1answer
2k views

Gauss-Seidel iterations node spacing

I am working on an assignment where I am determining the temperature distribution of a chip on a substrate. When I decrease the nodal spacing the results change drastically. The smaller the nodal ...
5
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2answers
489 views

Proving convergence of 5 point scheme for the Poisson equation

So, we are solving the Biharmonic equation ($\Delta^2 u = f$) on a rectangle by solving the Poisson equation ($\nabla^2 u = f$) two times. We have nice boundary conditions, $u = 0$ and $\Delta u = 0$ ...
10
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2answers
1k views

Which iterative linear solvers converge for positive semidefinite matrices?

I want to know which of the classic linear solvers (e.g Gauss-Seidel, Jacobi, SOR) are guaranteed to converge for the problem $Ax=b$ where $A$ is positive semi definite and of course $b \in im(A)$ (...
10
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1answer
292 views

Why do we have to rerun the CFD solver for higher Reynolds number?

I started to learn OpenFOAM from the Cavity tutorial which is provided at the web-site. When experimenting with different Reynolds numbers, in section "2.1.8.2 Running the code", tutorial says to ...
11
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1answer
464 views

How to establish that an iterative method for large linear systems is convergent in practice?

In computational science we often encounter large linear systems which we are required to solve by some (efficient) means, e.g. by either direct or iterative methods. If we focus on the latter, how ...
11
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3answers
1k views

Why does iteratively solving the Hartree-Fock equations result in convergence?

In the Hartree-Fock self-consistent field method of solving the time-independent electronic Schroedinger equation, we seek to minimize the ground state energy, $E_{0}$, of a system of electrons in an ...
5
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2answers
1k views

Sufficient conditions to ensure convergence of the conjugate gradient method

I know that a conjugate gradient method is guaranteed to converge to the solution of a linear system if the matrix is positive definite. I'm working with a family of matrices that have the following ...
25
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3answers
3k views

What is the principle behind the convergence of Krylov subspace methods for solving linear systems of equations?

As I understand it, there are two major categories of iterative methods for solving linear systems of equations: Stationary Methods (Jacobi, Gauss-Seidel, SOR, Multigrid) Krylov Subspace methods (...
7
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1answer
365 views

Richardson extrapolation for strong rate of convergence of SDE

Is it possible to apply Richardson extrapolation with Euler-Maruyama scheme to improve strong rate of convergence of stochastic differential equations?
5
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1answer
215 views

Discretization of Classical Density Functional Theory (CDFT) problems

I formulate questions about ion densities in biological and materials problems using Classical Density Functional Theory, as in this paper which is also on the arXiv. The discretization in that paper ...
19
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2answers
2k views

How to determine if a numerical solution to a PDE is converging to a continuum solution?

The Lax equivalence theorem states that consistency and stability of a numerical scheme for a linear initial value problem is a necessary and sufficient condition for convergence. But for nonlinear ...

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