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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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30 votes
3 answers
5k 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 (...
Paul's user avatar
  • 12k
7 votes
3 answers
451 views

Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method

I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally. The first subsystem includes ...
Johann's user avatar
  • 71
4 votes
2 answers
11k views

Error in result of finite-difference approximation when refining

I have calculated the first derivative of following equation using Euler method (first order), Three point Finite Difference method (second order) and Four point Finite Difference method (third order)....
AGN's user avatar
  • 544
1 vote
0 answers
221 views

Oscillating convergence in my Resilient BackPropagation (RPROP) implementation

I have implemented in matlab a neural network that uses rprop's algorithm to update its weights. Strangely the error on the training set does not converge to a local minimum, but oscillates. Here is ...
Teo Teo's user avatar
  • 11
13 votes
2 answers
7k 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}...
Paul's user avatar
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11 votes
2 answers
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 $b \in im(A)$ (...
olamundo's user avatar
  • 599
10 votes
1 answer
1k views

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

I know that the piecewise linear finite element approximation $u_h$ of $$ \Delta u(x)=f(x)\quad\text{in }U\\ u(x)=0\quad\text{on }\partial U $$ satisfies $$ \|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)} $...
Bananach's user avatar
  • 799
9 votes
1 answer
2k views

Convergence rate vs convergence order

I'm a bit confused about the concepts of convergence rate and convergence order. Let me first give you the definitions we use: [sorry for the English, it's all self translated] Let $x^{*}$ be our ...
xotix's user avatar
  • 241
8 votes
1 answer
359 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)...
Paul's user avatar
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8 votes
1 answer
603 views

Finite element convergence rates for mixed problems

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh. ...
Lukas Bystricky's user avatar
7 votes
1 answer
468 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?
learningmath's user avatar
4 votes
1 answer
316 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
Dan's user avatar
  • 116
4 votes
1 answer
608 views

Finite element error for second order ODE at nodes equal to zero

I coded a finite element method with linear basis elements for the problem $$-u'' = f(x), x\in[0,1], u(0) = u(1) = 0$$ The nodes are uniformly spaced and I will denote them as $x_i$. I initially ...
Kurt's user avatar
  • 513
3 votes
3 answers
621 views

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

I wanted to verify my FEM-program by applying the method of manufactured solutions, while solving the Poisson equation in two dimensions using the continuous Galerkin method $$-\nabla^2u=f$$ with $$u=...
arc_lupus's user avatar
  • 563
1 vote
1 answer
7k views

Numerically determining convergence order of Euler's method

I need to numerically determine the convergence order of Euler's method for various step-sizes. I am unsure how to go about doing this. Here is the question: Problem statement: $\frac{dy}{dt}=\alpha ...
user avatar
1 vote
2 answers
3k views

Finding rate of convergence by curve fitting in Matlab

I have some data: number of nodes $N$ and error in energy norm corresponing to it. I have seen in some references that the rate of convergence is reported by $$\| u-u_h\| _E=CN^{\alpha} $$ How can ...
Rosa's user avatar
  • 533
1 vote
1 answer
245 views

Converge rate analysis: issue with time convergence

I have written a code which solves the incompressible formulation of the Navier-Stokes equations. It uses high-order methods both for time and spatial derivatives. I have been conducting convergence ...
solalito's user avatar
  • 337