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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3
votes
2answers
919 views

Convergence problem for Poisson equation with periodic BC

I have written Poisson solvers using two different methods: A classic Jacobi scheme and one using the multigrid solver Hypre. I made up a couple of test cases ensuring the validity of those solvers. ...
0
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0answers
119 views

Getting euclidean distance between vector A and C without anyway of retrieving them when their distances with a common vector B is known

Motivation: My plan is to get the overall euclidean distance matrix for all the vectors in N number of dataset. Each dataset is basically an array of n-dimensional points. For e.g: A dataset can be ...
0
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1answer
107 views

interface value on the error equation

https://www.jstor.org/stable/pdf/2157482.pdf, here I have a problem in last equation of (2.6) in section (2.1). When they are considering error equation on the interface $\Gamma$ they get $e_v^{(n)} = ...
2
votes
1answer
95 views

Discontinuous Galerkin order of convergence on arbirary refined mesh: step-12 deal.ii tutorial

I'm learning DG methods and in order to practice a little bit I'm using the deal.ii library. In particular, I'm looking at step-12, where they solve $$\operatorname{div}(\beta u) = 0$$ $$u = g_D \text{...
8
votes
3answers
335 views

Accuracy issues with Arpack in Julia for eigenvalues of smallest magnitude

Following the documentation of Julia's Arpack package (Cf. https://julialinearalgebra.github.io/Arpack.jl/stable/eigs/) I have computed some largest and smallest magnitude eigenvalues of sparse ...
6
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1answer
224 views

What causes periodic humps in residual plots?

When using many iterative methods, whether for solving linear systems, looking for steady-state convergence in CFD, etc., the semilog plot of the residual often shows "humps" as the residual decays. ...
4
votes
3answers
400 views

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

I saw many stopping criteria for Newton's method all around Web and books. Some are defined from the residuals: of either current iteration only: $$ \|f(\mathbf{x}^{(k)})\| \leq \epsilon $$ (https://...
1
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0answers
59 views

Confused on how Method of Manufactured solutions works?

I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique so my starting equation ...
2
votes
2answers
154 views

Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous

I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too. ...
1
vote
1answer
136 views

FEM solution for Poisson is not exact at nodes

Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$ I know that the FEM solution (...
1
vote
1answer
126 views

Step3 in deal.II - Convergence of the mean

I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
2
votes
1answer
231 views

Solving Cahn-Hilliard equation using semi-implicit Fourier spectral methods

So, I have written both a C and python code to solve the 2D Cahn-Hilliard equation: \begin{equation} \frac{\partial c}{\partial t} = \nabla^2\left(c^3 - c - \kappa\nabla^2c\right) \end{equation} ...
0
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1answer
64 views

2 point BVP solver: how to compute errors

Background I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/ I have build my own solver in Python to solve the 2 point BVP: $$ \epsilon u''+u(u'-1) =0 , \\ u(0)...
0
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0answers
25 views

Convergence of the Roothaan-Hall equations

Suppose that we are given a time-independent quantum mechanical system whose wavefunction depends on three space coordinates. Let $F$ be the Fock operator of the system. Suppose also that we have a ...
15
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1answer
3k views

Comparing Jacobi and Gauss-Seidel methods for nonlinear iterations

It is well known that for certain linear systems Jacobi and Gauss-Seidel iterative methods have the same convergence behavior, e.g. Stein-Rosenberg Theorem. I am wondering if similar results exist for ...
3
votes
4answers
326 views

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

Recently, I have been studied my lessons about gmres iteration, probably the most popular iteration method for general large sparse linear system of equations Ax=b. And the convergence is obtained ...
1
vote
1answer
66 views

Rate of convergence - Stochastic Euler Method

The absolute error criterion of the pathwise approximation of an Ito process $X$ by an Euler approximation $Y$ is: $$ \epsilon=E\left(\left|X_{T}-Y(T)\right|\right) $$ We shall say that a time-...
2
votes
2answers
145 views

Accelerating convergence of a generalized continued fraction

I wish to compute $$ \frac{1}{1 + \frac{1^3}{1 + \frac{2^3}{1 + \frac{3^3}{1+\cdots} } } } $$ to high accuracy. To start, I tried computing $$ \frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
0
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0answers
64 views

L2 norm optimization problem

I have an optimization problem where i need to find an image x, that is very close to x' such that: monitor(x') is valid but monitor(x) is invalid. (output is valid when the neural network output is ...
0
votes
1answer
113 views

Finite elements with CFL condition - How to obtain correct order of convergence

I have discretized a PDE with continuous finite element method in spatial variable and with implicit Euler or Crank-Nicolson in temporal variable. In both cases, I have error estimates in $L_2$ norm ...
0
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0answers
38 views

Devising Convergent Numerical Scheme for PDE

I'm currently looking at the PDE \begin{align*} u_t + \left[x(1-y) - (1-x)\right]u_x - (1-y) u_y + (z-xy) u_z = (z-xy) u_{xy} - (1-x)u& \\ \end{align*} with \begin{align*} u(x,y,z,0) = 1& \\ ...
1
vote
1answer
82 views

Finite Difference for Advection Equation With Source

I'm trying to find a convergent finite difference scheme for the PDE \begin{equation} \begin{split} u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\ u(x,0) &= 1 \\ u(1,t) &= 1. \\ \...
1
vote
1answer
63 views

Where could error terms that blow up in SWE come from?

I have been working on a solver for shallow water equations with reflective boundary conditions. I have found that it diverges very fast. As a workaround I noticed yesterday that if I smooth the ...
4
votes
1answer
148 views

When does L-BFGS outperform GD?

In practice, L-BFGS is frequently held comparably to other inexact QN methods, and it provides a middle ground of sorts between Hestenes–Stiefel CG and BFGS as memory goes from zero to infinity (...
0
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0answers
37 views

How can the choice of coarsening factor affect Multigrid's convergence?

The linear system $Ax=b$ is coming from the discretization of an elliptic PDE. Multigrid method is used in order to solve it. Suppose $c_0$ is the coarsening factor on level 0 and $c_m$ the coarsening ...
1
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0answers
52 views

Multigrid Reduction In Time Convergence

I am trying to solve a 2D dynamic linear elasticity model parallel in time using Xbraid. The spatial domain is [0,1]x[0,1] and time domain [0,1]. For time integration I am using a backward Euler ...
3
votes
1answer
83 views

Numerical integration in time for finite elements

I am trying to solve $M\ddot{u}=-Ku+F_\text{ext}$ for a 2D linear elastic model with $M$ be the mass matrix,$K$ the stiffness matrix and $F_\text{ext}$ the external load vector coming from a uniformly ...
1
vote
0answers
68 views

Convergence of Conjugate Gradient Algorithm

I am trying to solve a linear elasticity model using finite element discretization in a rectangle domain [0,1]x[0,1]. For the solution of the the linear system $Ku=F$ I am using the CG algorithm. ...
3
votes
1answer
112 views

Choosing an appropriate time step for a discrete & continuous dynamics simulation

I have inherited of a flight dynamics simulation in C++ which represents a small drone with it's autopilot, actuator dynamics and a solid state IMU. Hence, it is composed of a few models, some ...
1
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1answer
74 views

Issue solving nonlinear equation containing a quotient

I have a coupled set of PDEs that need to be solved as part of a larger problem. I am currently approaching this by computing spatial derivatives with finite differences and using PETSc's nonlinear ...
5
votes
1answer
153 views

Accurate computation of Gauss-Kuzmin entropy

The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as $$ P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
1
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0answers
41 views

Effect of reducing flux consistency order at boundary on convergence order

Consider the 1D nonstationary convection-diffusion PDE $$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
1
vote
1answer
430 views

what is non-asymptotic convergence?

I guess convergence in general means it is in asymptotic sense but what does non-asymptotic convergence mean?. Can someone please explain with an example?
2
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0answers
35 views

Convergent Finite Difference Scheme for Parabolic Equation

Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$ where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$. In Strikwerda's book, the ADI scehme \begin{align*} \left( 1 - \frac{...
2
votes
1answer
130 views

Best way to check if SOR solution has converged for 2d matrix

I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10. I can obtain the fully converged solution for ...
5
votes
1answer
106 views

Non-conforming bi-linear finite element

The four-noded bi-linear rectangle element, which sometimes goes under the name Melosh element, is non-conforming unless the element sides are aligned. Out of curiosity I have implemented this element ...
2
votes
4answers
171 views

Testing the time dependent Schrodinger Equation with an analytical solution?

I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later, but I want to know the convergence rate of my numerical solver in different grid size and numerical methods....
3
votes
1answer
110 views

Determine stability of an algorithm?

This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of ...
1
vote
1answer
98 views

Unexpectedly Slow Convergence Implicit Euler

I'm solving the coupled ODE $$ \left[\begin{array}{c}x^\prime(z)\\p_x^\prime(z)\end{array}\right] = C(z)\cdot\left[\begin{array}{c}x(z)\\p_x(z)\end{array}\right] = \left[\begin{array}{cc}0& A(z)\\...
1
vote
1answer
79 views

How the number of pre-smoothing and post-smoothing steps affect the asymtotic convergence rate of geometrical Multigrid?

Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately?
3
votes
2answers
1k views

Numerical Sensitivity in Density of States of Tight-binding model

I'm working with the tight-binding model, and I'm trying to learn the basics of how to compute the Density of States (DOS) $N(E)$ numerically. The DOS is given by $$N(E) = \frac{1}{N}\sum_k \delta(...
0
votes
0answers
63 views

A name for a numerical phenomena when using numerical methods

I have a nonlinear solver for equation $g= c_1f(x_1,y_1)+c_2f(x_2,y_2)$. Note that $c_1$ is much bigger than $c_2$. So after using Levenberg–Marquardt algorithm, I could only get $x_1$, $y_1$ and $...
2
votes
2answers
746 views

Implementing Gelfand’s formula for the spectral radius in Python - lack of convergence

For context: Gelfand's formula for the spectral radius is $\lim_{k\rightarrow \infty}|A^k|^{1/k}$ where $|\cdot|$ is any well-defined operator norm. I naively coded a function to calculate the $k$th ...
3
votes
0answers
64 views

Why GA convergence curves continue as two parallel lines?

I'm working on a optimization problem and using GA algorithm (in MATLAB, ga function). As you know MATLAB plots GA result with two curves, one for the best values and other to show the mean values ...
11
votes
1answer
199 views

Implications of thermodynamic inconsistency in CFD calculations

During my PhD work, I had to use tabulated values of thermodynamic properties of gases in some Computational Fluid Dynamics (CFD in short) simulations. My tables are discretized in temperature and ...
3
votes
1answer
173 views

Conjugate Gradient for singular 2D poisson finite element with Neumann Boundary Conditions

Heavily edited question after I realised partly what the problem was I have programmed a simple 2D square finite element solution to the Poisson equation $-\Delta u = f$ The source function ...
2
votes
1answer
253 views

Number of GMRES iterations increase when stepping forward in time, using the Newton method

I am solving a system of nonlinear time-dependent equations using the Newton method in a finite-element-setting, i.e. first I create the jacobian matrix for the current time, and afterwards I try to ...
7
votes
0answers
81 views

Why not use the preconditioned residual as termination criterion for preconditioned CG?

I have a Poisson equation with wildly varying material parameters (1 .. 1000), wildly varying element sizes (5 nm .. 100 um) and some quite anisotropic (tetrahedral) elements (100 nm x 100 um). I use (...
8
votes
1answer
244 views

Should we always expect FEM error plots to be straight lines?

The error estimates in FEM are usually of the form $$||u^h-u||\leq Ch.$$ Taking logarithm on both sides, we obtain $$\log ||u^h-u||\leq \log C + \log h.$$ This estimate implies that the error lies ...
0
votes
0answers
105 views

Why does GMRES converge much slower for large Dirichlet boundary conditions?

I'm trying to numerically solve a simple Laplace equation in 2D, with a nonlinear source term: $\nabla^2 u = u^2$ with boundary conditions as $u=0$ everywhere except for $y=1$ where $u=u_0$. I'm ...