Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

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.

1
vote
1answer
38 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D. In order to both test the timestepping and the spatial discretisations I had a look at using ...
3
votes
0answers
68 views

Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
3
votes
0answers
30 views

Preconditioned residual converges, but true residual doesn't

I'm using Albany w/ Trilinos to solve an elasticity problem with thermal expansion mismatch. I'm using block GMRES with MueLu preconditioning. It works for problem size of several million dofs, but ...
4
votes
1answer
64 views

Convergence rate Jacobi/Gauss-Seidel with mesh resolution

In the book A Multigrid Tutorial - Briggs, Henson. McCormick in the beginning of Chapter 3, it is mentioned that ...because the convergence factor behaves as 1-$O(h^{2})$, the coarse grid ...
0
votes
0answers
25 views

Artificial Neuron, impossible for convergence?

I'm trying to learn AI and my idea was to build an artificial neuron network to construct the nonlinear generator. I initially tried to let each vertex have 5 dendrites to the next vertex, each ...
0
votes
1answer
77 views

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

I was using Metatrader5 and have designed a strategy for trading using MQL5 programming language. While I was running a Strategy Optimization process, I saw the it will need 10,00= iterations or ...
0
votes
1answer
43 views

Conflicting definition of limit point

This question was raised at a different place without sufficient answers. Definition 1: We say that a vector $x \in R^n$ is a limit point of a sequence $\{x_k\}$ in $R^n$ if there exists a ...
1
vote
0answers
44 views

Decrease in slope during convergence analysis

I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides. I used 5 refinements: $dx = dy = dz = ...
0
votes
0answers
60 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
31 views

Convergence rate of alternate convex search

Is there any reference regarding the convergence rate of alternate convex search? I tried to find it on google but got no luck. Alternatively, do u think it can be found/proved just like gradient ...
3
votes
1answer
66 views

Numerical Lax-Wendroff scheme order of convergence on Burgers equation

I was suggested to move that question here. The question to be as follows. Statement of the problem Is it possible to achieve the second order of convergence (OOC) of Lax-Wendroff (LxW) scheme ...
1
vote
0answers
32 views

Convergence rate assesment in space/time

I am solving a hyperbolic PDE (e.g. the shallow water equations) which depends upon $x$ and $t$. Typically, the overall convergence rate is calculated by comparing the numerical error in different ...
0
votes
1answer
58 views

How to report non-monotonic runtimes in convergence plots

Let's say I have an algorithm that can be tuned by a parameter $h>0$ and is expected to converge as $h\to 0$. I want to study the computational complexity of this algorithm, i.e., how the ...
1
vote
0answers
50 views

BFGS convergence problem

I would like first to state that it is beyond my capability to identify whether this is a BFGS issue or a R package problem. I've been doing some mixed logit regression using the R package "mlogit". ...
1
vote
0answers
70 views

Successive iteration method for solving eigenvalue ploblem

I have a question concerning the branch of successive iteration methods (Newton, Runge-Kutta). I definitely know (or can read in Wikipedia) the implementation of these methods. But I was wondering ...
3
votes
1answer
281 views

Scaling/nondimensionalization for numerical optimization

I have a numerical optimization problem that I am trying to scale appropriately, in order to allow for the solver to achieve faster and more accurate results. I found a paper here that had a short ...
2
votes
2answers
280 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. ...
1
vote
2answers
88 views

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

I am solving the initial value problem $$ \frac{d}{dt} (E C_g) = -\delta, \quad E(0) = E_0, $$ for $E$, where $E$ and $C_g$ are functions of $t$, $C_g$ is completely known, and $\delta$ is a function ...
1
vote
0answers
123 views

Accuracy of finite difference method for heat equation on a disk

To study an approximation for the heat equation $$\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}=f(r,\theta)$$ on the ...
5
votes
0answers
231 views

Convergence rate of Picard iterations

Given a first order ODE $y'(x)=f(x,y)$ with the initial condition $y(x_0)=x_0$ such that it satisfies Picard thoerem of existence and uniquness, one can compute the solution by Picard iterations : $$ ...
0
votes
1answer
150 views

Line search bracketing for proximal gradient. Is it good idea?

Maybe my question is obvious but i cannot find any good source which answers it I trying to learn about proximal gradient. One thing which is not clear for me is particular algorithm for line search. ...
2
votes
2answers
90 views

Relationship between global and local error?

In some cases I have seen that if the local error is: $Err = O (\Delta t^{p+1})$ where the global error is p. So if local error is 3, global will be 2. Does somebody know where it comes? For ...
4
votes
2answers
114 views

In numerical methods, eg, finite differencing approaches, does there exist convergent schemes that are not both consistent and stable?

In a book that our course is following this semester, the theorem given is only in one direction: if the scheme is both consistent and stable, then the scheme is convergent. However, since this ...
1
vote
1answer
119 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 ...
12
votes
1answer
193 views

Non-monotonic convergence in fixed-point problem

Background I am solving a variant of the Ornstein-Zernike equation from liquid theory. Abstractly, the problem can be represented as solving the fixed point problem $A c(r)=c(r)$, where $A$ is an ...
0
votes
0answers
90 views

how to do von neuman stability analysis for given equation

I have two coupled differential equations. Their implicit schemes were given below. is there anyone who will do "von neuman stability analysis". because i need to know in which conditions my solution ...
9
votes
1answer
220 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)} $...
3
votes
1answer
122 views

Computing rate and order of convergence

this is a follow up question to Convergence rate vs convergence order I guess the whole confusion about how rate vs. order of convergence also came from the implementations I saw. For example, for ...
8
votes
1answer
291 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 ...
3
votes
0answers
84 views

Finite difference method for coupled PDEs: optimizing performance (time step, iterations per step)

I'm solving coupled PDEs using finite difference method: Incompressible Navier-Stokes and the divergence-free induction equation (Maxwell's equations) with non-uniform electrical conductivity. The ...
1
vote
0answers
127 views

Diverged HDG solution for 2D incompressible Navier-Stokes test case at SMALL time step. Why?

I wrote a hybridizable Discontinuous Galerkin code for transient Navier-Stokes flow, with thanks to Martin Kronbichler and Scott Miller's step-51 code in Deal.II. The main algorithm is from Nguyen ...
2
votes
0answers
331 views

Numeric integration over Dirac delta

I'm trying to solve the following integral numerically. $$H(y) = \int dx \, f(x) \, \delta(g(x,y)).$$ For this I chose a representation of the delta-function and employ convergence with respect to $\...
0
votes
1answer
65 views

2nd order accurate finite difference method variable material properties near boundary

I'm aware that a 2nd order accurate finite-difference method using variable properties for central differencing can be written in a finite-volume type way: $$ \nabla \bullet (k \nabla f) = \frac{\...
1
vote
2answers
206 views

Does the convergence of finite element have limit?

I have been trying to solve a nonlinear PDE related to structural mechanics (nonlinear Timoshenko beam to be precise). I am doing both h-refinement and p-refinement to reach the solution. The ...
2
votes
0answers
108 views

Does applying the Newton-Raphson iteration for matrix reciprocal refine a matrix inverse from LU/GE?

This is a follow-up to this answer. Suppose you have a possibly very ill-conditioned matrix $A$, and you compute its inverse with LU/GE to get $X_{\text{lu}}\approx A^{-1}$. The Newton-Raphson ...
1
vote
0answers
43 views

Decreasing - increasing - stabilising $l_{2}$ norm

Let $\bar{x}$ denote the analytical solution of a PDE. Let $x^{(k)}$ be the solution at the $k^{th}$ iteration. The initial guess for the solution is $x^{(0)} = 0$. Let $r_{0} = ||\bar{x}-x^{(0)}||_{2}...
2
votes
2answers
195 views

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

I am trying to solve an elliptic PDE in 2-D: $$-\nabla^{2} u = 20tanh(10x-5)(10-10tanh(10x-5)^2) = f$$ I know that the solution is $u = tanh(10x-5)$ but I am unable to get $O(h^2)$ solution with a ...
3
votes
0answers
91 views

Strange convergence behavior of WENO5 for Hamilton--Jacobi equations

I have the following question. I have a code function that computes right-biased and left-biased approximations of the derivative of a function using WENO5 for Hamilton--Jacobi equations as described ...
5
votes
1answer
266 views

Global convergence in trust region algorithm

I was reading about TR methods and there are some terms, which are confusing for me. It says, method is globaly convergent. What does it really mean? Converges to global minima, or converges for ...
3
votes
1answer
194 views

Measure the convergence rate of a discretization of a wave equation

I'm currently trying to approximate the following type of wave equation (in weak formulation): Let $\Omega \subset \mathbb{R}^d$ ($d=2$) be some polygonal domain. We seek a function $u \in L^2\left(0,...
2
votes
1answer
80 views

Interpreting convergence study results, fixed CFL

I am trying to determine the order of my numerical method for resolving a fluid-structure interaction problem using the immersed boundary method. I am using Crank-Nicolson to resolve the fluid ...
2
votes
0answers
47 views

Dynamic Successive Over/under Relaxation (SOR) with several variables

I am solving a partial differential algebraic equation (PDAE) system which has the following dependent variables: $f=f(X,T)$ and $g=g(T)$, along with a few others My current method for coupling is ...
5
votes
1answer
138 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 ...
0
votes
1answer
72 views

Linear stationary iteration method

Suppose you are attempting to solve $Ax = b$ using linear stationary iteration method defined by $$x_k = G x_{k-1} + f$$ that is consistent with $Ax = b$, i.e., for which $f = (I - G)A^{-1}b$. Suppose ...
3
votes
1answer
75 views

Efficient solution of large systems of non linear algebraic equations

I have quite simple problem of FEM solution of 1D differential non-linear equation. Altough the problem itself is simple, the numerical solution of the arising system of non-linear algebraic equations ...
1
vote
0answers
60 views

Manufacturing a solution for non-smooth coefficients in elliptic problems

This question is a continuation of this answer (I can't comment) If we were going to manufacture a solution for a problem with discontinuous coefficients, I understand that the solution should have ...
5
votes
0answers
128 views

Nonlinear conjugate gradient restart threshold 1/10

Nocedal and Wright on Conjugate Gradient Methods, p. 123, describe a restart strategy ... whenever two consecutive gradients are far from orthogonal $\qquad {{| \nabla f_k^T \ \nabla f_{k-1} |} \...
0
votes
1answer
105 views

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

I am using FEM to do an assignment on a heat conduction problem on a complex domain, which needs me to get the variation of the temparature distribution subject to the variation of boundary conditions,...
3
votes
1answer
106 views

Problem with Richardson extrapolation method for weak convergence in SDE

I have implemented the Richardson extrapolation of the Euler-Maruyama method to 4th order, to estimate the moments of SDE. The Euler-Maruyama works, and I would expect the Richardson extrapolation to ...
6
votes
0answers
123 views

Are there any benefits of computable analysis to numerical algorithms

Computers can work only with computable numbers, while most of the algorithms are based on analysis of real numbers (real analysis). When I heard of the existence of computable analysis I ...