# Questions tagged [error-estimation]

For questions about determining the error caused by specific computational procedures, approximations, or numerical representations.

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### A priori estimates in finite elements for inhomogeneous heat equation

Consider the problem $$\partial_t u-\Delta u = f\\ u(\Sigma_1)=f_D\\ \partial_\nu u (\Sigma_2)=f_N\\u(0)=u_0$$ where the sides of the space-time cylinder $\Sigma_i$ are disjoint (one of them could be ...
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### How to maximize effectiveness of 2 check bits overseeing 6 data bits?

Having a single parity bit gives a Hamming distance of 2, so only one single bit can be corrupted to reliably detect an error. The next step I know of would be a Hamming code, which uses at least 3 ...
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### "A posteriori" estimates for finite difference methods

Suppose I have a PDE in a rectangular domain that I am solving numerically via a finite difference method. How do I answer the question, "How fine do I need to make the grid to so that my ...
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### Multigrid/Two-grid method restriction and prolongation of residual

Starting from the problem $Au=f$, I'm not sure that I understand why a coarse grid solution is implemented on the coarse grid residual $r_c^{(k)}=P^Tr^{(k)}$, with $r^{(k)}=f-Au^{(k)}$ and $P^T$ ...
143 views

### Accuracy loss in single-precision Euclidean norm computation

I do hydrodynamics simulations with Fortran and recently I met with this issue: I have a single-precision array b of length ...
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### What is the ERRCON parameter in rkqs?

Ive take a course in computational physics and was asked to implement some numerical methods to solve ODES. I was reading up on the algorithms described in the textbook: NUMERICAL RECIPES IN FORTRAN. ...
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### Initial condition precision

Is there a way to have an estimate of the error propagated on an ode numerical solution by the error of the initial conditions? I suppose this depend on the numerical method used and on the problem ...
1 vote
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### 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 (...
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### Perturbation problem using Runge-Kutta 4

I'm trying to evaluate the perturbations magnitude between 2 body orbiting a central one in three dimensions. In order to do this I need to have an estimate of the error, which I did using Richardson ...
217 views

### Comparing numerical solutions with very different time grids

I've read an article (Long-term integrations and stability of planetary orbits in our Solar system) in which the authors solved the problem of the absence of an analytical solution for the solar ...
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### Error too large in leapfrog method for solving the wave equation of a vibrating string

I have been trying to figure out what I did wrong for the last two days. I do not know if I actually did something wrong or if the error is supposed to be this large in usual leapfrog problems. I ...
137 views

### What are the advantages and disadvantages of using norm error control in the MATLAB ODE suit?

In MATLAB's ODE suit, there seem to be two basic methods of controlling the Local Truncation Error (LTE) of the ODE which the user can choose from, namely: The absolute error control (default), ...
87 views

### Dividing functions over a wide range-

I try to solve a system of coupled equations, where a very nasty division operation occurs. In fact, I need to compute a derivative of two exponential decaying functions. Let's illustrate this with ...
1 vote
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Let $\Omega\subset \mathbb{R}^{2}$ and $\tau_{h} = \{\Omega_{k}\}_{k=1}^{N}$ be a triangulation of $\Omega$. The $L^2$ error for a FEM approximation $u_{h}$ is given by: $|| u-u_{h} ||_{L^2} = \sqrt{ ... 3 votes 1 answer 90 views ### Cauchy Lorentzian simulation on FFT with oscillation Recently I do simulation on Lorentzian Function with FFT Lorentzian Function is 2a/(x**2+a**2) ... 0 votes 0 answers 83 views ### Error in Monte Carlo integration I am looking for a concise description to help me understand the error for Monte Carlo Integration using Uniform Sampling and Importance Sampling For Importance Sampling I have that the error is just ... 3 votes 1 answer 483 views ### Calculating Error for Poisson Equation using Successive Over-Relaxation technique, Python I am trying to solve the Poisson Equation$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 32(x(x-1) + y(y-1))$for a 61x61 grid using Python3 with boundary conditions being$T=... 1 vote
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### Recovery of smoothed continuous stresses using the Z^2 error estimator

I have trouble implementing the $Z^2$ error estimator (The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique by Zienkiewicz and Zhu). For this, I am ...
262 views

### Implementation of Z^2 error estimator in Abaqus for adaptive mesh refinement

Currently, I am working on a remeshing routine for my simulations (Abaqus 6.14-1) using python scripts. The simulation deals with the Brinell indentation test and as the remeshing software I use Gmsh ...
1 vote
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### About the discrete $H^1$ norm

I need to understand what is the right expression for the "discrete $H^1$-norm of a function v(which of course need to be in $H^1$). By definition, $$||v||_{H^1}^2 = ||\nabla v||_{2}^2 + ||v||_2^2$$ ...
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### Correct relative error for Comparison of levelsets outside a constrained region

I am working on the estimation of tumor extent outside of the region which is visible in MRT/CT/DTI images. I want to compare two methods, which approximate the tumor density profile. Lets say the ...
293 views

### The proper way to assess the error of Jacobi iteration (for 2D Poisson equation)?

Motivation: I'm using 2D regular grid (it's actually a quadtree but I can still treat it as a finite difference thing if I weight-average the solution over smaller scale cells for the purpose of ...
1 vote
I am trying to fit some data points $d_i$ to a non-linear model function $m_i$, which depends on a number of fit parameters $f_k$ (I want to determine these) and also on some known, constant values $... 2 votes 1 answer 113 views ### Step size updating scheme adaptive embedded RK methods If I have a RK method$y$of order$p$and a RK method$z$of order$p-1$I have read I can estimate the local error as$r_{n+1} = y_{n+1} - z_{n+1}$. First of all I don't see how this estimates the ... 3 votes 2 answers 297 views ### Asymptotic error of forward Euler I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS).... 1 vote 1 answer 376 views ### Lower bound for bilinear form in FEM I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries. I did some research and found:$\$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(... 