Questions tagged [error-estimation]
For questions about determining the error caused by specific computational procedures, approximations, or numerical representations.
183 questions
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Does the Aubin-Nitsche trick apply for time-dependent PDEs?
I have the occasion to be reviewing some of the "basics" of the finite element method. In particular, I am interested in several technical details related to the combination of the FEM with ...
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What’s the advantage of knowing the exact error in polynomial integration over estimated bounds?
What is the significance of having an explicit symbolic error formula for polynomials, instead of relying on traditional numerical error bounds?
Take a simple example: integrating the function x² over ...
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How do you assess convergence or error when using quasi-random Monte Carlo?
When using standard pseudo-random Monte Carlo integration, we can estimate the error using the Central Limit Theorem, and the convergence rate is typically proportional to $1/\sqrt{N}$. However, when ...
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Euler method error estimate for heat equation for constant time-step
I am solving the heat-equation $\frac{\partial u}{\partial t} = \alpha\nabla^2 u$ on the domain $\Omega = [0,1]^2$ and interval $t\in[0,1\times10^{-2}]$ with homogeneous Dirichlet boundary conditions ...
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residual error of traveling wave: runga kutta vs generalized alpha
Obtaining 1D traveling wave solution using FEA of $u_t+u_x=0$ 0<t<1 with periodic boundary conditions, a satisfactory steady shift of a wave using runga kutta is obtained as opposed to a ...
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books or resources of error estimates of convection and wave PDEs solved by FEA
For FEA solutions of convection PDE $u_t+u_x=0$ and wave PDE $u_{tt}+u_{xx}=0$, how are the errors estimated with various element orders and time integrations? Looking for books or resources.
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How to decrease the error for the solution of my high dimensional ode?
I am dealing with a very high dimensional (think thousands of dimensions) non-linear (not even Lipschitz) ode system. I am experiencing some phenomenon which I am trying to explain by means of an ...
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Computing global error of finite difference approximation without knowing the exact solution
I am reviewing my notes of finite-difference approximations and would like to clarify how global error is calculated. Consider an ODE system of the form $Au = f$. Per my textbook, the difference $$E = ...
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How to derive the $\tau$ in the SUPG method for compressible flow?
I want to understand the formulation and derivation of $\tau$ matrix in the SUPG stablization term, and we sometimes get(there are many variation of SUPG stablization)$$
\begin{aligned}
& \tau_c=\...
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How to estimate a finite element function with exponential matrix
I would like to know if there are any similar theorems related to the mass lumping finite element method:
For any function $\boldsymbol{u} \in H^m_{per}(\Omega)$
$$
\|I_h e^{\tau \Delta} \boldsymbol{u}...
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Understanding proof of the error bound for Simpson's quadrature rule
I have found the following proof of the error bound for Simpson's quadrature rule:
Using Newton's interpolation method, we derive a cubic polynomial $p_3(x)$ that interpolates $f(x)$ at the points $a, ...
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How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE
Setting
I am solving for $u(x,y,t)$ the wave equation $\partial_{tt} u - \partial_{xx} u = f$ on $(x,y)\in\Omega=[0,1]\times \mathbb{R}$ by splitting it into an equivalent first order system:
$$\...
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numerical schemes for 1D PDE: for smaller grid size there is an increased roundoff error, larger size more truncation, so sweet spot in between?
I had a discussion with a colleague today. He claimed that usually for a general numerical scheme for solving a general 1D PDE, for smaller grid size there is an increased roundoff error because of ...
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Can I reduce my simulation error with a staggered grid, postprocessing and compatibility equation feedback?
What I did
Using the finite difference method, I solved with a certain amount of error the following system of hyperbolic partial differential equations in cylindrical coordinates (the problem is ...
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Order of Error - Confusion: Clarifying Constraints on Constants and Determining Order of Error
I'm struggling to determine the order of error when considering the error value denoted by $\text{err}$ in relation to the variable $h$. Specifically, I aim to ascertain the value of $x$ in the ...
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