# Tag Info

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### What are differences between 'a priori' and 'posteriori' error estimate in numerical analysis?

Error estimates usually have the form $$\|u - u_h\| \leq C(h),$$ where $u$ is the exact solution you are interested in, $u_h$ is a computed approximate solution, $h$ is an approximation parameter you ...
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### What are the relative benefits of using Adams-Moulton over Adams-Bashforth algorithm?

Adams-Moulton method is significantly more stable. The analogy used when I was taught the difference is the same as extrapolation and interpolation. Interpolation is relatively safe numerically. ...
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### What does optimal convergence rate mean? How to prove whether a numerical method, e.g. FEM, has an optimal convergence rate or not?

The term is commonly used in the following way in the finite element context: Let's assume that $u\in V$ is the exact solution of the PDE, and $u_h\in V_h$ the finite element approximation. Then we ...
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### What is “tolerance” in ODE45 in Matlab?

And, it is my understanding that the 4 and the 5 are for the order of the global and local error, respectively. Your understanding is wrong. The local error of a Runge–Kutta method of order $n$ is ...
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### Difference between l2 norm and L2 norm

The 2-norm for sequences is denoted by $\ell^2$. For functions on the real line $L^2$ is the standard notation of the 2-norm.
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### discrete definitions of curl $\nabla \times F$?

Continuous It looks like you only need 2d curl, so let's start with a simpler continuous definition: $$\omega = \nabla \times \mathbf{u} = \frac{\delta v}{\delta x} - \frac{\delta u}{\delta y}$$ ...
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### Step size updating scheme adaptive embedded RK methods

First of all I don't see how this estimates the local error defined as the error we make in a single step using correct previous values. What does a lower order method have anything to do with ...
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### How to compute the $L^{2}$ error of the gradient in the Finite Element Method

Like any other integral, you evaluate the error integral through quadrature. Your suggestion at the end of the question is exactly what you would do: The norm of the gradient error square is a sum of ...
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### Initial condition precision

In general, the initial error will already grow exponentially even if you solved the ODE exactly. The Lyapunov exponent (related to the Lipschitz condition) will tell you have fast. Separately, though,...
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### Error in Simpson's 3/8 rule is higher than that of Simpson's 1/3 rule

The simple segment-$[a,b]$-with-midpoints error formulas are for the 1/3 rule $$E=-\frac{(b-a)^5}{90·2^5}f^{(4)}(\zeta)$$ and for the 3/8 rule $$E=-\frac{(b-a)^5}{90·72}f^{(4)}(\zeta)$$ Now the ...
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### Different sources of error in Finite Element computations

The triangle inequality is your friend. Let's ignore the issue of boundary approximation for a moment, then you are computing a solution with inexact linear solver. Let's call it $u_h$. We will call ...
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Although this thread is quite old, I thought it might be useful to have a reference to a peer-reviewed paper for generalizations of some common quadrature rules. Nenad Ujevic, "A generalization of ...
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### Measurement error library

The two basic approaches I can think of are: interval arithmetic something like polynomial chaos expansions I'm sure there are other approaches out there. In brief, interval arithmetic redefines ...
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### Machine precision and local error

I'm using double precision values in the calculation. Is it feasible of me to demand of my integrator that the difference between the 4th and 5th order estimates be < 1 x 10 ^-16 if machine ...
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### Calculation of error

After a many days of discussions, your problems are 1) you choose too simple problem for which you reach the machine zero and therefore cannot observe normal properties 2) perhaps you do not use the ...

### How to avoid the round-off errors in the larger calculations?

Below are some tips to reduce the effect of round off errors. A short method is to increment the floating point precision, for example from float to double, but many times this is too expensive or not ...
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### What should I put on the paper to show the correctness and convergence of my solution?

You should definitely do a mesh refinement study (some conferences and journals require this). You should compare the results of this study to a known, analytical solution. You can either get this ...
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### $L^2$-error in FEM: how to compute integral over reference element?

You can write $$\int_{0}^1 (f - \Pi_h^1(f))^2 dx = \sum_{i=0}^{N_e} \int_{e} (f - \Pi_h^1(f))^2 dx$$ where $e = [x_k, x_{k+1}]$ is an interval (triangle as you say). Method A (simple, no reference ...
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