# Tag Info

Accepted

### 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 ...
• 55.9k

### 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 ...
• 2,032
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### Accuracy loss in single-precision Euclidean norm computation

The squares are harmless (as long as they don't overflow/underflow), because the relative perturbation they introduce is of the order of the machine precision $u\approx 10^{-8}$. Your troubles here ...
• 11.8k
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### How can I derive the a priori error estimate for a symmetric bilinear form using lagrange finite elements?

To extend VorKir's basically correct answer, to derive a priori error estimate for finite element approximations, you need to stack the following three Lego blocks: The well-posedness of the weak ...
• 12.3k
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### Understanding the diffusion error of numerical schemes

Just figured it out. The solution diffuses BY A FACTOR OF $|G|^n$ after n time steps. So in my example, for the low frequency case, if let's say the original signal was a square wave advecting to the ...
• 111

### 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 ...
Accepted

• 55.9k
Accepted

### Relationship between global and local error?

This is not a formal proof, but intuitively you're making a truncation error at each time step of $O(\Delta t^k)$. Your global error will be the error at the time horizon $T = N\Delta t$. This means ...

### How can I derive the a priori error estimate for a symmetric bilinear form using lagrange finite elements?

I am not pretending to give the best point of view, but I think of the following general lines. To obtain the error estimates in classical cases, you use as ingredients: interpolation error (which ...
• 254
Accepted

### How can I compute the difference between shape function and dual solution in dwr?

You seem to be confused about what is a vector and what is a scalar. $z_h$ and $\phi_i$ are both scalars if evaluates at quadrature points. The fact that the $z_i$ form a vector really has no ...
• 55.9k
Yes. This behavior is to be expected and normal. When you are computing with a small value for $\mathrm dx$ then, to compute the difference quotient, you are subtracting two numbers that are nearly ...