30 votes
Accepted

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 ...
user avatar
17 votes
Accepted

Difference between l2 norm and L2 norm

Both norms are similar in that they are induced by the scalar product of the respective Hilbert space, but they differ because the different spaces are endowed with different inner products: For $\...
user avatar
14 votes
Accepted

Finite elements $W^{1,\infty}$ error estimates

Chapter 8 of Brenner and Scott's Mathematical Theory of Finite Element Methods is devoted to this subject. In particular, theorem 8.1.11 and the corollary give you that $ \|u - u_h\|_{W^1_\infty} \le ...
user avatar
12 votes
Accepted

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. ...
user avatar
  • 4,567
12 votes
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 ...
user avatar
11 votes

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 ...
user avatar
  • 1,839
10 votes

diagonalization of matrix - omitting small matrix elements

It's not implementation-dependent in the sense that this is a mathematical operation performed on your matrix. However, it is very much matrix-dependent. If your matrix is diagonalizable and $A=XDX^{-...
user avatar
  • 11.4k
10 votes
Accepted

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 ...
user avatar
9 votes
Accepted

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 ...
user avatar
8 votes
Accepted

How do error estimates scale for multidimensional cubature?

Using product quadrature rules for multi-dimensional integrals suffers from the so-called curse of dimensionality. An $O(N^{-2})$ accurate rule using N evaluations in one-dimension is generally $O(N^{...
user avatar
  • 221
7 votes
Accepted

Correct way of computing norm $L_2$ for a finite difference scheme

Use equation (2), equation (1) is wrong. Technically the $L^2$ norm (upper case "L") is an integral norm of a function defined as $$ \left|\left|f(x)\right|\right|_2 = \sqrt{ \int_\Omega |f(x)|^2 dx}....
user avatar
7 votes
Accepted

Error of interpolating polynomial

For a polynomial interpolant we have the formula for the pointwise error $$ E_n(x) = f(x) - p_n(x) = \frac{f^{(n+1)}}{(n+1)!}\prod_{i=0}^n(x-x_i),$$ where the $x_i$ are the interpolation knots. In ...
user avatar
6 votes
Accepted

diagonalization of matrix - omitting small matrix elements

There is a field of study known as eigenvalue sensitivity analysis or eigenvalue perturbation analysis that allows you to estimate the effect of small matrix perturbations on the eigenvalues and ...
user avatar
  • 3,013
6 votes
Accepted

Error on a integral quantity with noise

Assuming that you mean the following inequality in your prompt $$ |F_\mathrm{true}(x) - F(x)| \le |dF(x)| \qquad \forall x,$$ a simple bound for $dM$ is the following $$ dM \equiv | M_\mathrm{true} - ...
user avatar
5 votes

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.
user avatar
  • 151
5 votes

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: $$ \omega = \nabla \times \mathbf{u} = \frac{\delta v}{\delta x} - \frac{\delta u}{\delta y} $$ ...
user avatar
5 votes
Accepted

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 ...
user avatar
5 votes
Accepted

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 ...
user avatar
5 votes
Accepted

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,...
user avatar
5 votes
Accepted

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 ...
user avatar
  • 3,601
5 votes
Accepted

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 ...
user avatar
4 votes

Numeric Quadrature with Derivatives

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 ...
user avatar
4 votes

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 ...
user avatar
4 votes

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 ...
user avatar
4 votes
Accepted

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 ...
user avatar
4 votes

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 ...
4 votes
Accepted

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 ...
user avatar
  • 10.8k
4 votes
Accepted

$L^2$-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 ...
user avatar
  • 1,246
4 votes
Accepted

error estimator VS error indicator in the context of FEM error estimation

In general, for a finite element approximation $u_h$ of the exact (but unknown) solution $u$ of a partial differential equation, it would of course be nice if we could compute something like $\|u-u_h\|...
user avatar
4 votes
Accepted

Taylor expansion of error - Finite elements approximation

The question you ask is simpler than you may think. For any mesh size $h$, you can compute $\lambda(h)$. If this dependence is continuous, then you can do a Taylor expansion of $\lambda(h)$ around $h=...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible