Also called numerical integration, quadrature refers to the approximation of an integral made by evaluating the integrand at a finite number of points.
3
votes
2answers
70 views
FEniCS: custom quadrature rule
For the numerical integration of reaction terms in my PDE on a 2D triangular mesh, I would like to use the scheme
$$
\int_T \Phi(x) \approx \frac{|T|}{3} (\Phi(x_0) + \Phi(x_1) + \Phi(x_2))
$$
where ...
5
votes
0answers
59 views
Finding quadrature weights for a given set of points? How to select points such that all weights are positive?
Currently, I fit a Finite Element solution of a PDE on a spectral basis. The matrices ($R^{25000\times 2000}$) of the corresponding system of linear equations are highly ill-conditioned ($\kappa ...
5
votes
1answer
78 views
Method selection for numeric quadrature
Several families of methods exist for numeric quadrature. If I have a specific class of integrands how do I select the ideal method?
What are the relevant questions to ask both about the integrand ...
11
votes
3answers
165 views
Numeric Quadrature with Derivatives
Most numerical methods for quadrature treat the integrand as a black-box function. What if we have more information? In particular, what benefit, if any, can we derive from knowing the first few ...
7
votes
1answer
69 views
Integrating a harmonic function over a tetrahedron
Say I have a function $f : \mathbf{R}^3 \to \mathbf{R}$ that I wish to integrate over a tetrahedron $T \subset \mathbf{R}^3$. If $f$ was arbitrary, Gauss quadrature would be a good solution, but I ...
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votes
1answer
355 views
Finite Element Method: 2-D Poisson's Equation in Matlab, Gaussian quadrature
I'm having trouble understanding how to code 2-D Poisson's Equation with Dirichlet boundary conditions. What I have thus far is
Constructed square mesh with triangular elements
Assembled stiffness ...
13
votes
1answer
144 views
Does transforming $J_0(x)\to\int\cos(x\sin\theta)$ help with numerical integration?
I've heard anecdotally that when one is trying to numerically do an integral of the form
$$\int_0^\infty f(x) J_0(x)\,\mathrm{d}x$$
with $f(x)$ smooth and well-behaved (e.g. not itself highly ...
14
votes
3answers
385 views
Method for numerical integration of difficult oscillatory integral
I need to numerically evaluate the integral below:
$$\int_0^\infty \mathrm{sinc}'(xr) r \sqrt{E(r)} dr$$
where $E(r) = r^4 (\lambda\sqrt{\kappa^2+r^2})^{-\nu-5/2} ...
10
votes
2answers
237 views
Numerical Integration - handling NaNs (C / Fortran)
I am dealing with a tricky integral that exhibits NaNs at certain values near zero and at the moment I am dealing with them quite crudely using an ISNAN statement which sets the integrand to zero when ...
10
votes
1answer
165 views
How to integrate polynomial expression over 3D 4-node element?
I want to integrate a polynomial expression over a 4-node element in 3D. Several books on FEA cover the case where integrating is performed over an arbitrary flat 4-noned element.
The usual procedure ...
8
votes
3answers
327 views
Numeric integration of multi-dimensional integral with known boundaries
I have a (2-dimensional) improper integral
$$I=\int_A \frac{W(x,y)}{F(x,y)}\,\mbox{d}x\mbox{d}y$$
where the domain of integration $A$ is smaller than $x=[-1,1]$, $y=[-1,1]$ but further restricted by ...
5
votes
2answers
315 views
Methods to solve a double integral
I want to solve the following expression (used to obtain an analytic solution to a current distribution inside a workpiece):
$$a_{mn} = -\frac{\frac{4}{ab} \int_0^a \int_0^b ...
11
votes
5answers
438 views
How can I approximate an improper integral?
I have a function $f(x,y,z)$ such that
$\int_{R^3} f(x,y,z)dV$
is finite, and I want to approximate this integral.
I'm familiar with quadrature rules and monte carlo approximations of integrals, ...
6
votes
2answers
357 views
Higher-order numerical integration on a triangle/tetrahedron/simplex
Let $T$ be a triangle and let $f$ be a smooth function on $T$.
We can use mid-point quadrature $\int f dx \approx |T|\cdot f(x_M)$, where $x_M$ is the middle-point of $T$.
Can you provide me with (a ...
6
votes
3answers
175 views
Numerical integration of compactly supported function on a triangle
as the title suggests I'm trying to compute the integral of a compactly supported function (Wendland's quintic polynomial) on a triangle. Notice, that the center of the function is somewhere in 3-D ...
2
votes
1answer
403 views
Error calculation in trapezoidal rule
If we use the composite trapezoidal rule, then what is the least number of divisions $N$ for which the error of the integral $\int^1_0{e^{-x}}dx$ doesn't exceed $\frac{1}{12}\times10^{-2}$.
My guess ...
5
votes
0answers
93 views
Suggestions for numerical integral over Pólya Distribution
This problem arises from a Bayesian statistical modeling project. In order to compute with my model, I need to perform an integration in which part of the integrand is the "Pólya" or ...
9
votes
1answer
153 views
Quadrature rules, methodologies, and references
There is at least one quite comprehensive encyclopaedia of quadrature rules that doesn't seem to have been updated in quite a while and has restricted access. This source refers to several classical ...
6
votes
1answer
216 views
What numerical quadrature to choose to integrate a function with singularities?
For example, I would like to numerically compute the $L^2$-norm of $\displaystyle u = \frac{1}{(x^2+y^2+z^2)^{1/3}}$ in some domain that includes zero, I tried Gauss quadrature and it fails, it is ...
5
votes
2answers
271 views
numerical integration with possible division by 'zero'
I am trying to integrate
$$\int^1_0 t^{2n+2}\exp\left({\frac{\alpha r_0}{t}}\right)dt$$
which is a simple transformation of
$$\int^{\infty}_1 x^{2n}\exp(-\alpha r_0 x)dx$$
using $t = \frac1{x}$ ...
15
votes
3answers
504 views
What's the state-of-the-art in highly oscillatory integral computation?
What's the state-of-the-art in the approximation of highly oscillatory integrals in both one dimension and higher dimensions to arbitrary precision?
