Questions tagged [laplacian]
The laplacian tag has no usage guidance.
28
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2D cavity flow: Adding zero Dirichlet condition to the top boundary leads to NAN
I am following this guide to create a simple cfd solver for incompressible 2D fluid in a square cavity. The following Matlab code sample from the guide creates a Laplacian (coefficient) matrix with ...
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1
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Computing the Fiedler vector of a large, sparse graph
I have a sparse, undirected and unweighted graph $G$ of size $n$, with $n$ on the order of say several million. I would like to compute the Fiedler vector $f$ of $G$, which is the eigenvector ...
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Reference request: graph Laplacian approximation for domains/manifolds
Is there any reference on solving the heat equation on irregular geometries via creating a mesh and using the graph Laplacian instead of FEM techniques? (Convergence to real solution).
That is to say, ...
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Fast Fourier Transform on Meshes
I have a (closed, manifold, oriented) triangular mesh for which I build a matrix $L\in\mathbb{R}^{n\times n}$ discretising the negated Laplace-Beltrami operator. The matrix $L$ is symmetric positive ...
- 2,352
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Solving the wave equation for a circular membrane in polar cordinates
As you see this mode is not right, unless for what i understand
And the initial conditions were
...
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Constructing generalized Laplacian matrix?
I am staring intently at this paper by Botsch and Kobbelt.
In particular, I want to make the matrix specified in equation 5. I am trying to understand the specific computations I must instruct a ...
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Discrete Laplacian operator with finite element discretization
Let a function $u \in H^1_0(\Omega)$ defined by its values at the mesh nodes. Can we compute its Laplacian using the matrix resulting from the finite elements discretization of Laplace's equation? I ...
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Discrete laplacian 9 point
I am trying to write a code for 9 point discrete laplacian. I would like to write a matrix and solve the linear system $AU=F$ with gradient conjugate method.
I wrote the matrix this way
...
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1
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Applying spectral method to a damped, driven 2D bending-mode wave equation on an irregular domain with heterogeneous boundary conditions
I'm trying to model some two-dimensional waves, and am unsure how to combine my boundary conditions with spectral methods. The PDE I'd like to explore resembles the equations for damped, driven ...
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4
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ON the Kronecker product form of the laplacian matrix
It's well known that if we use 2nd order, centered, finite differences for the Laplace operator, we have that the matrix can be written as $$K=I \otimes A + A \otimes I $$ where $I$ is the identity ...
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Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?
If one has the Poisson problem (assume $\int_{\Omega} f = 0$ and $\int_{\Omega} u = 0$):
\begin{alignat}{3}
\Delta u(x) &= f(x), &\quad&x\in\Omega \\
\partial_nu(x) &= 0, &\quad&...
- 2,352
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Numerical estimation of eigenfunctions of Laplacian
Consider the Laplace equation,
$$
\nabla^2 f(r,\theta,\phi) = 0
$$
in spherical coordinates. We know that the solution to this equation can be derived analytically, and is given by,
$$
f(r,\theta,\phi)...
- 1,068
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253
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Numerical instability in the inverse Laplace transform
I have a problem with Laplace inversion and my function is not numerically stable for the Laplace inverse, but I do not understand the cause of this problem.
Here is my code and graph of this problem. ...
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Eigenvectors of Laplacian
I am studying introduction to Multigrid methods.
In all tutorials, authors write that eigenvectors of Laplacian (1D, finite difference) are given as
$w_k(x_i) = \sin(k \pi x_i),$
where $x_i$ is a ...
- 13
1
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3
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preconditioner for Laplace "without" boundary values
I'm looking at solving systems with the FEM discretization
$$
-\int_\Omega (\Delta u) v = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot\nabla u) v.
$$
without applying Dirichlet- or ...
- 3,136
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3D laplacian operator
I have been unable to find the equivalent of the 5-point stencil finite differences for the Laplacian operator.
In 2 dimensions for me it is clear that, using the finite difference method: $$
\nabla_{...
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How avoid square shape with Laplacian operator in reaction diffusion calculations?
I have used different variants of the Laplacian operator (div grad) using 4, 8, 12, 20 and 24 of the closest points. I get problems due to the chosen coordinate system and the discretization of the ...
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What happens when I use a conjugate gradient solver with a symmetric positive semi-definite matrix?
I have a symmetric positive semi-definite matrix, i.e., a laplacian and wonder what may happen when I use a CG solver, that is an algorithm for positive definite matrices.
What happens when the ...
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2
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Kronecker product representation of the finite difference laplacian
The laplacian equation when discretized gives a system of linear equations that can then be solved. See the answer to this question: https://math.stackexchange.com/questions/3120948/discretization-...
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How to account for a corner node with zero-flux condition at an extrapolated distance
I am trying to implement a numerical solver and am having troubles dealing with boundary conditions, especially in the corners.
I have a 2D mesh, and on the left I have a Dirichlet condition, on the ...
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510
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High-accuracy numerical differentiation
I have a $200 \times 200$ matrix representing the values taken by a function over an equally spaced grid. I would like to perform derivatives on it.
I am interested in its gradient (i.e. its ...
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Factorize laplacian in terms of first derivative matrix
I am trying to factorize the following Laplacian matrix in terms of $ D^TD$, D is the first derivative matrix.
The tridiagonal form of the secon derivative matrix using Neumann boundary condition is ...
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Access optimized data structure for representing integer lattice
Consider the integer lattice in $2d$, namely the set $\mathbb{Z}^2 = \{(x,y): x,y\in \mathbb{Z}\}$, and let $u:\mathbb{Z}^2 \to \mathbb{R} $ be a function defined on some bounded subset of $\mathbb{Z}^...
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Simultaneous update to barycenters
Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex.
Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...
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What is the finite-difference representation of the Laplacian operator with periodic boundary conditions? [duplicate]
I am using a central-difference scheme to solve the eigenvalue problem
$$\frac{d^2}{dx^2}u = \lambda u$$
on a unit interval with periodic boundary conditions. My understanding is the eigenvectors $...
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Correct eigenfunctions of Laplace operator by Finite Differences
I am trying to compute the eigenfunctions of the Laplace operator, i.e. finding $u$ in
$$ -\nabla^2 u = \lambda u .$$
For now I am trying to do this in 1D, so
$$ \nabla^2 = \partial_{xx} .$$
I am ...
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3
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2
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Solving Ax = b with sparse A and sparse b
Let's suppose I'm numerically solving the Poisson equation for a delta function source:
$$ \nabla^2 f(x) = \delta(x-x') $$
I can represent the Laplacian $\nabla^2$ using the finite difference method ...
- 203
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How to discretize Laplacian near refinement boundary
Given a block structured grid with a refinement factor of two, what are some of the common ways to discretize a Laplacian near a refinement boundary? See also the picture below, in which $u_H$ and $...
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