# Questions tagged [numerical-analysis-of-pde]

The numerical-analysis-of-pde tag has no usage guidance.

100
questions

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16
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### Coonvergence rate for the clamped plate problem when approximating with polygonal domains

I'm trying to understand Ridgeway Scott's "A survey of displacement methods for the plate bending problem" [1]. In chapter four he is talking about polygonal approximation and states that ...

0
votes

0
answers

25
views

### Peak search for NLS equation solutions

By solving the initial value problem of the NLS equation, I can get a matrix of numerical solutions. I hope to perform a peak search on the matrix of this numerical solution, that is, there are ...

3
votes

0
answers

75
views

### Rate of convergence of mollified distributions in Besov spaces with negative regularity

Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...

0
votes

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47
views

### Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)

What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...

1
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0
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24
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### Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)

Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...

0
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0
answers

33
views

### Convergence of numerical scheme for HJB equation

Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is:
Consistent
Stable
Monotony
...

0
votes

0
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69
views

### How to solve with FEM a semilinear elliptic equation?

I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...

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133
views

### Relative bounds for vorticity

Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...

2
votes

0
answers

67
views

### Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?

Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...

3
votes

1
answer

240
views

### How to generate a random function with conditions?

The background is as follows:
I consider the following differential equation
$$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$
where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...

1
vote

0
answers

292
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### Solutions of a Gauss–Codazzi-like system of nonlinear PDEs

Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...

3
votes

2
answers

424
views

### Inconsistency in determinability of the solution of a linear first order PDE

Consider the following differential equation:
$$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$
with $u(x,0)=f(x)$. The solution of \eqref{1}, ...

7
votes

2
answers

731
views

### What are dissipative PDEs?

I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...

1
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0
answers

37
views

### Studying the evolution of laplacian in NS equation

The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...

0
votes

1
answer

173
views

### Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?

The split-step method is a numerical method that can be used to solve a nonlinear PDE (https://en.wikipedia.org/wiki/Split-step_method). Even Wikipedia does not refer to the original authors (F.D. ...

1
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0
answers

51
views

### Error estimates for inhomogeneous semidiscrete PDE

I have the following semidiscrete problem on a meshed domain $U_h$. Let
$V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and
$V_{h\partial}$ be ...

3
votes

0
answers

69
views

### Confusion with implementation of PDE constraint Bayesiain inverse problem

Consider a PDE,
$$\partial_t u -a \nabla u - ru (1-u) = 0$$
at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...

1
vote

1
answer

105
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### How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]

Crossposted at SciComp SE
I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method.
I find ...

2
votes

0
answers

28
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### Free programs suggestions to simulate parabolic EDPs

I'm interested in learning how to computationally simulate the behavior of parabolic partial differential equations, but I don't know where to start, what are the best free programs to use and where ...

2
votes

0
answers

315
views

### Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]

Here it is an Advection-Diffusion equation in 2D:
$$
\frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...

2
votes

0
answers

116
views

### Approximating solutions to Monge-Ampere from optimal transport plans

I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...

0
votes

0
answers

66
views

### Computing and isotopy of curves in $\mathbb{R}^3$

Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length.
The (stationary) string can be modelled by a unit speed curve:
$$[0,1] \...

1
vote

1
answer

209
views

### PDE involving curl

Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE
$$\dfrac{\partial}{\partial t}\...

4
votes

2
answers

555
views

### How to compute $\sin(\frac{d}{dx})f(x)$?

Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...

2
votes

0
answers

84
views

### Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications

What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...

1
vote

0
answers

75
views

### Highy non-linear PDE involving directional derivative

Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed
\begin{equation}\label{ConvoDef}
\left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...

2
votes

2
answers

304
views

### $H^s$ norm of non-integer power of functions

Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.
My ...

1
vote

0
answers

61
views

### Flux that can be represented by low and high resolution schemes

In the wiki page of Flux limiter, it writes:
If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...

2
votes

0
answers

96
views

### Derivation of the Cahn-Hilliard PDE from the point of view of finite difference methods

Consider the Cahn-Hilliard equation
$$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$
defined on your favorite domain. I'm looking for a literature reference that formally ...

0
votes

1
answer

110
views

### FEM based solution to parabolic problem

Consider the problem
$$
\begin{cases}
u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega
\end{cases}
$$
...

2
votes

0
answers

125
views

### Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition

$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$
\begin{cases}
\partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...

3
votes

0
answers

61
views

### How I can distibute values over the computational cells?

I am an engineering student and I try to solve the fluid equations over a given set of computational cells. I have a mathematical question about a field I am currently studying, precisely the ...

1
vote

0
answers

44
views

### P1-finite element as convolution of P0-finite element

For a vector $u\in\mathbf{R}^N$ let's denote $\pi_N(u)$ the unique piecwise linear and $1$-periodic function matching the components of $u$ on the discretization $x_k = \frac{k}{N}$ of the unit ...

1
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0
answers

58
views

### How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...

1
vote

0
answers

147
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### Semilinear PDE - BSDE presentation via Feynman Kac formula

For a semilinear PDE, we usually have this FBSDE representation:
$\mathcal{X}_t=\mathcal{X}_0+\int^t_0 \mu (s,\mathcal{X}_s)\, ds\, +\int_0^t \sigma (s,\mathcal{X}_s)dW_s,\quad 0\leq t\leq T, \\
Y_t = ...

1
vote

1
answer

92
views

### Typo in error a-priori estimate in a discontinuous Galerkin paper?

I'm looking at this famous paper which is available in the link below:
Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...

3
votes

0
answers

60
views

### Tuning parameters of PDEs given a set of data

I am interested in doing statistical inference in the context of PDEs. Loosely speaking, the kind of problem I have in mine is the following.
Problem setting
Let $(t_i, x_i, y_i) \in \mathbb{R} \...

2
votes

0
answers

71
views

### Dense matrix vs sparse matrix, when they have same number of nonzero elements

I came across a new way in the literature to solve PDE problems numerically, which is called 'Patch Reconstruction'. One example paper is: Li, R., Sun, Z., Yang, F., & Yang, Z. (2019). A finite ...

3
votes

2
answers

359
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### Questions for the non-linear PDE $2u_t=\log(-u_{xx})$

Consider the PDE as follows :
$$2u_t=\log(-u_{xx}), \quad \forall (t,x)\in [0,1)\times (-1,1)$$
with the terminal and boundary conditions
$$u(1,x)=0,\quad \forall -1<x<1 \quad\quad \mbox{and} \...

3
votes

0
answers

79
views

### What is the purpose of converting a level-set function into a signed distance function?

In the paper Electrical impedance tomography using level set representation and total variational regularization, the authors tried to implement an iterative algorithm to find the interface of two ...

3
votes

0
answers

115
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### Smoothly connecting PDEs with finite differences

A PDE with non-smooth inhomogeneity
Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$.
I'm numerically solving the inhomogeneous PDE
\begin{...

6
votes

1
answer

653
views

### What is the big-O complexity of solving the sparse Laplace equation in the plane?

In MATLAB, you can get a 2d Laplacian via A = delsq(numgrid('S',N)); yielding a matrix $A$ that is $n \times n$ with $n = O(N^2)$, for a square domain discretized ...

1
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0
answers

86
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### Shape derivative at manifold $M$ in direction $v$ is equal to the shape derivative at $\partial M$ in drection $\langle v,n\rangle n$

Let $\tau>0$ and $d\in\mathbb N$.
Definiton 1$\:\:\:$If $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)\in C^0([0,\tau],\mathbb R^d)$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...

3
votes

1
answer

283
views

### Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold

Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...

1
vote

0
answers

74
views

### Show that the support of the shape gradient $\nabla\mathcal F(\Omega)$ is contained in $\overline\Omega$

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta)...

0
votes

0
answers

134
views

### Non linear second order PDE involving max operator (Dynamic Programming)

I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$)
$$ v(x,t) = \max\Big\{|x|\,,\,v(x,t)+dt\Big(v_t(x,t)+\frac{1}{2(t+1)}v_{xx}(x,t)\Big) \Big\} - \...

3
votes

1
answer

400
views

### Weird claims and conclusions in "Introduction to Shape Optimization"

I'm trying to understand the notions of Euler and Hadamard derivatives of shape functionals. All the lecture notes and papers on this topic that I've found seem to build up on the books Shapes and ...

1
vote

0
answers

122
views

### Von Neumann analysis on a finite difference hyperbolic scheme

I am doing a Von Neumann analysis on a staggered finite difference scheme (for Maxwell's Equations).
The finite difference scheme is:
$$
\mathbf{u}_v|^{n+2}_{i,j} - \mathbf{u}_v|^{n}_{i,j} = - A \frac{...

1
vote

0
answers

115
views

### Differentiation under the integral sign for a $L^1$-valued function (shape derivative)

Let
$d\in\mathbb N$;
$U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$
$E:=\bigcup_{\Omega\...

3
votes

1
answer

406
views

### Prove of the shape-derivative identity relating the shape and material derivative of a shape-dependent function

I've started reading about shape optimization. Most of the concepts I've encountered so far (such as the shape derivatives of domain and boundary integrals and the corresponding) seem to be complex, ...