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

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

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### On singularly perturbed external differentiable equations with canards

Let $X$ be an internal set. A flexible function is a function $F:X \rightarrow E$. An internal function $f: X\rightarrow R$ is called a representative of $F$ if $f(x)\in F(x)$ for all $x\in X$. A ...

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17 views

### Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...

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44 views

### Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...

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109 views

### How to solve this integral equation?

$$g(p) = \frac{1}{p^3} + \frac{2}{p^2} \int_{p}^{1}{(u-p)g(u)du}$$
Need to find $g(p)$ for $p > 0$.
If there is no explicit solution, how to solve it numerically? Maple13 calculates bad results, ...

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**1**answer

68 views

### Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...

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32 views

### Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...

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**3**answers

515 views

### Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...

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22 views

### Role of polyhedral domain in convergence of finite element method

I am reading a paper by Diening and Kreuzer where they consider the convergence of finite element approximations for $p$-Laplace equation when using a certain algorithm.
In the paper, they assume ...

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**1**answer

249 views

### Naviers Stokes equation and machine learning

I am looking for a reference explaining how to solve Navier-Stokes numerically using Machine learning algorithms .
Thank you in advance for your help .

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**1**answer

69 views

### Numerical iterative methods for Poisson equation

Given a domain $\Omega \subset \Bbb R^n$ and $\Delta\varphi=f$ where $\varphi:\Bbb R^n \to \Bbb R$ is unknown and $f:\Omega\to \Bbb R$ is a blackbox function (for each $\bf x$ it provides $f({\bf x})$,...

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50 views

### $L^{1} $ estimate for $2^{nd}$ order elliptic boundary value problem

This is probably a classical question in numerical analysis of PDE (but I don't know the answer).
Suppose you are solving a traditional elliptic problem, for example, $u\in H^1_0(\Omega)$ (in a nice ...

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**2**answers

72 views

### Iterative method for $p$-Laplacian

Consider the following iterative procedure for solving the $p$-Laplace equation $\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$ with fixed Dirichlet boundary data:
$u_0$ is our initial guess, for ...

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**1**answer

52 views

### Simulating Fisher Equation (FKPP)

I'm researching about microbial growth (on 2D). I think that a microbial population can be modeled by the Fisher equation (any other suggestion is welcomed). My doubt is about how can I solve ...

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110 views

### A discrete-to-continuous approach to the Dirichlet principle?

Dirichlet principle: Let $\Omega \subset R^n$ be a compact set with $C^1$ boundary. Then, there exists a unique solution $f$ satisfying $\Delta f = 0$ in $\Omega$ and $f=g$ on $\partial \Omega$.
We ...

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30 views

### Truncation error of product and composition of functions?

I'm trying to solve the following problem numerically $$\int_0^a\alpha(x)\frac{dw(x)}{dx}\frac{d v(x)}{dx}dx.$$ For any $\alpha$,w and v functions of x.
The problem is that I don´t know what the ...

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226 views

### Steklov eigenvalue problem for a planar region bounded by ellipse

The Steklov problem for a compact planar region $\Omega$ is
\begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$},
\end{...

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30 views

### Coarse grid correction

Let $A_h \in \mathbb{R}^{n \times n}$ be the matrix corresponding to a finite element discretization of some nonselfadjoint, bounded and indefinite bilinear form corresponding to a second order ...

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**1**answer

80 views

### A mathematical motivation for Lax-Friedrich type of Numerical Fluxes

A Lax-Friedrichs (LF) type of flux for a conservation law $\partial_tU+\partial_xf(U)=0$ is given by
\begin{align}
F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^...

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106 views

### Solving numerically an equation involving exponentials [closed]

I met an equation of the following form:
$$\sum_{i=1}^nk_ip_i e^{-k_i\lambda}~~=~~b,$$
where $p_i\ge 0$, $k_i$ and $b$ are known for $i=1,\cdots, n$. I'd like to know how to find the solution $\...

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223 views

### How do I solve a 3D poisson equation with mixed neumann and periodic boundary conditions numerically?

The PDE is being solved over a cube. Four of the faces are periodic, and one set of opposing faces have no-flow Neumann boundary conditions:
$\nabla^2 u = f(x,y,z)$
$\frac{\partial u}{\partial z} = ...

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110 views

### Time discretization of the variational formulation of the Navier-Stokes equation

Let
$T>0$
$I:=(0,T]$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$$ and $$V:=\overline{...

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48 views

### Time discretization of the (stochastic) Navier-Stokes equation

Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonnempty and open
$\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Lambda,\:\mathbb R^d)}$
I've found a thesis where ...

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50 views

### Ask for reference about finite difference method on HJB equation

I am a fresh PhD student in numerical analysis. Recently I am considering finite difference methods and their error analysis for solving HJB equation of the following form:
$$
v_t=g(a(x)v_x),\quad x\...

**3**

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**1**answer

135 views

### Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?

I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...

**7**

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**1**answer

1k views

### Review paper/book on Finite Difference Methods for PDEs

I am looking for a good, relatively modern, review paper/book on Finite Difference Methods for PDEs with a theoretical emphasis in mind. By theoretical emphasis I mean that I care about theorems (i.e. ...

**3**

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**1**answer

218 views

### Solving a differential system

Let $\mu$ be a probability measure on $\mathbb R$ with Lebesgue density, i.e. $\mu(dx)=\mu(x)dx$. We aime to find increasing and decreasing functions $\phi_{+}: \mathbb R_+\to \mathbb R_{+}$ and $\...

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**1**answer

96 views

### Finite Element Method on a single triangular element

Consider the Laplace equation on a single triangular domain with a Dirichlet condition on two of the sides and a Neumann condition on the remaining side. I am using a linear element ... $\mathbb{P}_1$ ...

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**1**answer

135 views

### a condition for Laplacien

Let $u\in L^{2}(R^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(R^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(R^{2})$?
Thank you in advance.

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109 views

### Fundamental solution matrix of a linear PDE

I've asked a very similar question also at math.stackexchange, but I've not received any answer.
A vectorial function $\boldsymbol{x}:\mathbb{R}^D \rightarrow \mathbb{R}^N$ satisfies the following ...

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74 views

### When can an analytical solution for the heat equation be obtained?

I am currently trying to model a system with a time varying heat flux. It seems most researchers are using FEM to obtain the heat distribution (solve the heat equation).
When can the heat equation be ...

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**0**answers

123 views

### A mathematical biology reference request

Is there any mathematical articles that describe the differential equation modelling of locomotion of amoeba using pseduopodia? I am looking for physics based models of pressure difference modeling of ...

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63 views

### About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines".
For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...

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74 views

### Properties of a Sobolev bound

I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...

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503 views

### What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I've asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well.
I am reading a paper [1] where they solve the following non-...

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**1**answer

715 views

### Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible:
$$\partial_t \...

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**0**answers

131 views

### Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)

Edit: The question is solved !! The code is actually correct. There is not error in the codes. I miss-used it. Thank you for your attention : )
This problem arises when I tried to compute the valua ...

**6**

votes

**1**answer

103 views

### Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$.
Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...

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**1**answer

178 views

### Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of:
$$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$
$$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ,u(0,t)...

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110 views

### Level Set Advection with Extension Velocity

We're studying the following system of PDEs for a scalar function $F(x, t)$ with $x \in \mathbb{R}^3$ and $t \in \mathbb{R}$. The function $F(\cdot, t)$ is a level set function for a time-dependent ...

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**1**answer

170 views

### Solving a nonlinear PDE numerically

I want to solve numerically the following PDE:
$$ u_x + u_t - (u_{xt})^2 = u(x,t) $$
The boundary conditions are no concern of mine, pick the ones that work.
So which numerical method will work for ...