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Questions tagged [numerical-analysis-of-pde]

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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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0answers
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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0answers
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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0answers
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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1answer
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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0answers
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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3answers
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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1answer
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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1answer
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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0answers
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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2answers
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 ...
2
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1answer
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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0answers
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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0answers
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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0answers
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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1answer
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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2answers
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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0answers
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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0answers
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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0answers
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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1answer
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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1answer
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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1answer
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 $\...
2
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1answer
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 ... $\mathbb{P}_1$ ...
0
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1answer
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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0answers
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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0answers
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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0answers
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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0answers
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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0answers
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-...
2
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1answer
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 \...
3
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0answers
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
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1answer
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 ...
0
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1answer
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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0answers
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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1answer
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 ...