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For questions about the numerical analysis of partial differential equations, the intersection of top-level tags [tag:ap.analysis-of-pdes] and [tag:na.numerical-analysis].

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Suppose we trained a neural network to fit a solution of a PDE, but we want to do something in a Finite Element Space, so we need transform our neural network to the latter. What is the way to do this ...
Hao Yu's user avatar
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Consider the following PDE: $$ -\Delta u + \alpha u + \beta (x \cdot \nabla) u = 0. $$ Is there any nonzero weak solution of this equation on $\Bbb R^n$, in $H^1$ or other function spaces, for some ...
Hao Yu's user avatar
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I’m exploring a reaction–diffusion-type scalar field equation of the form $$ ∂_t K=D\nabla^2 K+SK(1-K)(K-K_*), $$ where $D>0$, $S>0$, and $0<K_*<1$. Numerical simulations in 2D produce the ...
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Consider the generalized eigenvalue problem: $$ [- \nabla \cdot (D(\mathbf{x}) \nabla) + \Sigma_a(\mathbf{x})] \phi(\mathbf x) = \lambda \Sigma_f(\mathbf x) \phi(\mathbf x)$$ Some specifications: The ...
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I'm looking for a Python solution to compute a conformal map that transforms an entire 2D domain (given as a shape; convex or not, by giving its binary mask for example) onto the unit disk or another ...
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I'm working on the finite difference scheme to solve $-\Delta u(x) = f(x)$ in the unit square $\Omega= (0,1) \times (0,1)$ with $u=0$ on $\partial \Omega$. I want to check whether convergence of ...
zcmw's user avatar
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To be very short (before explaining more), I am trying to build an efficient and stable numerical scheme for the following systems of coupled PDEs: $$\partial_t \rho + \partial_x[\rho v] = 0,$$ $$\...
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Let $H: \mathcal{D} \rightarrow \mathcal{H}$ be a densely defined, self-adjoint, non-negative operator. Let $P: \mathcal{H} \rightarrow \mathcal{H} $ be an orthogonal projection onto a subspace. We ...
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Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$. We will assume the following: we are ...
Plemath's user avatar
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Consider a densely defined, self-adjoint operator $$ H: \mathcal{D} \rightarrow \mathscr{H}. $$ Assume for simplicity that $H$ is nonnegative. We want to effectively restrict this operator $H$ to a ...
Qualearn's user avatar
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The PDE I am working on comes from geology, which I do not have much background on. Said equation aims to describe describes the erosion by describing it as an advection phenomena: the advection ...
betelgeuse's user avatar
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I am researching numerical methods for PDEs. I particular, I am looking at methods for the linear hyperbolic PDE $$ u_t+au_x=0. $$ This is a common approach, because successful methods for this model ...
Philip Roe's user avatar
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For whatever reason, I stubbornly decided to use tetrahedral elements and find myself needing to use P3 elements with bubble functions ("P3b3d" in FREEFEM-style denomination). The 2d case is ...
Sébastien Loisel's user avatar
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I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book. The idea is to model the market price of risk as a ...
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Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
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What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
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Intro Suppose we have the following static linear equations (e.g. of an elastostatic problem): $$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$ We want a multipoint constraint of the type $$\boldsymbol{\...
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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(\...
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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\...
0xbadf00d's user avatar
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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 ...
miao zhengwu's user avatar
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Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$. $$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
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Consider the following differential equation: $$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$ with $u(x,0)=f(x)$. The solution of \eqref{1}, ...
Mirar's user avatar
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I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ​​...
kumquat's user avatar
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The split-step method is a numerical method that can be used to solve a nonlinear PDE (https://en.wikipedia.org/wiki/Split-step_method). Even Wikipedia does not refer to the original authors (F.D. ...
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I have the following semidiscrete problem on a meshed domain $U_h$. Let $V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and $V_{h\partial}$ be ...
Lilla's user avatar
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Consider a PDE, $$\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(...
Jarwin's user avatar
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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 ...
Hari Sam's user avatar
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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 ...
Ilovemath's user avatar
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Here it is an Advection-Diffusion equation in 2D: $$ \frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
Edric Jonathan's user avatar
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I am interested in finding numerical solutions to a Monge-Ampere type equation for applications in physics. Due to the close connection between Monge-Ampere and optimal transport and the well ...
Yly's user avatar
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Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length. The (stationary) string can be modelled by a unit speed curve: $$[0,1] \...
sitiposit's user avatar
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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}\...
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644 views

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, $\...
Mirar's user avatar
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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 ...
Hiro's user avatar
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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 ...
Chushamm's user avatar
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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 ...
Ho-Oh's user avatar
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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 ...
ithmath's user avatar
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Consider the problem $$ \begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases} $$ ...
Lilla's user avatar
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$\Omega\subset \mathbb{R}^N$ 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\...
Tibeku's user avatar
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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 ...
mohammad fazli's user avatar
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For a vector $\vec{u}\in\mathbb{R}^N$ let's denote $\pi_N\left(\vec{u}\right)$ the unique piecwise linear and $1$-periodic function matching the components of $\vec{u}$ on the discretization $x_k = \...
Ayman Moussa's user avatar
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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 ...
Ma Joad's user avatar
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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 = ...
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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 ...
bobinthebox's user avatar
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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} \...
Onil90's user avatar
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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 ...
陳Keefe's user avatar
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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} \...
GJC20's user avatar
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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 ...
Ken Hung's user avatar
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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{...
Alex's user avatar
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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 ...
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