Questions tagged [numerical-analysis-of-pde]

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$H^s$ 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 ...
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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 ...
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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 ...
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  • 53
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1 answer
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FEM based solution to parabolic problem

Consider the problem $$ \begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases} $$ ...
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2 votes
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68 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\...
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  • 121
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44 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 ...
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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 ...
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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 ...
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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 = ...
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Low order quadrature for low order terms in finite element method- convergence analysis

Consider the PDE $$\frac{\partial u}{\partial t} = \Delta u - \kappa u,$$ posed over a bounded domain $\Omega$ (say, $\Omega \subseteq \mathbb{R}^2$). Assume we have a family of triangulations $\...
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1 vote
1 answer
81 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 ...
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sign for Galerkin product of a monotone nonlinearity?

Let $M$ be a smooth, compact manifold without boundaries. Let $(e_n)_{n\geq 0}\subset L^2(M)$ be the Hilbert basis generated by the Laplacian eigenfunctions, i-e $-\Delta e_n=\lambda_n e_n$ (no ...
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3 votes
0 answers
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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} \...
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57 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 ...
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3 votes
2 answers
259 views

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} \...
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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 ...
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  • 161
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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{...
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  • 245
6 votes
1 answer
148 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 ...
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1 vote
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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)...
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  • 297
3 votes
1 answer
250 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],\...
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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)...
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  • 297
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123 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\} - \...
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3 votes
1 answer
306 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 ...
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82 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{...
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  • 11
1 vote
0 answers
87 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\...
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3 votes
1 answer
282 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, ...
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3 votes
1 answer
155 views

More important or relevant progress in discretizing hard problems in physics in last decade

This is a reference request, and soft question as companion. I'm curious to ask, from an informative point of view, what about the more important progress in the goal to discretize hard problems in ...
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1 vote
0 answers
29 views

Choice of finite element spaces in plasticity

I am planning to run numerical simulations in metal elastoplasticity (von-Mises yield condition with and without isotropic hardening). However, I am completely new to this subject and I am unsure ...
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  • 111
2 votes
1 answer
119 views

Discrete curve-shortening flow – numerical implementation

I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the ...
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0 votes
0 answers
53 views

Fokker-Planck equation with zero diffusion coefficients on the boundaries

I am currently working on a Masters project in which I will be looking at numerical methods for a specific PDE. The PDE that I will be considering is a Fokker-Planck equation on a unit hyper-cube. ...
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1 vote
0 answers
129 views

Rigorous error estimate for semi-discrete heat equation in bounded domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of $$ \begin{cases} \partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\ u_h=0 &\text{ in } \...
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  • 177
3 votes
2 answers
463 views

Representing a nonlinear elliptic PDE as an energy minimization problem

I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation: $$\...
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  • 245
3 votes
0 answers
71 views

Shooting method still relevant?

I'm preparing to teach a "Numerical Analysis II" course next term, and in previous years this course involved a section on the shooting method for solving one-dimensional boundary value problems. This ...
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  • 223
3 votes
0 answers
48 views

Spectrum of a symmetric saddle point matrix

Let $C=\left[ {\begin{array}{cc} A & B^{T} \\ B & O \\ \end{array} } \right]$, where $A\in \mathbb{R}^{n\times n}$ is SPD, $B\in \mathbb{R}^{m\times n}$ and $m\leq n$. The matrix $B$ ...
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1 vote
0 answers
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discrete Fourier transform for composition of differential operators on a grid

This question pertains to stability analysis of finite difference methods using the discrete Fourier transform. Suppose I have a convection diffusion equation of the form: (1) $\hspace{.5in}u_t + \...
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  • 111
2 votes
0 answers
29 views

Discrete maximum priniciple for parabolic operators

While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ...
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13 votes
4 answers
2k views

Is there a connection between representation theory and PDEs?

As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
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  • 149
5 votes
1 answer
94 views

Is the minmod limiter energy stable?

It is well-known, that upwind scheme and Lax-Wendroff scheme are energy stable for the linear advection equation $u_t +a u_x = 0$ with periodic boundary conditions, if the CFL condition is satisfied, ...
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  • 61
4 votes
1 answer
126 views

Numerics for continuity equation with Sobolev vector field

Has any work been done about numerical methods for the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ where $...
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  • 643
2 votes
0 answers
76 views

How we can do the derivative for this equation w.r.t.to time t>0

Let $x\in[0,L]$ and consider the following equation, $$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
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2 votes
1 answer
210 views

Numerical methods for IDE [closed]

I would like to read a popular literature on the topic "Numerical methods for integro-differential equations". Could you recommend me any articles or book with a brief overview of some methods (maybe ...
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0 votes
0 answers
84 views

Why does this numerical scheme work on this nonlinear PDE?

i am currently solving a nonlinear PDE of mixed parabolic/hyperbolic type of the Form \begin{align*} \frac{\partial}{\partial x} \left(A \frac{\partial p}{\partial x} \right) +\frac{\partial}{\...
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1 vote
1 answer
294 views

Time discretization in the Feynman-Kac formula with boundary conditions

I am applying the Feynman-Kac theory for solving a PDE with boundary conditions. For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...
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2 votes
1 answer
131 views

Construct examples satisfying some inequalities [closed]

How do I construct two vectors $a,b\in \mathbb{R}^{n}$, $a=(a_1,a_2,\ldots, a_n)^T$ and $b=(b_1,b_2,\ldots, b_n)^T$ which satisfy in the following conditions ‎\begin{align} & a_ib_i\geq 1,a_ib_j&...
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  • 37
2 votes
0 answers
182 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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  • 21
2 votes
0 answers
122 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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0 votes
1 answer
224 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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11 votes
3 answers
601 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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  • 532
2 votes
1 answer
582 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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3 votes
1 answer
128 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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