Questions tagged [numerical-analysis-of-pde]
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67
questions
0
votes
0answers
39 views
Making rigorous a suggested Newton-Raphson like approach for finding root of a functional on $H^1_0$
I am trying to figure out the reasoning behind the proposed method in the following document. So we have the following non-linear equation on the unit square $\Omega=[0,1]^2$ given by
$$-\mu \Delta u+\...
3
votes
1answer
106 views
Questions for the non-linear PDE $2u_t=\log(-u_{xx})$
Consider the PDE as follows :
$$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} \...
3
votes
0answers
46 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
0answers
68 views
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
1answer
64 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 ...
0
votes
0answers
17 views
Derivative $\left.\frac{\rm d}{{\rm d}t}\nu_{\partial T_t(\partial M)}\right|_{t=0}$ of outer normal field on a transformed boundary $T_t(\partial M)$
Let $d\in\mathbb N$, $v\in C_c^1(\mathbb R^d,\mathbb R^d)$, $X^x\in C^0([0,\infty),\mathbb R^d)$ denote the solution of $$T_t(x):=X^x(t)=x+\int_0^tv(X^x(s))\:{\rm d}s\;\;\;\text{for all }t\ge0\tag1$$ ...
0
votes
0answers
45 views
How can we calculate this tangential differential?
Let $\tau>0$, $d\in\mathbb N$ and $T_t$ be a $C^1$-diffeomorphism on $\mathbb R^d$ for $t\in[0,\tau]$ with $T_0=\operatorname{id}_{\mathbb R^d}$. Assume $$T_t(x)=x+\int_0^tv(s,T_s(x))\:{\rm d}s\;\;\...
0
votes
0answers
27 views
Relation between the (pointwise) material and shape derivative
Let $\tau>0$ and $d\in\mathbb N$.
Definition 1$\:\:\:$If $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ with $v(\;\cdot\;,x)\in C^0([0,\tau],\mathbb R^d)$ and $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,...
1
vote
0answers
76 views
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 $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)...
3
votes
1answer
246 views
Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold
Let
$\tau>0$;
$d\in\mathbb N$;
$v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be Lipschitz continuous in the second argument uniformly with respect to the first with $v(\;\cdot\;,x)\in C^0([0,\tau],\...
1
vote
0answers
64 views
Show that the support of the shape gradient $\nabla\mathcal F(\Omega)$ is contained in $\overline\Omega$
Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $(T^{(\theta)}_t)_{t\ge0}$ denote the $C^1$-diffeomorphism from $E$ onto $E$ with $$T^{(\theta)...
0
votes
0answers
121 views
Non linear second order PDE involving max operator (Dynamic Programming)
I'm trying to solve the following Dynamic Programming equation in continuous time ($dt \rightarrow 0$)
$$ v(x,t) = \max\Big\{|x|\,,\,v(x,t)+dt\Big(v_t(x,t)+\frac{1}{2(t+1)}v_{xx}(x,t)\Big) \Big\} - \...
3
votes
1answer
272 views
Weird claims and conclusions in “Introduction to Shape Optimization”
I'm trying to understand the notions of Euler and Hadamard derivatives of shape functionals. All the lecture notes and papers on this topic that I've found seem to build up on the books Shapes and ...
1
vote
0answers
76 views
Von Neumann analysis on a finite difference hyperbolic scheme
I am doing a Von Neumann analysis on a staggered finite difference scheme (for Maxwell's Equations).
The finite difference scheme is:
$$
\mathbf{u}_v|^{n+2}_{i,j} - \mathbf{u}_v|^{n}_{i,j} = - A \frac{...
1
vote
0answers
69 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
1answer
261 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, ...
3
votes
1answer
143 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 ...
1
vote
0answers
28 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 ...
2
votes
1answer
83 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 ...
0
votes
0answers
47 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.
...
1
vote
0answers
126 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
$$
\begin{cases}
\partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\
u_h=0 &\text{ in } \...
3
votes
2answers
278 views
Representing a nonlinear elliptic PDE as an energy minimization problem
I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...
3
votes
0answers
62 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 ...
3
votes
0answers
46 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 $B$ ...
1
vote
0answers
63 views
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 + \...
2
votes
0answers
27 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 ...
13
votes
4answers
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 ...
5
votes
1answer
89 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, ...
4
votes
1answer
116 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 $...
2
votes
0answers
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,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
2
votes
1answer
186 views
Numerical methods for IDE [closed]
I would like to read a popular literature on the topic "Numerical methods for integro-differential equations".
Could you recommend me any articles or book with a brief overview of some methods (maybe ...
0
votes
0answers
83 views
Why does this numerical scheme work on this nonlinear PDE?
i am currently solving a nonlinear PDE of mixed parabolic/hyperbolic type of the Form
\begin{align*}
\frac{\partial}{\partial x} \left(A \frac{\partial p}{\partial x} \right)
+\frac{\partial}{\...
1
vote
1answer
232 views
Time discretization in the Feynman-Kac formula with boundary conditions
I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.
For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...
2
votes
1answer
129 views
Construct examples satisfying some inequalities [closed]
How do I construct two vectors $a,b\in \mathbb{R}^{n}$, $a=(a_1,a_2,\ldots, a_n)^T$ and $b=(b_1,b_2,\ldots, b_n)^T$ which satisfy in the following conditions
\begin{align}
& a_ib_i\geq 1,a_ib_j&...
1
vote
0answers
131 views
Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
2
votes
0answers
119 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, ...
0
votes
1answer
163 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 ...
11
votes
3answers
576 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{...
2
votes
1answer
475 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 .
3
votes
1answer
103 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 $f({\bf x})$,...
2
votes
0answers
54 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 ...
2
votes
2answers
99 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
votes
1answer
110 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 ...
1
vote
0answers
135 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 ...
6
votes
0answers
293 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{...
2
votes
0answers
98 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 ...
1
vote
1answer
165 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^...
2
votes
2answers
108 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 $\...
0
votes
0answers
125 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{...
1
vote
0answers
58 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 ...
