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

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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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### 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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### 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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### 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 $$T_t(x):=X^x(t)=x+\int_0^tv(X^x(s))\:{\rm d}s\;\;\;\text{for all }t\ge0\tag1$$ ...
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### 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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### 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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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 $u=\text{... 1answer 475 views ### Naviers Stokes equation and machine learning I am looking for a reference explaining how to solve Navier-Stokes numerically using Machine learning algorithms . Thank you in advance for your help . 1answer 103 views ### Numerical iterative methods for Poisson equation Given a domain$\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})$,... 0answers 54 views ###$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 ... 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 ... 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 ... 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 ... 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{... 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 ... 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)=0is given by \begin{align} F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^... 2answers 108 views ### Solving numerically an equation involving exponentials [closed] I met an equation of the following form: $$\sum_{i=1}^nk_ip_i e^{-k_i\lambda}~~=~~b,$$ wherep_i\ge 0$,$k_i$and$b$are known for$i=1,\cdots, n$. I'd like to know how to find the solution$\...
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{...
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 ...