# 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, ...
1answer
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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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### 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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### 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 ...