# 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 ...
1 vote
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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 ...
49 views

### 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 ...
76 views

### FEM based solution to parabolic problem

Consider the problem $$\begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases}$$ ...
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\...
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 ...
1 vote
25 views

### 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 ...
1 vote
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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 ...
1 vote
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1 vote
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### 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 ...
34 views

### 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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1 vote
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### 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
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### 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 ...
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}{\...
1 vote
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 ... 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&...
182 views