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# Questions tagged [elliptic-pde]

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### Elliptic PDE finite volume method with Dirichlet boundary condition

I want to discretize the following equation using a Finite Volume Method $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ I'm using Voronoi cells here: ...
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### Increasing V-cycles for constant Coarsest Grid Size and increasing Fine Grid size

Problem statement I implemented geometric multigrid for $-\nabla^{2}=f$ where $f=\frac{3\pi^{2}}{4}sin \frac{\pi x}{2} sin \frac{\pi y}{2} sin \frac{\pi z}{2}$ on $\Omega \in [0,1]$ on a unit cube. ...
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### FEM solution for Poisson is not exact at nodes

Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$ I know that the FEM solution (...
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### Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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### 1D Poisson equation and quadratic basis functions assembly

I'm solving the simple Poisson problem $$-u''(x)=1$$ in the interval $[0,1]$ with $u(0)=u(1)=0$. I discretised my domain as done here, i.e. with ...
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### General case Kutta condition

I'm working on a 2D inviscid fluid simulation using a "panel method", with Potential being used to enforce the no-through boundary condition. I'm trying to incorporate the Kutta condition, which says ...
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### Elliptic equation with finite volume and unstructured high order geometry

I have found that in unstructured mesh, discretizing the laplacian operator with finite volumes requires special care, as given in An Introduction to Computational Fluid Dynamics: The Finite Volume ...
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### Neumann-Neumann boundary intersection

I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ...
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### Reference Request: Raviart Thomas with hanging nodes

I am interested in reading about the analysis (existence, uniqueness, error estimates) of elliptic problems solved with a Mixed method that uses the Raviart Thomas elements (so far so good, easy to ...
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### Comments needed on the doubts of PDEs in moving boundary problems

We know that in classical two-phase Stefan problems, let's say in the temperature distribution of ice-water problem here, the governing PDEs are: \left. \begin{aligned} C_1\frac{\...
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### Do DG methods for the Helmholtz equation always return positive quantities?

Helmholtz Diffusion equation with reaction term: $$k\Delta u + u = f ~ \text{in} ~\Omega$$ $$\nabla u \cdot \mathbf{n} = 0 ~ \text{in} ~\partial \Omega$$ For sufficiently small $k$ (relative to ...
I have the biharmonic equation on a 2D rectangular domain $\Omega$ with the following boundary conditions: $\Delta^2 u = f$ on $\Omega$ $\nabla u \bullet \mathbf{n}=0$ on $\partial \Omega$ (1) $u_{... 1answer 8k views ### What is the general idea of Nitsche's method in numerical analysis? I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ... 0answers 613 views ### Numerical implementation of the Dirichlet-to-Neumann map I am solving the Dirichlet problem $$\begin{cases} \Delta u = 0, \\ u|_{\partial D} = f, \end{cases}$$ in a$2d$domain$D$using the finite element method. What I want to get is the ... 2answers 605 views ### Correctly setting boundary condition for periodic linear elasticity problem From an old, wise engineering book Peterson's Stress Concentration Factors (http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470048247.html page 324) I've got the following problem: There is 2D ... 1answer 272 views ### An example of mixed elliptic problem using lowest-order Raviart Thomas element I try to solve the following mixed second order elliptic PDE in the domain$D=[0, 1]^2\begin{eqnarray*} v+\nabla p=&0 \quad &\text{in} \quad D,\\ \text{div}(v)=&1/2 \quad &\... 1answer 241 views ### How to correctly define the flux in a finite volume method for Poisson's equation with a piecewise constant material How do we correctly define the flux in a finite volume method applied to Poisson's equation where we have a piecewise constant material? Specifically, say we have the equation \begin{align*} -\nabla\... 1answer 759 views ###L^2$-convergence of finite element method when right hand side is only in$H^{-1}$(Poisson eqn) I know that the piecewise linear finite element approximation$u_h$of $$\Delta u(x)=f(x)\quad\text{in }U\\ u(x)=0\quad\text{on }\partial U$$ satisfies $$\|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)} ... 4answers 2k views ### biharmonic equation I want to solve the biharmonic equation numerically, that is:$$\Delta^2 u=f~~in~~\Omegau=g_1~~on ~~\partial \Omega\frac {\partial u}{\partial n}=g_2~~on ~~\partial \Omega$$Using Green's ... 2answers 291 views ### O(h^2) convergence for Elliptic PDE I am trying to solve an elliptic PDE in 2-D:$$-\nabla^{2} u = 20tanh(10x-5)(10-10tanh(10x-5)^2) = f$$I know that the solution is$u = tanh(10x-5)$but I am unable to get$O(h^2)$solution with a ... 0answers 57 views ### Decreasing - increasing - stabilising$l_{2}$norm Let$\bar{x}$denote the analytical solution of a PDE. Let$x^{(k)}$be the solution at the$k^{th}$iteration. The initial guess for the solution is$x^{(0)} = 0$. Let$r_{0} = ||\bar{x}-x^{(0)}||_{2}...
Assume we use FEM with piecewise linear finite elements to discretize the BVP over $\omega = (0,1)$: $-u''+ bu' + u = 2x$, $u(0) = u(1) = 0$ for parameter $b\in R$. Given a mesh \$T = \left\{x_i\right\}...