Questions tagged [elliptic-pde]

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24 questions with no upvoted or accepted answers
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8
votes
1answer
328 views

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. ...
7
votes
0answers
482 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 ...
5
votes
0answers
150 views

Elliptical problem with Robin BC

Working in finite differences, I am using a transformation on the temperature variable $\Theta = \int_{T0}^T \kappa(T)dT$ to linearize the steady-state heat equation into a Poisson equation $-\...
3
votes
0answers
103 views

$L^2$ norm error estimates 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{{\...
3
votes
0answers
107 views

Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
3
votes
0answers
60 views

Solving numerically a linearized system of elliptic (?) Navier-Stokes equation (Shallow Water Derived)

For my PhD Thesis, my advisor asked me to build a solver inspired from the article "Optimal Control Theory Applied to an Objective Analysis of a Tidal Current Mapping by HF Radar, J-L Devenon, 1989". ...
3
votes
0answers
353 views

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 ...
3
votes
1answer
318 views

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 ...
3
votes
0answers
328 views

Solving PDE or eigenvalue problems without FEM

Do you know any methods/solvers for PDE or eigenvalue problems like $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ (...
2
votes
0answers
82 views

Are there known accuracy issues between 2D axisymmetric and 3D solutions?

In my full 3D solutions I am solving for the potential throughout a $100\times 200\times 200$ grid. Inside is a ring electrode set to -5V via a Dirichlet boundary condition, and surrounded on all ...
1
vote
0answers
63 views

Determine truncation error of PDE discretization

The equation is $$\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)=f(x)\\ 0<x<1, u(0)=u(1)=0$$ I'm discretizing this PDE using FVM as follows: $0=x_0=x_{1/2}<x_1<x_{...
1
vote
0answers
78 views

PDE discretization on triangular domain

Given the 2D Poisson equation $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\ \partial_n u (x, 1-x) =0, 0<x<1$$ defined on the domain $\Omega := \{(x,y) \in \...
1
vote
0answers
57 views

scaling in discretized PDE system

I want to solve the following system via Matlab $\Omega=(0,1)^2$ $$\Delta y=\frac{1}{\alpha} p$$ $$ -\Delta p= y -1 $$ $$p|_{\partial \Omega}=0,~y|_{\partial \Omega}=0$$ using ...
1
vote
0answers
118 views

FiPy with derivative source terms

I have a coupled nonlinear PDE system in 1 spatial dimension, in which I want to solve using FiPy. The dependent variables are $n$ and $T$: \begin{align} \frac{\partial n}{\partial t} \,&=\, D\,\...
1
vote
0answers
96 views

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 ...
1
vote
0answers
51 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}...
1
vote
0answers
70 views

Manufacturing a solution for non-smooth coefficients in elliptic problems

This question is a continuation of this answer (I can't comment) If we were going to manufacture a solution for a problem with discontinuous coefficients, I understand that the solution should have ...
1
vote
0answers
29 views

Gradient convergence on a checkerboard domain in finite element

I've got an elliptic problem in a domain with two materials in a checkerboard arrangement. Both materials are highly dissimilar, being one much more compliant than the other. If I use classic linear ...
1
vote
0answers
112 views

understanding interior penalty jump of basis function

If cardinal basis functions are used (i.e. $\psi_{ij}=1$ iff $i=j$, and 0 otherwise) in interior penalty methods for elliptic equations such as SIPG & NIPG, shoudn't the jump in basis functions ...
1
vote
0answers
429 views

Finite Difference in Polar Co-ordinates

Background Please note that I am duplicating the question on scicomp. I have already asked this in math. I am trying to come up with a scheme in Polar Co-ordinates for the following PDE: PDE I am ...
0
votes
0answers
46 views

Existence and uniquness of solution of FVM for Poisson equation

I'm discretizing the following Poisson equation using FVM where the domain $\Omega$ of the solution is a regular hexagon of side $1$ centered about the origin. $$\Delta u =k,\text{ $k$ constant}\\ \...
0
votes
1answer
96 views

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: ...
0
votes
0answers
65 views

Finite element method for Surface integrals using polar coordinates

I am trying to solve a 2D elliptic PDE (see complete electrode model for electrical impedance tomography) using the finite element method (FEM) over a circular region $\Omega$. I have discretized the ...
0
votes
0answers
178 views

How to form the stiffness matrix for the Poisson equation using a spectral method

This is a follow-up question to How do I form the Chebyshev differentiation matrix in MATLAB? The goal of the following code is to solve the Poisson problem: ...