Questions tagged [elliptic-pde]

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2
votes
1answer
160 views

Boundary elements method — calculation of solid angle

I am developing a BEM code based on a deal.ii tutorial. Consider the Poisson equation $$ \Delta u=-f\,, $$ and its Green's function $G\left(\mathbf{x},\mathbf{x}'\right)$ with the property $$ \Delta ...
3
votes
1answer
141 views

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: \begin{equation} \left. \begin{aligned} C_1\frac{\...
0
votes
1answer
133 views

Calculation of error

I have written a code in which I find the approximation of the solution of this elliptic problem. I calculated the error using the following part of code: http://pastebin.com/7b5mmuRW but I get the ...
1
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0answers
121 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 ...
4
votes
1answer
276 views

Points on the interface

We consider the problem $\left\{\begin{matrix} k(x)\Delta u(x)=f(x) & \text{ in } \Omega\\ u=0 & \text{ in } \Gamma \end{matrix}\right.$ where $\Omega \subset \mathbb{R}^2$ open and ...
9
votes
1answer
391 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. ...
1
vote
2answers
109 views

Building minimization optimization problem for 2nd-order elliptic PDE

I am solving elliptic PDE problem, for which, Euler scheme looks as following: $$ \nabla [\gamma ( |\nabla u|^2) \nabla u] = 0,$$ where $$\gamma(|\nabla u|^2) = (1 + |\nabla u|^2)^{-1/2}. $$ I am ...
4
votes
1answer
135 views

Computing Fourier representation of space dependent advection operator via FFT

Consider the following equation on the circle: $$\dfrac{\partial p(x,t)}{\partial t} = a(x)\dfrac{\partial p(x,t)}{\partial x} \equiv L(p) \enspace ,$$ where $L$ is the operator acting on $p(x,t)$. ...
1
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0answers
91 views

Monotonic convergence of Newton's method for boundary value problems [closed]

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
1
vote
1answer
123 views

A doubt in Multigrid V-cycle

Assume I have 3 levels of grids. Finest Grid = level 2, Coarser Grid = level 1, Coarsest Grid = level 0. Relax $u$ on $Au = b$ at level 2 for 3 times. Find residual $r2$ at level 2, then restrict to ...
20
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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 ...
1
vote
2answers
173 views

Specific questions for 2-D Multigrid

I am simulating $\nabla^{2}u=0$ with mixed Dirichlet-Neumann boundary conditions on 2-D using 2-Grid method. Dirichlet ...
2
votes
1answer
154 views

Finding the frequencies of vibration of a circular and square drum

I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have researched ...
2
votes
1answer
141 views

Equivalence of linear elasticity and biharmonic equations: variational formulation

Wikipedia tells me that the equations for linear elasticity and biharmonic equations have the same solution for Dirichlet boundary condition. How do you show the equivalence in the variational ...
0
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1answer
160 views

Simulation of Laplace Equation in 3-D with mixed BC of Dirichlet-Neumann

I am simulating a Mixed-Boundary value (Dirichlet-Neumann) problem using Finite Difference Method on a unit 3-D cube such that the left, lower, and front plane have $u=u(x,y,z)=1$ (Dirichlet) and ...
0
votes
1answer
730 views

How to impose Neumann boundary conditions in interior penalty DG method

Consider the following two point BVP: $$ -u''(x)=f(x),~~~u(0)=u(1)=0. $$ An interior penalty DG method for this BVP that weakly imposes homogeneous Dirichlet boundary conditions is of the form: $$ \...
1
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0answers
579 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
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0answers
54 views

What is a reasonable manufactured solution to test finite difference method? [duplicate]

What is a reasonable manufactured solution to test the following equation against its finite difference approximation? I want it to look like a Cosine function about $0$, rotated about $Z$ axis (...
2
votes
2answers
115 views

Infinite Function Value on Dirichlet Boundary

I have been working on a multigrid solution to a non-homogeneous Dirichlet boundary value problem. However, the function goes to infinity on the boundary. This causes numerical overflow errors to be ...
3
votes
1answer
101 views

Fast methods to solve an elliptic PDE if high accuracy is needed only in part of the domain

Does someone know a method to get cheap approximation of harmonic problems (and possibly local approximations)? Let me explain: I need to compute the solution of an harmonic problem \begin{equation} ...
3
votes
1answer
305 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + f(x,t)...
0
votes
1answer
212 views

jump conditions for Poisson/Darcy equation in primal form versus mixed form

Consider the Darcy equation, $$\mathbf{v} + \dfrac{k}{\mu_0}\nabla p = \mathbf{f} \\ \mathrm{div}\; \mathbf{v} = 0$$ If the coefficient $k$ is piecewise constant across an interface $\Gamma$ in the ...
3
votes
1answer
527 views

Jump condition for elliptic equation in standard finite element method

Consider the elliptic PDE $ -\mathrm{div}(k(x)\nabla u) = f(x) \in \Omega$ and $u = 0$ on $\partial \Omega$ If $k(x)$ is piecewise constant across an interface $\Gamma$ in the domain, then we ...
5
votes
0answers
163 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 $-\...
7
votes
4answers
2k views

biharmonic equation

I want to solve the biharmonic equation numerically, that is: $$\Delta^2 u=f~~in~~\Omega$$ $$u=g_1~~on ~~\partial \Omega$$ $$\frac {\partial u}{\partial n}=g_2~~on ~~\partial \Omega$$ Using Green's ...
3
votes
1answer
92 views

Library to compute eigenvalues of the Laplace operator in a polyhedral domain

What library can one use to compute efficiently the lowest eigenvalues of the Laplace operator in a polyhedral domain in $R^3$? For the application I have in mind one has to consider very acute ...
2
votes
0answers
395 views

assembly matrices in finite element method [closed]

I'm trying to construct the right–hand side of my 2D Poisson's equation in Matlab. I used the vertex rule in order to approximate the integral: ...
5
votes
2answers
401 views

Finite element discretization of Reaction-diffusion problem with Dirac source term

I'm writing a code using continuous piecewise linear finite elements on triangular grids to solve the diffusion-reaction problem. the source function f is a Dirac mass at the center. How can i compute ...
8
votes
2answers
897 views

Is there a good tutorial or textbook-like source on implementing ENO/WENO with limiters in one (and more than one) dimension?

I've inherited a finite volume code that does a second-order discretization of flux terms for a set of mixed parabolic-elliptic equations with discontinuous diffusion coefficients. The impression I ...
26
votes
3answers
5k views

What is the purpose of using integration by parts in deriving a weak form for FEM discretization?

When going from the strong form of a PDE to the FEM form it seems one should always do this by first stating the variational form. To do this you multiply the strong form by an element in some (...
2
votes
1answer
159 views

Solving the elliptic eigenproblem with periodic boundary conditions

Given an energy level $\mu$, I'm looking to calculate the eigenvectors corresponding to the time-independent Schrodinger operator on the torus (that is, periodic boundary conditions)-- $H = -h^2 \...
8
votes
3answers
4k views

Role of boundary conditions (e.g. periodic) in Poisson equation

Given 3D Poisson equation $$ \nabla^2 \phi(x, y, z) = f(x, y, z) $$ and the right hand side and the domain, am I free to impose any boundary conditions (BC) on the function $\phi$, or do they have to ...
5
votes
3answers
2k views

Online Poisson Solver

I'm wondering if anyone can point to a browser-based FEM (or other) 2D PDE solver for simple elliptic problems. It seem like there ought to be a javacript implementation that would allow for the ...
3
votes
1answer
440 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 ...
8
votes
1answer
323 views

Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde

Suppose I have the parabolic system $$u_t=\nabla\cdot(k(x)\nabla u)+f,\quad (x,t)\in\Omega\times I$$ with Dirichlet boundary conditions $$u=g, \quad x\in\partial\Omega$$ and initial condition $$u(x,t)...
3
votes
0answers
345 views

Solving PDE or eigenvalue problems without FEM

Do you know any methods/solvers for PDE or eigenvalue problems like $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ (...
4
votes
1answer
2k views

variational formulation of linear elasticity

First I'm not 100% sure I'm on the good stack for asking my question. I would like to get a bilinear form for linear elasticity that separate a rotational part from a pure divergence part, so starting ...
8
votes
2answers
624 views

Helmholtz and Biharmonic equation examples with exact solution

I'm looking for examples of Helmholtz and Biharmonic equations in Cartesian co-ordinates with exact solutions, in order to compare my numerical solutions with it. I was able to find quite a few ...

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