Questions tagged [elliptic-pde]

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83 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
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1answer
111 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 ...
16
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1answer
6k 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
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2answers
154 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
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1answer
128 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 ...
1
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1answer
116 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
153 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
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1answer
576 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
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
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0answers
51 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
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2answers
100 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
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1answer
100 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
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1answer
260 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
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1answer
187 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
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1answer
438 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
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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 $-\...
5
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4answers
906 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
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1answer
76 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
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0answers
391 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
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2answers
360 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 ...
7
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2answers
742 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 ...
24
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3answers
4k 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
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1answer
149 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
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3answers
3k 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 ...
4
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3answers
1k 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
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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 ...
8
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1answer
281 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
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0answers
328 views

Solving PDE or eigenvalue problems without FEM

Do you know any methods/solvers for PDE or eigenvalue problems like $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ (...
4
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1answer
1k 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
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2answers
542 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 ...