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Questions tagged [elliptic-pde]

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3
votes
1answer
277 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 ...
1
vote
0answers
57 views

How to solve potential flow with FEM, stream function, and the Kutta condition?

I'm trying to solve two-dimensional potential flow over airfoils with the finite element method, using the stream function formulation ($\Delta\psi = 0$, $u = -\partial\psi/\partial y$, $v = \partial\...
8
votes
1answer
311 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. ...
2
votes
0answers
64 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 ...
0
votes
1answer
38 views

Elliptic PDE: Proving that a 2nd order accurate discretization of the 2nd derivative of the unknown is 2nd order accurate for the unknown itself

For an ODE: $\frac{dy}{dt}=f(y(t),t)$ The Euler Explicit scheme reads: $y_{n+1}=y_{n}+\Delta tf_n$ and it can be easily shown with a Taylor expension that: $y_{n+1}=y_{n}+\Delta t \frac{dy}{dt}|...
2
votes
2answers
79 views

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 ...
3
votes
0answers
83 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{{\...
4
votes
1answer
66 views

Stabilization parameter for an elliptic equation

I simply want to solve the elliptic equation: $$ -\kappa \nabla^2 u + u = f $$ where $f\in [0,1]$. When using continuous Galerkin with Lagrange elements, I have noticed that $\kappa$ has to be greater ...
3
votes
0answers
86 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 ...
0
votes
1answer
148 views

Discontinuous Galerkin - Inhomogeneous Dirichlet B.C. for 1D Poisson Equation

I am trying to get some code working for the 1D Poisson equation using the textbook: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications. I use the following formulation (for ...
3
votes
0answers
57 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". ...
0
votes
1answer
179 views

Bifurcation of linear PDE

I have a linear elliptic PDE (unfortunately not allowed to be shown here) with a constant parameter $\epsilon$ giving the stable solutions qualitatively shown by the functions below. As we smoothly ...
5
votes
1answer
87 views

Finite-volume method applied to a particular advection equation

I'm trying to apply the finite-volume method (FVM), with which I'm not so familiar, so a simple 1D PDE equation. I already asked this question in the math stackexchange, but was told that it could be ...
7
votes
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 ...
1
vote
0answers
52 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 ...
3
votes
1answer
199 views

FEM current toy problem

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 ...
1
vote
0answers
98 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
1answer
157 views

Raviart Thomas Mixed Finite Element with Mixed boundary conditions reference request

This is perhaps a more focused version of this question. Using standard notation, I have a code that works and solves the following PDE using a Raviart Thomas mixed method. $$\begin{align} 0 &= ...
4
votes
0answers
201 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 ...
4
votes
1answer
108 views

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 ...
4
votes
1answer
127 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{\...
2
votes
0answers
95 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 ...
5
votes
3answers
241 views

How to impose boundary condition with mixed derivatives?

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_{...
16
votes
1answer
5k 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 ...
8
votes
0answers
403 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 ...
0
votes
2answers
348 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 ...
1
vote
1answer
166 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 &\...
1
vote
1answer
161 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\...
9
votes
1answer
287 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)} $...
5
votes
4answers
694 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 ...
2
votes
2answers
210 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 ...
1
vote
0answers
48 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
1answer
83 views

3 questions on FEM to solve elliptic PDE with homogeneous and mixed boundary conditions

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\}...
1
vote
0answers
158 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: ...
4
votes
1answer
157 views

Finite element error for second order ODE at nodes equal to zero

I coded a finite element method with linear basis elements for the problem $$-u'' = f(x), x\in[0,1], u(0) = u(1) = 0$$ The nodes are uniformly spaced and I will denote them as $x_i$. I initially ...
1
vote
1answer
74 views

Does a mixed method solve this elliptic pde exactly if the source function is piecewise polynomial?

Let $\Omega\subset \mathbb{R}^d$, $d\in \{2,3\}$ be an open bounded polygonal/polyhedral set. Suppose I want to solve the following pde \begin{align*} \vec{q}+\vec{\nabla}u &=0\,&x&\in \...
1
vote
0answers
63 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 ...
6
votes
2answers
117 views

Stokes Equation in “two-fold saddle point” form?

Are there papers that deal with the (nondimensionalized) Stokes equation for incompressible fluid flow in a "doubly mixed" form like the following? \begin{align*} 0&=\underline{\epsilon} + \frac{...
2
votes
1answer
291 views

FreeFem user-defined function [closed]

I'm trying to solve the steady-state heat equation with a space-dependent thermal diffusivity in FreeFem. I.e. $$\nabla\cdot(\kappa(x,y)\nabla T) = 0$$ I'm hoping to read in from a file locations $(...
4
votes
1answer
81 views

In mixed elliptic formulation, what are the weakest requirements to ensure the flux is in $H^1$?

In the book Mixed Finite Element Methods and Applications by Boffi, Brezzi, and Fortin there is a pretty long discussion about why the Raviart-Thomas (RT) projection is only defined for functions in $...
3
votes
1answer
240 views

How do I avoid divide-by-zero when solving the Poisson equation with Fourier transforms?

I wanted to try to implement part of the method in the following article using Fourier transforms. http://www.shodor.org/media/content/jocse/student_submissions/nocito2010/nocito2010_pdf Right now I ...
1
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0answers
25 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 ...
2
votes
1answer
96 views

Boundary elements method — calculation of solid angle

I am developing a BEM code based on a deal.ii tutorial, see https://www.dealii.org/8.3.0/doxygen/deal.II/step_34.html . Consider the Poisson equation $$ \Delta u=-f\,, $$ and its Green's function $G\...
0
votes
1answer
120 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 ...
4
votes
1answer
240 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 ...
1
vote
0answers
107 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 ...
8
votes
1answer
273 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)...
1
vote
2answers
94 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
121 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
vote
0answers
78 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. ...