Questions tagged [boundary-conditions]

For questions regarding the choice and/or appropriateness of conditions necessary to model a particular phenomenon with a partial differential equations.

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27
votes
1answer
5k views

Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation

I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density)...
23
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4answers
3k views

How to incorporate the boundary conditions with the Galerkin method?

I've been reading some resources on the web about Galerkin methods to solve PDEs, but I'm not clear about something. The following is my own account of what I have understood. Consider the following ...
18
votes
1answer
7k 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 ...
16
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1answer
12k views

How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the domain (...
15
votes
2answers
18k views

Writing the Poisson equation finite-difference matrix with Neumann boundary conditions

I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone ...
14
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3answers
2k views

How to deal with curved boundary condition when using finite difference method?

I'm trying to learn about numerically solving PDE by myself. I've been beginning with finite difference method(FDM) for some time because I heard that FDM is the fundament of numerous numerical ...
14
votes
4answers
7k views

Boundary conditions for the advection equation discretized by a finite difference method

I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs. The books and notes which I currently have access to all say ...
14
votes
2answers
10k views

Periodic boundary condition for the heat equation in ]0,1[

Let us consider a smooth initial condition and the heat equation in one dimension : $$ \partial_t u = \partial_{xx} u$$ in the open interval $]0,1[$, and let us assume that we want to solve it ...
11
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4answers
6k views

solving coupled ODEs with initial-value and final-value constraints

The essence of my question is the following: I have a system of two ODEs. One has an initial-value constraint and the other has a final-value constraint. This can be thought of as a single system with ...
11
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2answers
2k views

Solid mechanics with finite differences: How to handle “corner nodes”?

I have a question concerning coding boundary conditions for solid mechanics (linear elasticity). In the special case I have to use finite differences (3D). I am very new to this topic, so perhaps some ...
10
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2answers
4k views

How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices

I am wondering how Dirichlet boundary conditions in global sparse finite element matrices are actually implemented efficiently. For example lets say that our global finite element matrix was: $$K = \...
10
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3answers
4k views

Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
9
votes
2answers
5k views

How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?

In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
9
votes
1answer
1k views

Schrodinger equation with periodic boundary conditions

I have a couple of questions regarding the following: I am trying to solve the Schrodinger equation in 1D using the crank nicolson discretization followed by inverting the resulting tridiagonal ...
9
votes
2answers
657 views

Boundary conditions Chebyshev differentiation

I was wondering if anyone has any experience dealing with boundaries when implementing chebyshev differentiation. I am currently trying to implement a no slip boundary condition to solve the ...
9
votes
1answer
1k views

Which fourier series is needed to solve a 2D poisson problem with mixed boundary conditions using Fast Fourier Transform?

I have heard that a fast fourier transform can be used to solve the poisson problem when the boundary conditions are all one type... Sine series for dirichlet, cosine for neumann, and both for ...
8
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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 ...
8
votes
1answer
2k views

Example of a PDE model with nonlinear Dirichlet boundary conditions

Is there any application for PDEs with nonlinear Dirichlet boundary conditions? That is, I am looking for an example of a partial differential equation for a state $u$ posed on a domain $\Omega$ with $...
8
votes
1answer
263 views

Adaptive mesh refinement with perfectly matched layers?

We have an adaptive mesh refinement (AMR) code for solving the elastic wave equation with frictional fault interfaces (based on Chombo for those that are interested). One of the things that we have ...
8
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0answers
564 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 ...
7
votes
2answers
2k views

FEniCS: boundary conditions for electrostatic problems with dielectrics

I carefully read all circa 70 pages of FEniCS tutorial and I still do not understand how to solve electrostatic problems when I have materials with different dielectric constant. The self contained ...
7
votes
2answers
3k views

FEniCS: separate boundary conditions in normal and tangential direction of mesh boundary

Given a vector-valued PDE, I'd like to enforce the boundary conditions $$ \vec{n}\cdot u = g\\ \vec{n}\cdot \nabla (\vec{t}\cdot u) = 0 $$ on the solution $\vec{u}$. If the boundary happens to align ...
7
votes
1answer
798 views

Are we free to choose the position of ghost cells on a non-uniform finite-volume mesh?

Following Hundsdorfer approach the finite volume discretisation of the advection-diffusion equation (conservative form) on non-uniform cell centered grid can be written as, $$ w_j^{\prime} = \frac{w_{...
7
votes
1answer
376 views

Potential flow around a non-symmetric obstacle using stream functions

I've seen that there is a way to use the finite differences method, on a cartesian orthogonal grid, to perform calculations on potential flow about an obstacle without using the Neumann conditions, ...
7
votes
1answer
476 views

Discrete wave simulation - absorbing boundaries?

I wrote a simple 2D wave simulation using the following equations: $$\frac{\partial^2 u}{\partial t^2}=c^2\nabla^2u$$ Where $\nabla^2$ is the discrete laplace operator using a Von Neumann neighborhood ...
6
votes
2answers
777 views

FEM: Possible to have boundary conditions “inside” the domain?

I work on geological problems and I use the Finite Element Method. But this question can be applied on other classical mechanical problems. I work on implicit 3D surfaces (which represent the limits ...
6
votes
3answers
3k views

Open boundary conditions with the advection-diffusion equation

Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term), $$ \frac{\partial \phi}{\partial t} = \frac{\partial}{\...
6
votes
1answer
215 views

Neumann BCs in cylindrical geometry (FEM)

I was wondering where I could get a detailed account (either in print or online) on applying a Neumann/mixed Boundary condition along the $r=0$ axis in an axially symmetric geometry. Though this is a ...
6
votes
2answers
207 views

Why naively chopped finite difference matrix works for different ODE boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. One naive way to write down the matrix of the differential operator is like the following, ...
6
votes
1answer
263 views

Do the class of PDEs that lack initial conditions have a name?

I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation. Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...
6
votes
2answers
1k views

Finite-volume method: can Dirichlet boundary conditions be applied to the integral form?

I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?" ...
6
votes
1answer
4k views

FEniCS: how to specify boundary conditions on a circle inside 2D mesh

I would like to numerically find a mutual capacitance of two stripes of metal on the opposites sides of a cylinder. The problem is obviously a 2D Laplace equation. I would like to find the potential ...
6
votes
1answer
2k views

Trouble implementing Neumann boundary conditions because the ghost points cannot be eliminated

Neumann boundary conditions are implemented by introducing ghost points outside the domain and then using the boundary conditions to eliminate the ghost points. For example, see this question. I ...
6
votes
3answers
6k views

Poisson equation finite-difference with pure Neumann boundary conditions

I'm trying to solve a 1D Poisson equation with pure Neumann boundary conditions. I've found many discussions of this problem, e.g. 1) Poisson equation with Neumann boundary conditions 2) Writing the ...
6
votes
1answer
731 views

how to visualize lattice with periodic, helical, etc. boundary conditions?

I am trying to write a special hexagonal lattice generator, with several kinds of boundary conditions, such as helical BC, periodic BC, and I find it hard to verify whether it works correctly. I tried ...
6
votes
3answers
366 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_{...
6
votes
0answers
528 views

Understanding Boundary Condition in FEM

I am trying to understand Dirichlet and Neumann boundary conditions in FEM and I wanted to know if my inference is correct. To articulate my understanding, lets consider a simple case of TE and TM ...
6
votes
0answers
133 views

Do practice and theory differ substantially when implementing Neumann Boundary Conditions using a Mixed Method?

I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both ...
5
votes
2answers
427 views

Solving Poisson equation with current BC using FEM

I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$ Insulator (Neumann BC) Electrode set at some ...
5
votes
3answers
915 views

Computing accurate fluxes with FEM

I have solved Poisson equation on a 3d domain with neumann and dirichlet boundary condition. I get the potential, take the gradient for each element and integrate on a surface of an element, I do this ...
5
votes
2answers
326 views

Transparent boundary conditions for finite element simulation of TDSE

I have implemented a version of Visscher's method for numerically solving the TDSE (A fast explicit algorithm for the time-dependent Schrödinger equation) (also described in Are there simple ways to ...
5
votes
2answers
1k views

Do I need to impose boundary conditions in the Jacobian matrix?

In the framework of Finite Element Method, when the Newton method is used, we solve $J(x^k) \delta x = -f(x^k)$, and the increment $\delta x$ would not change some entries from $x^k$ related to ...
5
votes
3answers
255 views

which numerical method for ode with mixed BCs

I've got a second order nonlinear ODE (nothing fancy), but the BC are a little odd to me: $y'(0) = 0$ $y \rightarrow y_a$ as $x \rightarrow \infty$ What's a good numerical method for solving this? ...
5
votes
2answers
639 views

Absorbing boundary conditions for acoustics in Discontinuous Galerkin

Note: I'm trying to implement a Discontinuous Galerkin method, as kind of a way to learn about these things. As of now, I've taken the acoustic wave equation $c^2 \nabla \cdot \nabla u(x,t) - \frac{\...
5
votes
3answers
3k views

Discrete Poisson Equation with Pure Neumann Boundary Conditions

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...
5
votes
1answer
381 views

The Lax-Milgram Lemma in FEM with non-homogenous Dirichlet BC

How can show that the prerequisites for the Lax-Milgram Lemma holds if I have different test and trial spaces (which I think is the natural thing to have if at least part of the boundary is non-...
5
votes
2answers
172 views

Is the diffusion equation with Neumann and Dirichlet BCs well-posed?

I am considering the following diffusion equation: $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$ over a grid ...
5
votes
2answers
866 views

Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions

I would like to numerically solve a hyperbolic PDE of the form $\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t \...
5
votes
1answer
1k views

boundary conditions for electrostatic problem

I'm solving an electrostatic problem governed by Laplace equation $$-\nabla \cdot (\rho^{-1} \nabla u) = 0$$ in the following domain: a brick ($\Omega_1$) with a cylindrical inclusion ($\Omega_2$), ...
5
votes
1answer
195 views

What Linear Equation Solver should be used for a problem with many dirichlet conditions?

I am solving a laplace equation on a finite-element mesh (tetrahedral, triagonal) and have many say 99% dirichlet conditions compared to the number of unknowns. Is there an efficient way to solve this ...

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